# Analyses of a group of pre-tests for students of first year college mathematics

код для вставкиСкачатьANALYSES OF A GROUP O F :P R I E S T S FOR STUDENTS OF FIRST YEAR^COLLEGE MATHEMATICS hj Henry Vernon Price A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, In the Department of Mathematics, In the Graduate College of the State University of Iowa August$ 1940 P ro Q u e st N u m b e r: 10311004 All rights rese rv ed INFO RM ATIO N TO ALL USERS The q u a lity o f this re p ro d u c tio n is d e p e n d e n t u p o n th e q u a lity o f th e c o p y s u b m itte d . In th e unlikely e v e n t th a t th e a u th o r did n o t sen d a c o m p le te m an u scrip t a n d th e r e a re missing p a g e s , th e s e will b e n o te d . Also, if m a te ria l h a d to b e re m o v e d , a n o te will in d ic a te th e d e le tio n . uest. P ro Q u est 10311004 Published by P ro Q u est LLC (2017). C o p y rig h t o f th e Dissertation is h e ld by th e A uthor. All rights reserved. This w ork is p r o te c te d a g a in s t u n a u th o rize d c o p y in g u n d e r Title 17, U n ited States C o d e M icroform Edition © P roQ uest LLC. P ro Q u est LLC. 789 East E isenhow er P arkw ay P.O. Box 1346 A nn Arbor, Ml 48106 - 1346 \ \D4-v ii C o p *-2»— The writer takes this opportunity to express his sincere appreciation to Professor E* W* Chittenden for his guidance in the preparation of this thesis and also for making available both materials and IOTA help* Acknowledgment la also due to Professor E* F* Lindquist for his friendly criticisms and suggestions* in TABLE OF CONTENTS CHAPTER I. PAGE IHTKODtJCTXON * # # # # # The pre-tests # # * . * „ . . * * * « * « • * * • • * ........ 5 Analyses and results • * » * » « • * * • » » Some suggestion® for further research IX* RELATED HESEAHCH 5 • « • 14 .......................... 15 The report of the Committee on Tests • • » • 15 Test scores and success in college • • • » « 17 Studies of the Iowa Placement Examinations « IS The effect of additional preparation In mathematics upon retention and college ............ achievement 21 Variability In item difficulty from school to school III* • » • * • » • • • • » » » • • • 24 ANALYSES OF THE 1955 PRE-TESTS........... 25 Analyses made by the Cooperative Test Service • • « • • * * • • • ! » * * . * • 25 Preparation groups * » « ♦ * • • • * • • * # 55 Partial and absolute scores » * * • • • * * 55 Distributions of scores made by preparation groups * « * • • » * .................... 57 Effect of preparation on first semester grades • • • • • « • • • • • • • • * • • • 45 iv CHAPTER IV* PAGE ANALYSES OF THE 193© PRE-TESTS 48 Analyses made by the Cooperative Test Service * * * * * # • « * » * • * • • • * 48 Distributions of scores made by preparation groups • * * * .......... • * . * . • • Relative difficulty of the 1936 forms V* 51 « • « $6 Mean scores mad® at each institution • • • • 37 # @2 ANALYSES OF THE ITEMS OF THE 1935 BRE-TESTS Analyses made by the Cooperative Test Service • « * • • • • • Selection of samples ........ @2 .......... .. , « * » 65 * g© Difficulty indices for preparation groups The standard error of a percentage as com* puted from a stratified sample 81 Comparison of the difficulty Indices found for preparation groups # • 84 ............ Effect of preparation upon knowledge of the concepts and processes covered by the 1935 pre-tests ........ * ........... 39 Comparison of difficulties of related Items. The predictive power of individual Items • VI* ANALYSES OF THE ITEMS OF THE 1936 PRE-TESTS 93 * 95 # 105 1935 items which were repeated in the 1936 pre-tests ............ Cooperative Test Service analyses 105 » * . « * 112 V CHAPTER PACE Item dls crimination within preparation groups « * # ♦ • * « , » * * . . » * » . * • Difficulty indices for preparation groups 114 * 11© Comparison of difficulty indices found for preparation groups ...................... 124 Effect of preparation upon knowledge of the concepts and processes covered by the 1936 pre-tests VII* «•« .......... • • * • * « » THE DIFFICULT! OF AH ITEM FROM TEAR TO YEAH 127 # 151 • 131 Comparison of 1935 and 1936 difficulty Indices « • » • * * * . . , Comparison of difficulty indices found for high school freshmen « * « » * « BIBLIOGRAPHY ....... ................. .. APPENDIX............................................... 142 150 154 vi LIST OF TABLES TABLE 1# PAGE Humber of Students at Each of Six Institutions to Whom the 1935 Tests Were Administered » # * 2* Distributionof Pro-*Test Scores for Form A, 1935 3* Distribution 4* Distribution of Pre-Test Scores for Form C* 1935 5# Correlations between Scores on the 1935 Pre-Tests of Pre-Test Scores for Form B, and First Semester Grades* at Each Institution 6* 25 26 193527 28 31 Humber of Students at Each Institution and in Each Preparation Group to Whom Form A* 1935 Was Administered • # « « * • * • * • » . • » * 7* 34 Humber of Students at Each Institution and in Each Preparation Group to Whom Form B* 1935 Was Administered » » » • • » * « » • • • « • » 8* 34 Humber of Students at Each Institution and in Each Preparation Group to Whom Form C, 1935 Was Administered # * » « » • * * » • « • • • • 9* Total Distribution of Partial Scores by Preparation Groups * Form A f 1935 10# * ... 39 Total Distribution of Partial Scores by Preparation Groups 11# Form B f 1935 * • *• » * 40 Total Distribution of Partial Scores by Preparation Groups ** Form C f 1935 12# 54 42 Stability of Differences Observed between Preparation Groups on the 1935 Tests * *• * * 44 vi! TABLE 15# PAGE Percentages of* Students Earning Various Grades in 1935 14# • * • • * * * . « • * * • « ........ 46 Correlations between 1936 Test Scores and First Semes ter Grades in College Mathematics, at Each Institution * • • « * • * • • • • • « • • 15# Distributions of* Scores for 16* Distributions of the 1936 Tests • • • P ^ * Pg « • • • • * and Pg as Found for Form A, 1936* and Form B, 1936 • « * • * « « .................. 33 Comparison of Arithmetic Means Found for Form A* 1936* and for Form B, 1936 19* 51 Comparison of Means of the Preparation Groups ?1* 18* 49 Scores for Forms A and B (1936) by Preparation Groups P^, 17* 49 ................ 57 Mean Scores on the 1956 Pre-Tests for the Prepa ration Groups P^, P ^ , and Pg at Each Cooperating Institution 20* * * * « * * * • • « • 58 Mean Scores for Form A* 1936* at Each Cooperating Institution for Each Half Tear of Total Prepay ration in Mathematics 21# * * » • • • • • • • • • @0 Mean Scores for Form B, 1936, at Bach Cooperating Institution for Each Half Year of Total Prepa ration In Mathematics 22# * * • » » * * « • • * » Distributions of Validity and Difficulty Indices Found for the 1935 T e s t s ............ 23* 60 Humber of Papers in Each 1935 Sample 64 66 viii table page 24* 25# Humber of Papers in Each 1936 Sample Item Difficulties for Each Preparation Group f l* 26# *%k* F o m A * 1935 * ............... p2 ~ Form » » # * » # # * P2 * Form C, 1935 • • • • « • » • ......... 33 87 Items of Form C, 1955, which Yield Significant Differences between Preparation Groups • • » • 34* 30 Items of Form B, 1935, which Yield Significant Differences between Preparation Groups • • • « 33# so Items of Form A, 1935, which Yield Signifleant Differences between Preparation Groups • • • • 32* 70 Distribution of Difficulty Indices for Each Preparation Group - Form C , 1935 31# 77 Distribution of Difficulty Indices for Each Preparation Group - Form B, 1935 • « # # * « • 30* 76 Distribution of Difficulty Indices for Each Preparation Group - Form A, 1935 ............ 29# 74 Item Difficulties for Each Preparation Group Fl* *1%* 28* p8 * Item Difficulties for Each Preparation Group pl* pl§* 27# eg qq Correlations (Tetraohorlc) between Correct Solu tions of Individual Items in Form A, 1955, and Satisfactory (A,B,C) First Semester Grades for Each Preparation Group # # • * • • « » . « » • 99 ix SABLE 35* PAGE Correlations (Tetrachoric) between Correct Solu tions of Individual Items in Form B, 1935, and Satisfactory (A,B,C) First Semester Grades for Each Preparation Group • • • • » » * • « • # * 36* 100 Correlations (Tetrachoric) between Correct Solu tions of Individual Items in Form C, 1935, and Satisfactory (A,B,C) First Semester Grades for Each Preparation Group » « • « « « » 37* Items of Form A, 1936, which Were Taken from the 1955 Tests « * • * 38* + **« 110 • Items of Form B* 1936* which Were Taken from the 1933 Tests 39* • • • * « * • • » * • « • • * * • 111 Distributions of Difficulty and of Validity Indices Found for the 1936 Tests • • * » • « * 40* 102 113 Percentage of Correct Responses to Each Item in Each Approximate Third of the Preparation Group and Total Distributions - Form A, 1956 • . , • 41# 116 Percentage of Correct Responses to Each Item in Each Approximate Third of the Preparation Group and Total Distributions - Form B, 1936 • • . « 48* Item Difficulties for Each Preparation Group P^, Px^, and Pg - Form A, 1936 * • * • « • • • « » 43* 117 121 Item Difficulties for Each Preparation Group P^, P»i, and Pg - Form B, 1936 ..........* « 122 X TABLE 44# PAGE Distribution of Difficulty Indices for Each Preparation Group * Form A* 1936 « • • » * # • 46# IBS Distribution of Difficulty Indices for Each Preparation Group * Form B* 1936 « » • • • • * 46# 123 Items of Form A* 1936# which Yield Significant Differences between Preparation Groups « « « 47# « 126 Items of Form B, 1936# which Yield Significant Differences between Preparation Groups * * . 48# ♦ 126 Comparison of Difficulties of Items Given in 1935 and Repeated in Identical Form In 1936 49* « 162 Comparison of Difficulties of Items Given in 1935 and Repeated in Almost Identical Form in 1936 50* 1-35 Comparison of Difficulties of Items Givon in 1936 which Involved the Same Process but Were Hot Identical to 1935 Items 51* * « • * « * • * « 138 Difficulty and Validity Indices for Each Item of Form A* 1935# as Given by the Cooperative Test Service • • • 52* 155 * 156 Difficulty and Validity Indices for Each Item Of Form B# 1935 53# * Difficulty and Validity Indices for Each Item of Form C# 1935 157 xl TABLE 54* PAGE Difficulty and Validity Indices for Each 1950 Item as Given by the Cooperative Test Service 55* 159 Humber ©f Correct Responses (R), Incorrect Responses (W), and Omissions (Q), Arising from Each Sample, for Each Item of Form A, 1955 « • 56* 161 Humber of Correct Responses (E)* Incorrect Responses (W), and Omissions (0), Arising from Each Sample, for Each Itemof Form B, 1935 * * 57* 162 Humber of Correct Responses (K), Incorrect Responses (W), and Omissions (0), Arising from Each Sample, for Each Item of Form C, 1935 * * 58* 164 Humber of Correct Responses (R), Incorrect Responses (W), and Omissions (0 ), Arising from Each Sample, for Each Item of Form A, 1936 * * 59* 165 Humber of Correct Responses (R), Incorrect Responses (W), and Omissions (0), Arising from Each Sample, for Each Item of Form B, 1936 * « 60. 167 <r^2 = 3=2j£ 90 £■ apq Form A, 1935 169 / 61. 0~pZ Z 4 ^npq Form B, 1936 68. (Tpz i 170 4^npq Fora C, 1935 172 xii TABLE PAGE 63. s 2 Z np<j Form A* 1956 • • « • • • • • * • • • • » » • 64# Op S 2 1 nP1 P o m B, 1936 65* 173 ........................ 175 Coaparlaon of Observed Differences between Groups P1 and P-ti In Percentage of Correct Responses to Each Item in Form A* 1935* and the Standard Errors of 66 * the Differences * ............ 177 Comparison of Observed Differences between Groups P ^ and Pg in Percentage of Correct Responses to Each Item In Form A* 1935* and the Standard Errors of 67* the Differences # * * • * « • • • • 173 Comparison of Observed Differences between Groups P^ and Pg In Percentage of Correct Responses to Each Item in Form A, 1955* and the Standard Errors of 63* the Differences * » * * # « • » . « 180 Comparison of Observed Differences between Groups pi pi* In Percentage of Correct Responses to Each Item In Form B, 1955* and the Standard Errors of the Differences 69* • » ............* « 181 Comparison of Observed Differences between Groups P ^ and Pg In Percentage of Correct Responses to Each Item in Form B, 1935, and the Standard Errors of the Differences • » • « • • • * * * 185 xiii STABLE 70* PAGE Comparison of Observed Differences between Croups P^ and Pg In Percentage of Correct Responses to Each Item in Form B, 1935, and the Standard Errors of the Differences 71* ............. .. • 184 Comparison of Observed Differences between Croups pi ana pi§ in Percentage of Correct Responses to Each Item In Form C, 1935, and the Standard Errors of the Differences 79* * * • • * • • • • • 185 Comparison of Observed Differences between Groups P ^ and Pg In Percentage of Correct Responses to Each Item in Form C, 1935, and the Standard Errors of the Differences 73* * * • • .......... 187 Comparison of Observed Differences between Groups P^ and Pg in Percentage of Correct Responses to Each Item in Form C, 1935, and the Standard Errors of the Differences 74* ..........* * 188 Comparison of Observed Differences between Groups and P ^ in Percentage of Correct Responses to Each Item in Form A, 1936, and the Standard Errors of the Differences 75* * * • • « » . . » * 190 Comparison of Observed Differences between Groups Pl| •»* P2 In Percentage of Correct Responses to Each Item in Form A, 1936, and the Standard Errors of the Differences « # « « ..........* 191 xiv TABLE 76* PAGE Comparison of Observed Differences between Groups P^ end P^ Percentage of Correct Responses to Each Item In Form A* 1936* and the Standard Errors of the Differences 77* • ..........* * * 192 Comparison of Observed Differences between Groups P1 and piJ In Percentage of Correct Responses to Each Item In Form B* 1936* and the Standard Errors of the Differences 78* * * * « 194 Comparison of Observed Differences between Groups and Pg In Percentage of Correct Responses to Each Item In Form B, 1936* and the Standard Errors of the Differences 79* 195 Comparison of Observed Differences between Groups P^ and Pg In Percentage of Correct Responses to Each Item In Form B* 1936* and the Standard Errors of the Differences * « • • « • • • » • 196 i CHAPTER I IH TBODUC TI0H In December# 1934# the Mathematical Association of America approved a proposal of th© Committee on Educa tional Testing to collaborate in the construction of tests in first* and second-year college mathematics# A Committee on Testa* was thereupon appointed and given power to act for th© Association in the prosecution of the proposal# As a part of its program# the new committee^ constructed, In 1933, a battery of three pre-test forma for students of first year college mathematics* These examinations were revised and reassembled in two new forms in 1936# Seven colleges and universities administered the tests in these two years to 4*639 students# and it is the purpose of this dissertation (1 ) to present such analyses of the re sults as have been made# and to elaborate upon them? (9) to make further analyses of a nature to be specified subsequently! and (3 ) to point out some of the opportunities for further research# Items Included in th® pre-tests were selected because of th© subject matter they Involved and not becaus© they were thought to possess high discriminating power or ^ Ralph Be&tley# E* W. Chittenden (Chairman)# A# H* Crathom©* L* L* Dines# and H# 8 # Everett# ® Hereafter referred to as ^th© committee*” 2 would contribute appreciably to the prediction or college success* That the analyses which have been made show the pre-tests to be very satisfactory in this respect In no way alters the fact that such virtues are only accompanying by-products and not of first importance* The fundamental purpose of the pre-tests is to aid the teacher of first year college mathematics in determining the extent to which his students are familiar with the fundamental concepts and processes of secondary school mathematics* The analyses made by the writer, which constitute by far the greatest portion of this dissertation, consist of comparisons of the responses made to th© pre-tests by students with different degrees of preparation in mathe matics* Analyses are both statistical and graphical and are given for both tests and it ©as* It was expected by the committee that the tests would be useful for sectioning or placement, diagnosis, prediction of subsequent performance, and would provide a basis for the measurement of growth* A great amount of Information relative to the first three of these objectives is to be found In Chapters III to vilj the fourth has no validation In an Investigation of this type since end-tests were not given* 3 » ^Report of the Committee on Tests,” The American Mathematical Monthly, XLVTI (May, 1940), No* 5* 3 THE PRE-TESTS T!m 1935 test®^ were eons true ted in three nearly equivalent form® - Form® A, B, and C * and were administered in October of that year to 2*630 student® In six cooperating colleges and universities* Each form was divided into two parts* th© first part being composed entirely of formal questions dealing with routine skill® and with the emphasis on manipulation* The second part was Intended to contribute a subjective element and contained problems# Many of the items* particularly those in Fart XI, consisted of several related parts* Fart I contained more question® than Part II In each of the three forms and received a time allowance of twenty-five minutes$ Fart II was given an allowance of only twenty minutes* At the conclusion of th© testing program, all papers were returned to the Cooperative Test Service in Hew York City where a brief statistical analysis was made of the results* It was the Intention of the committee to eventually present th© pro-teats in two permanent forms* Hence for the purpose of revision more items were available than could be used* Selecting from this collection those most * ill of the tests were published by th© Coopera tive Tost Service located at 500 West 116th Street, Bow Yoazk City* ^ It was found impractical to devote more than on© class period to the administration of the tests. 4 suitable In t o m s of the analyses provided by the cooperative Test Service* and filling In a few gaps In difficulty with new items* the committee produced the 1936 forms* Two changes in organisation were embodied In the new examina tions - no ^multiple** items were Included* and no separa tion of a test Into parts was ef fee ted* Th© revised tests were administered In the fall of 1936 to 2*029 students in five colleges and universities* Four of these schools had also cooperated In th© 1935 program, and hence th© populations of students examined in th© two years were quite similar* As before, th© test papers were sent to the Cooperative Test Service for analysis# Th© content of the pro-tests was limited almost exclusively to algebra, partly because of th© limited administration time and partly because It Is generally recognised that success or failure In college mathematics is largely dependent upon preparation in algebra# concepts were Included which are Very few not a part of any coups In first year algebra* Th© ©ommltt©© considered it advisable to allow two weeks (eight lessons) for review, so the examinations were administered uniformly on th© ninth day# That this review period la valuable Is brought out In a totally unrelated paper written by Douglass (11)* In surveying ten studies concerned with retention of secondary school mathematics, he concludes by making th© following © 5 observations It i® quit® difficult to generally® from thea® studies* It is clear, however, that a large propor tion of the forgetting takes place within a very few months and that material® forgotten within a few years can be relearned within a few hour® of instruction* ANALYSES AND RESULTS The Cooperative feat Service reported th® follow ing statistical measures m derivatives of the 1955 program* 1 * The reliability of each test form* These coefficients were found to be as follows* for Form A, *9173? for Form B, *91801 and for Form 0* *8906* 9* The correlation between test score® and first semester grades* Coefficients were reported for each school and ranged from *42 to *74 with a median of *80 and a mean of *59* Similar coefficient® for 1988 ranged from *51 to *70 with a median of #57 and a mean of *59# 5* Item difficulties* 4* Item validities# Th® last two sets of Indices are listed, both for 1935 and 1986, in the appendix* No reliability coefficients were reported for the 1986 forms/ After th© Cooperative Test Service had completed its analysis, the test papers were sent to Professor 32# W* Chittenden at the State University of Iowa for further study* There, with th© help of several TOA students, he set up a program of research, part of which leads up to the present investigation# Samples were drawn and the Item analyses found in Tables 55, 50, and 57 of th® appendix derived from them# On th© basis of these samples, several correlation studies were made*® At about the same time, Kura T a m e r (58) made a brief study of 300 Form A papers* She confined her atten tion largely to a classification of th© errors mad® on individual Items# A few of her analyses, however, were concerned with the effect of preparation upon achievement, and these will be reviewed In the next chapter# In the summer of 1957 the writer became Interested w In the pro-tests, secured the 195© test papers9 from the Cooperative Test Service, and began th© study which consti tutes this dissertation# Analyses of the tests as a whole are given in Chapters XXX and IV, and Item analyses in Chapters V, VI, and VII# Test analyses embrace the ©ntir© ® Most of theso analyses are Included in th© com mittee report already mentioned# This report will b© reviewed later* ^ This is not to b© construed as meaning that th© 1936 pre-tests only were studied# Both batteries were analysed In th© same manner* 7 population involved In the study* whereas item analyses were made on the basis of stratified staples* The following conclusions result from the analyses discussed in Chapters III and XV* 1# The correlation between partial and absolute scores made on five Items of Form A* 1935 was found to b© #87 with a probable error of * 01 # Scores obtained by marking an Item consisting of several parts as right or wrong are termed partial scores are those obtained by considering each part a© a separate item# Seven such items are found In Form A* but only five were scored uniformly* A random sample of @00 papers was used in order to determine fee scores on these items* This relationship is of some interest but, because of th© small number of Items involved, is not very reliable# 2# Groups of students with one, on© and on© half, and two years of algebra have been designated as ?x* and Pg* respectively* Th© mean scores (partial in 1935) mad© by each of these groups on each test form are as follows: 1225 t> re Bom A B a 22*32 14*34 10 *43 28*28 20*51 23*08 Maximum score 32*66 23.78 53 46 50 20*12 1933 Form A B Px 22*31 20*04 hi 30*12 26*13 £g Maximum Score 35*31 33*4? 50 50 It has been shown statistically that all of these differences between preparation groups are signifleant* Mean scores made on the two 1936 forms are given for each preparation group in each cooperating school* Although not shown to be statistically significant, th® same differences between preparation groups were found* These differences are still present when total preparation In mathematics# rather than In algebra alone. Is considered* The total distribution of scores made by each preparation group Is shown on a single graph for each 1936 form* The position of the curves shows that the entire distribution is moved to a higher position on the scale by an increase 9 in preparation* 4* Assuming no appreciable differences to exist between groups wi th the same amount of prepara tion who wrote the 1930 forms, it has been shown statistically that Form A is easier than Form B for each preparation group and for all groups combined* 3* All first semester grades given at each insti tution in 1950 were classified by preparation groups and then reduced to percentages within each group* in some schools, there is a tendency for high grades to be associated with a high degree of preparation and low grades to b® associated with less preparation, but th® results are not uniform* The following oenelusl&ns result from th® analyses discussed in Chapters ¥ and VI* 1* Difficulty Indices for each item and distribu tions of these Indices were computed for each preparation group* The distributions illustrate as did th® graphs already mentioned, that th© entire distribution is forced upward as prepara tion increases* 2* In order to compare difficulty Indices found for the same Item but different preparation groups, It was necessary to make extensive us© 10 of the following formula for th© standard error of a percentage as calculated from a stratified samples (J^ : ^ ji niPiqi* 10 the 1=1 writer’s knowledge* this formula has not been used before with test data* Using this formula* differences between difficulty indices calculated for th© same item but different preparation groups were tested for significance* The following number of items were found to show a significant difference between th© groups listed* In all cases the group with more preparation was th© superior one® 1955 Form A B 0 *JksLk-ps A> Pl*Pg 9 2 7 29 17 50 21 10 11 1936 Form P1 Pl4 ?ii,p2 Pl aP2 A B 27 10 17 27 58 42 It is interesting to note that Form A* 1956 distinguishes principally between th© two groups with lesser preparation* whereas Form B* 1956 separates more frequently 11 the two groups with higher preparation* 3* The correlation (tetrachorie) between success on an Item and satisfactory (A* B# G) first semester grades was calculated for each 1935 item and for each degree of preparation* The results were not uniform# a great many items giving widely divergent coefficients for the three preparation groups# Two Interpretations of this phenomenon may be given* an Item is too small a unit to us® for predictive purposes# or els© different items should be used fbr students with different degrees of preparation# 4* The papers in the 1936 samples were arranged in descending order according to total score# Then each sample (also the combination of all three samples of a given form) was divided into three approximately equal parts# Finally# the percentage of correct responses to each item in each of those thirds was calculated* The purpose of this analysis was to determine how well each item discriminated within a given preparation group# Two characteristics of the tables stand out* a# The Indices usually given for an item and baaed on a heterogeneous population are inadequate to describe the discrimi nation within preparation groups# b* The difference between preparation groups is so great that items which discriminate for on© group seldom do so for another# 5# Groups of items pertaining to the same process (not to be confused with the ^multiple” items already mentioned) were constimeted in 1935 for the purpose of comparison# As expected# difficulty indices derived for these items decreased as the complexity of the problems Increased# Chapter VII is devoted to a comparison of the difficulty indices derived for those items which were given in 1935 and repeated in 1936* Forty such pairs of items were identicals thirty-nine were almost identical* differing only in directions or simple change of number* and fortyseven other pairs involved the same process# The apparent stability of the difficulty index was very marked* relation of #924 ± for Identical items* A cor *016 was found between indices given This coefficient is #885 dfc *024 for the ttalmost identical1* pairs# and *761 ± *042 for those Items dealing with the same process* Mention should be mad© of the fact that th© populations in 1935 and 1936# from which these indices were derived# were very much alike in so far as preparation is concemecU The percentage of students in each prepara tion group in each year Is as follows; Yea** 1935 1936 Pl» 39.9 28.4 39.0 43.0 2 Other® 21 ,5 6,8 22.2 6,4 SOME SUGGESTIONS FOR FURTHER RESEARCH Growing out of th© present study are many posslbillties fbr future investigation* A few of these are as follows: X* The protest Items should he sorted according to discrimination and predictive power for each of the preparation groups* and new tests experimented with ~ one for each degree of preparation* 2* Further research should be made to check the two methods suggested for scoring an item* 3# A very promising possibility for research is that w£ determining the manner in which success on a given Item affects success on other Items* 4* A study should be made to determine the extent to which the difficulty of an item is is constant} that Is* to find out whether or not it is universally true* 5* Results obtained through future use of the pre-testa^ should be studied as a check on the analyses of this investigation* ® The pro**teats are now offered for general use by the Cooperative Test Service and listed in their catalog* 15 CHAPTER II BELATED RESEARCH *■* report of the Committee on Tests#1 This report gives a general survey of those analyses made of the pre-tests by investigators other than this writer* Most of these analyses were in the form of ©orrelatlon studies based on the 1953 samples* In every case coefficients were derived for each of three preparation groups® and for all groups combined* Correlations between scores made on Part I of each test form and the corresponding scores on Part II were reported as follows? for Form A* th® coefficients ranged from *43 to *83 with a median of *55 and a mean of ♦54f for Form B, the rang® was from *23 to *>06 with a median of *45 and a mean of *46; and for Form G, the rang® was from *48 to *7© with a median of *63 and a mean of *61* Th® committee interprets these coefficients as an Indication that th© two parts measure different skills* Another study embodied in th© report was made to determine the predictive power of a short test# representative Items were selected sidered as a group* Ten from Form A and con Th© average correlation eosfficient 1 *Report of the Committee on Tests*** The American Mathematical Monthly* XLVII (May, 1940), Mo* 5* ® Groups with 1 year, 1§ years, and 2 years of preparation in algebra* reported between acores on this group of Items and scores on Fart I of the test Is *83* Th© correlation between the first semester grades of a group of 189 students and their scores on these ten questions was *39* This coefficient is somewhat lower than those reported by the Cooperative Test Service for scores on the entire test* The following things were found by Turner (33) In her study of 300 Fern A, 1935, papers# a* The bi-serial coefficient of correlation be tween test scores and unsatisfactory (B, P, or withdrawn) first semester grades is -*51, indicating that unsatisfactory grades decrease substantially aa the teat scores in crease* Mo unsatisfactory grades were reported for the 22 students who made scores of 39 or more on the pro-teat* h« There is a direct relationship between the amount ©f preparation in algebra and the percentage of correct responses to th© Items, and an inverse relationship between amount of preparation and th© percentage of Items omitted* The data Is given In tabular form for th® three preparation groups already mentioned, and no attempt was mad© to prove the differences significant* 1.7 <s» Superior students of ninth-year algebra scored higher on Part I of the pre-test than did any of the preparation groups# Th® first 33 of the 51 items In the 1936 Iowa State Scholarship Examination in Algebra were made to coincide with Part I of Form A* Th© examination was administered to 52 students who were completing their first year of algebra and who were competing as finalists -In a state wide testing program* These students responded correctly to 85 per cent of the Items under consideration and Incorrectly to the remaining 15 per cent* Bone were omitted* 2* Test scores and smeeeaa in college* An enormous amount of research has been carried on relative to the prediction of success in college* Most of these in vestigations* however* have made use of Intelligence exami nations rather than content examinations in a particular field* Brumbaugh (6 ) summarised a large number of these studies in 1936* and we quote that part of his paper which seems most significant In terms of our study* Only a few studies have boon made of the relation ship between achievement in colleg© and the scores mad,© in achievement tests such as th© College Entrance Board Examinations* the Iowa High School Content Exam inations * the Iowa Placement Tests, the New York Regents Examinations, and the Sones-Harry High-School Achievement Tests# The data so far available indicate that achievement tests, particularly the Iowa Placement Tests and the Sones-Harry High-School Achievement Tests, 18 are ms efficient as intelligence tests in forecasting suecess in college* They h&v© an additional value in that they afford valuable information for purposes of classifying and advising those who enter college* In support of this conclusion Holssinger says2 "The psychological examination Indicates the student1© general capacity to handle the various college courses| it does not furnish standards for academic achievement* Achievement tests* such as the Son©s-Harry Test, measure not only achievement hut Intelligence as well* The tmlverslty would do well to develop tests of the achievement type as a basis for standards of entrance and subsequent guidance In college departments#** Studies of the Iowa Placement Bx&mlnationa*^ This group of examinations has been involved in a number ©f Investigations*^ but In very few cases have the exami nations themselves been studied* The examinations were constructed by Stoddard and others In 1925 and are In two series «* aptitude tests and training tests* On© test in each series Is devoted to mathematics* and the MathematicsTraining Examination la designed for the same puipose as are th© pro-*tests* Th© content* however* Is considerably different* Stoddard (30) studied th© examinations exten sively, •principally for the purpose of determining and Improving their power as predictive and placement ® A description of these examinations and their construction is given by Stoddard (30)* ^ In a monograph published in 1928, Hammond and Stoddard (14) review studies of the uses of these exami nations In a number of engineering colleges* An annotated bibliography of most of these studies Is Included* 19 instruments* Hi® analyses included calculations of the reliabilities of the various tests, inter-part correla tions, correlations of scores made on each test and on the parts of each test with first semester grades, and corre lations of th© corresponding training and aptitude tests# Revision of the tests was then effected in accordance with th© information secured# Two criteria were used exten sively In passing judgment on items * if (a) student Items were eliminated responses were all correct or all incorrect; (b) they failed to discriminate (in the per cent of pupils passing) those students who subsequently succeeded5 In the course from those who subsequently failed® in the course* In a later report describing the placement examination®, Stoddard (Si) makes a remark which has bean quoted many times and which, In the writer1s opinion, is equally descriptive of th© pre-tests# He says: The Iowa flaeement Examinations ar© less a prognosis test than an educative procedure* Their aim Is not primarily to predict academic success * but to render its "attainment more likely; that is, to give aid in the setting up of educational condi tions such that sound principles of selection* classsectioning and curriculum organisation may be more effectively applied to tfoe securing of maximum per formance on the part of each student# In 1929, Miller (23) made a detailed study of the data which had accumulated through us© of the Iowa s Grades of A or B# ^ Grades of D, Conditioned, or Fd# 20 Placement Eliminations In 1925, 1926, and 1927. Th© pro cedure© he used were essentially the same as those already described for Stoddard1© study, and on the basis of the Information secured he effected a further revision of the examinations* As a check, th© new forms were tried out and the data studied In th© same manner as before* Conclusions are, for the most part, confined to technical descriptions of th© various examinations, and ar© relatively meaningless without a more detailed descriptien of the study than is given here* On© result, however, Is quit© general and of considerable importance* We quotes Th© result® of the Investigation In ©very subject indicate that th© best type of training examination for placement purposes is one mad© up of Items and parts which emphasis© the most common and fundamental principles of th® subject rather than th© less common and less essential* On# further study of th© Iowa Placement Examina7 tions should be mentioned* Hansen (15) studied samples drawn from th© papers written in twenty-eight colleges and universities in 1925 and 1926* He calculated a multitude ©f intercorrelations, principally to determine how well aptitude© ©r training in on® field predicted aptitude or training in another field. One of his conclusions was that mathematics training was the best of th© group for 7 Many other studies of these examinations are reviewed in Miller*s study* Since they have little bearing on the present investigation, they are not repeated her©* general predictive purposes* In general, it may be said that th© evidence Is in agreement that the Iowa Placement Examinations are very useful for the purposes of prediction and placement* 4* 2 *& M M SI of additional preparation in mathe matics, upon retention and college achievement* The final grades of 291 college algebra students at th© Morgan Park Junior College were classified by Bergen (3) according to the number of semesters of mathematics taken In high school* The purpose of this classification was to study th© effect of longer training In mathematics upon achievement in college algebra* All students involved in the study had at least four semesters of preparation and some had as many as eight* Bergen concluded that (grade) differences between the groups of students who had four and five semesters of preparation were negligible, but that there were large differences when the groups who had six or more semesters war© compared with those who had only four or five* Schoonm&ker (28) reports a study conducted with 128 freshman students at an unnamed college for women In 1925-24 and 1924-25* It had been the policy of th© school to group students with only on© year of preparation in algebra in a 11Sub Math11 class* During the investigation period this policy was abandoned and the sectioning was carried out on the basis.of- scares .raid®:;tsn* the Hotz Algebra Seales* She reports sectioning on th© basis of ability to be definitely th© better method in so far as conditioning achievement is concerned* This would imply that prepara tion and achievement arc not too closely related* Douglass and Michael son (11) have reported an investigation of 387 students of the class of 1930 In th© College of Arts and Science and the School of Business Administration at th® IMiversity of Oregon* They found a coefficient of #2 S ± *06 for the correlation between th© number of semester credits in high school mathematics ©nd average college marks in mathematics, and list among their eonclusions, *Prediction of success of students in college mathematics cannot be made with any high degree of accuracy from knowledge of the amount of high school training in mathematics, # * #* This study has been discounted and quit© soundly criticised by Hart (16) who points out that average grades were taken at the end of a long period, two years, and that in the meantime the students involved had become quit© homogeneous with respect to knowledge of mathematics* Four theses written very recently at the Uni versity of Iowa reveal pertinent facts concerning the effect of quantity of preparation In mathematics on reten tion of algebraic knowledge* All four present analyses of algebra tests given without warning to high school seniors* 09 Branham (5} gave the 195? Iowa Every-Fupil Test8 In ninth year algebra to SS9 aenlors In the high schools or Peoria* Illinois, while Allen {!}* Burch (7), and Olson (24) administered a somewhat similar test® to 2*798 seniors in 59 Iowa high schools* The conclusions of the four studies are almost identical# We quote from Burch* The amount of retention of algebraic knowledge varies directly with the amount of mathematics that the students have had* Advanced algebra seems to provide more retention than does any other mathematics course* Th© other three studies are in agreement with this statement except that they all deny that a course In plan© geometry adds to the retention* They hasten to add that even th© better groups do not retain enough knowledge to Insure success In M e t i n g problems of an algebraic nature* but offer some hope to th© college teacher by pointing out that th© college-bound group Is definitely superior to those not planning to continue In school# The studies of this section are In general agree ment with each other and also with the writer In regard to the following point©! 1* There Is little relationship between th© number of semesters of preparation in 8 A description Of this testing program was given by Lindquist (20) In 1955* ® The test is divided Into three parts and each man reported ©n one part# mathematics and first semester grades* unless the preparation is six semesters or more# 2# There is a definite relationship between the amount of preparation in algebra and retention of algebraic skills* V&ffiability i*^ Item difficulty from school to school* Two these®, written at the State Chlversity of Iowa* are concerned with item difficulties# Both utilised the results of the 1953 Iowa Every-Pupil Testing Program* and both studied the variability In Item difficulty from school to school* Atchison (2) used general science scores and Robb (27) algebra scores# Their procedure was to calculate th© difficulty Index of each item In each of several schools, and then to make comparisons from school to school* Both found large differences between schools In the sis© of the Index of a given item* and also in the rank of this Index in the distribution of all such Indices* CHAPTER III ANALYSES OF THE 1035 PRE-TESTS ©• Analyses made by the Cooperative Teat Service# Th© three forms of the 1935 tests were administered to 2*630 students in th© ©ix cooperating Institutions men-* tioned in th© introduction# The following tables show the number taking each form and the distribution of scores which resulted# The scores ©re well distributed over th® rang©* with considerable difference appearing between schools on Forms B and 0 # Tabl© 1 NUMBER OF STOTENTS AT EACH OF SIX INSTITUTIONS TO WHOM THE 1935 TESTS WERE AMINISTEKEB Gollme Form A EstssLB. Form.c X IX XIX XV V VI 330 114 462 149 444 93 85 299 103 71 376 104 - 566 565 Totals 1499 Table 8 DISTRIBUTION OF FEE-'TEST SCORES FOR FORM A, 1935 School X . . 42 41 40 39 30 37 36 35 34 33 32 31 30 29 28 27 26 85 24 23 15 14 9 23 23 16 17 22 81 School __ XI School 111 1 1 1 4 2 2 10 7 1 1 1 1 1 1 I 2 2 1 1 1 10 6 a 4 5 7 1 X 7 2 2 5 IS 2 10 6 6 13 6 8 12 81 4 12 2 19 14 16 24 10 12 5 17 16 15 14 13 9 16 6 12 11 10 2 4 4 3 7 5 0 6 6 5 9 a 2 0 5 3 25 19 15 26 31 24 25 29 m 25 23 IS 17 14 7 8 2 12 1 3 4 9 3 3 6 9 9 5 6 6 3 4 3 2 1 Totals .IV 1 1 3 20 School 330 114 13 1 1 8 3 2 4 7 3 7 4 6 4 4 13 10 5 9 9 4 10 @ 4 9 3 3 5 3 1 3 1 1 1 1 482 149 School 7 1 I Total 1 2 2 4 4 ? 1 1 1 1 1 7 2 2 7 5 8 9 9 13 18 10 12 12 16 12 13 19 89 87 17 21 31 29 24 24 15 18 13 11 8 2 3 1 1 5 a 23 18 22 19 23 37 38 43 54 55 47 60 59 62 61 67 66 85 76 54 71 80 64 03 56 43 41 50 16 18 5 6 3 1 ijjAA 1499 Table 3 DISTRIBUTION OF FBB*TEST SCORES FOR FORM B, 1935 BOOS’© 42 41 40 39 38 37 36 33 34 33 32 31 30 29 28 27 26 23 24 23 22 21 20 19 18 17 16 13 14 13 12 11 10 School 1 . 1 1 s School 1 X1 1 1 1 8 1 2 3 3 4 3 7 3 7 7 7 7 2 6 2 6 6 2 1 8 7 4 7 1 2 4 3 1 11 12 11 7 13 17 30 17 21 2 10 22 8 9 7 6 7 4 1 1 1 2 1 8 1 0 3 4 7 7 3 6 6 0 7 7 3 3 3 2 1 2 1 2 7 3 4 4 4 4 2 0 1 1 1 2 Total© 95 52 31 40 28 34 29 28 £1 29 11 9 10 a 1 2 I 20 17 23 17 30 23 2 4 8 5 4 3 4 4 7 7 B 12 19 10 1 9 82 17 1 1 2 10 5 Total 2 5 1 3 3 3 2 3 4 3 VI 2 3 9 6 School 3 1 2 3 School X? 1 1 1 3 299 103 71 306 Table 4 PKS*TBST SCORES DISTRIBUTION OP BOfoOOl Score 43 48 41 40 39 38 37 m 38 34 33 32 51 30 29 28 27 28 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 "W* J k 8 School X? 8 8 2 3 © 4 3 5 7 7 3 1 1 3 1 8 2 5 4 5 7 7 5 7 1 11 10 3 9 1 2 10 3 20 2 6 1 2 26 83 22 33 4 20 2 1 2 2 1 1 23 27 3 6 612 a 7 5 8 6 10 12 5 9 © 22 11 2 5 10 2 3 3 85 4 2 13 14 © Total 1 2 1 1 POKM C, 1938 1 a 8 7 6 Totals School XIX FOR 376 4 5 11 11 11 19 17 17 15 18 21 28 40 37 30 40 32 31 38 19 17 25 1© a 1 12 3 3 10 5 3 8 104 565 OQ After such, examination of results as each Institu tion desired to make, the tests were sent to the Cooperative Test Service in Hew York City where they were rescored and subjected to a statistical analysis* At the end of the first semester (or quarter) of work* a list of the grades earned by students participating in the testing program was also forwarded to the Cooperative Test Service to he included in the analysis* This analysis consisted of the computation of a* th® power of each form to predict first semester grades, b* the reliabilities of the various forms, c* the percentage of correct responses to each item, commonly known as the ndifficulty” of the item, and d* the degree to which an item discriminated between superior and Inferior students as measured by their total score on the test* This will be referred to as the 11validity11 of an item* The results of the last two parts, inasmuch as they are item analyses rather than test analyses, will be presented in Chapter V* Th® reliability coefficient found for Form A was ,9173, for Form B, *9166, and for Form C, *8926* ??n The following table shows the relation found at each institution between scores on th® 1935 tests and first semester grades* schools* This was reported for only four of the Three observations should b© made in connection with this table* First, the correlations found compare quite favorably with those reported for similar tests*^ Stoddard (50) reports finding a coefficient of *28 for the correlation between scores on the Iowa High School Content Examination in Mathematics and first semester grades in college mathematics* He also reports coefficients ranging from *53 to *70, with a median of *60, for the correlation between scores on the Iowa Placement Examinations, Mathe matics Training, and first semester grades in college mathe matics s Miller (23) and Rammers (25) report coefficients of *48 and *66 respectively for the same examinations* Crawford and Burnham (9), who studied the relationship be tween scores on the Comprehensive Mathematics test of the College Entrance Examination Board and first year grades in mathematics mad© by the Yale freshman classes of 1933 and 1954, report a median correlation coefficient of *36* Gilkey (15), in a somewhat similar investigation, found a coefficient of *34 for the correlation between scores made at New York State College for Teachers on the New York 1 Segel (29) lists several tables of such corre lation coefficient® on pp* 59-70 of his monograph, * # • ^ to 10 iO * * * * rH |H »H H*HiHi CO £-* ^ t£> ■• * • • Hr! W W oa oa to to t- o> • * * 03 ^ 01 0 *0*0 • * * *> tO <0 too ^ • • * fr-to *o •# * •* to to CD 10 0» 10 *0 BO >* * * ■* • ih ♦ * m o o too too • 03 CO ID # ■* • *Q 05 f**4Hi |f*i lOO&H 03 •• • * • to to to H CM 03 03 O! CO ^ ^ 03 O COCO 05 H Hi mi * to m o i•» ^ *H ft 03 to +% to to ^ ft to 03 to l* ID o # h to • # H CD 03 03 H 03 05 CD tD H O O H *«at* A A © m ■• * to 356*6 79*8 524*3 88*9 * 6*5 4.7 * •* 23*9 13*5 <0 ID 0* 10 M +*v go Oe eO Og -* as o H © H O O Q oa o to *• m «P © 1 +» © I «* © 4* S3 'O £ © P3 0 o o E g‘a .jQu *1 '■f fyn ’ l r*^ &i i"'! 0? CO +«fc £Q 05 I* <$ O Hi «* <*} H H VA H Pi 03 O 10 GQ ^ (XJ ^ <q ID < Hi Hi hi Hi to 1«* ft 0* &t j&»&« ** m g (S to *i > H ggg p oo &. £V| Ce* Fora Fora O ID <3 ft O «4 00 O 01 6 ““ F,plT**t students* students: s Advanced He&ientary Hi to <0 gs§ig VOtOtO'V 144 258 10 to to to H oa 0> ^ *6239 *5811 31 32 Hegents l&amination in Mathematics and all college grades In mathematics* Ha remarks that the coefficient diminishes as the number of years of college work increases* and hence the correlation reported is probably less than it would have been had first semester grades only been used in th® study* The second observation to b© made In connection with Table 5 is due to th© fact that there is a definite positive relationship* as will be demonstrated very shortly, between th© amount of preparation in algebra and scores on these tests* Some of the institutions involved in th© analysis adhere to th© policy of assigning those students with only one year of preparation in algebra to a special class* Th© tendency to grade on the class average hence effects a general increase in the ratings assigned to members of this group in comparison with those in th© other groups* This* in turn, tends to force down the correlations presented in Table 5» The third observation is that th© rather high correlations found for School 1 are associated with rela2 tlvely high mean scores* Whether till3 has some signifi cance, or is merely a coincidence, we are unable to say* In view of the preponderance of highly trained students enrolled in this school, it is quit© likely that the ® A similar phenomenon was noted by Hammond and Stoddard (14) In studies of the Iowa Placement Examinations* 33 detrimental effect mentioned in the preceding paragraph does not have a chance to operate* Preparation croups* The analyses presented and discussed thus far have been due to th© Cooperative Test Service* The writer has confined his attention to the test papers written by those students who presented one, one and one half, or two years of preparation in algebra* The reason for this Is that these three groups are the ones usually found in college freshman classes* The tests were given before the recent trend toward elimination of college entrance requirements in mathematics had begun to operate, and hence the typical student had on© and one half years of preparation in algebra* It was the one and one half year group which the committee kept in mind whilo constructing the tests* Th© three preparation groups, which we shall label ^1 * ^li* Pg, comprise almost the entire population and are distributed among the various institutions as shown In Tables 6 , 7, and 8 * A comparison of these tables with Table 1 will show how nearly equal in sis;© the two popula tions are* 34 Table 6 NUMBER OF STUDENTS AT EACH INSTITUTION AND IN EACH PREPARATION GROUP TO WHOM FORM A, 1935 , WAS ADMINI STERKD Ins titration I II III IV V Totals £l M Es. 22 72 45 250 75 55 497 194 19 42 18 73 346 46 145 45 291 549 Total 288 110 437 138 419 1392 Table 7 NUMBER OF STUDENTS AT EACH INSTITUTION AND IN EACH PREPARATION GROUP TO WHOM FORM B, 1935, WAS ADMINISTERED Institution I III XV VI Totals £i 7 146 24 19 196 M 22 118 56 8 204 Total E& 60 16 11 89 280 91 41 128 528 68 Table 8 NUMBER OF STUDENTS AT EACH INSTITUTION AND IN EACH PREPARATION GROUP TO WHOM FORM C, 1935, WAS ADMINISTERED Ins tl tut ion I III IV Totals El Eli 5 79 35 119 14 252 60 326 Po Total 48 35 2 85 67 366 97 530 35 8 * Partial and absolute scores.# Before concern ing ourselves with th© distributions of scores made by these preparation groups, let us observe that there are some Items In each test which consist of several parts# For example, item 3 of Form A Is a. 3 (-6 ) -4(-S) » b* 2 * 0 * c• 0 * 3 * a d* 0• s • 4 Directions accompanying the tests stated that such a ques tion was to be scored as right only if correct answers were exhibited for every part; otherwise It was to b© marked wrong* This procedure gives rise to what we shall term an absolute score* However, these Items were also scored as If th© above example were fotir separate exercises* Scores thus obtained w© shall label partial* The question w© wish to raise now Is, "How do the two methods compare in so far as ranking the student® is concerned?” An attempt has been mad© to answer this question by a study of items which were subjected to both scoring procedures. A brief examination of th© tests found on pages 6 8 , 69, 70, 71, 72 and 73 will disclose that each of the three forms of the 1935 tests contained about th© same number of "multiple" Items* However, since Form A was administered to more 38 students than were Forma B and 0 combined confined to this form* attention was In order to select a sample with which to work, all copies of Form A were drawn from the three preparation groups and placed In a single pile in a haphazard fashion* Then, using Tippettfs table of random numbers (32), a random sample of 200 papers was drawn* The next logical step was to pair partial and absolute scores* Since only a few items were of the "multiple” type, it was obvious that if scores for the entire test were used, the correlation would necessarily be spuriously high* Hence only those Items which were scored by both methods were considered* Seven such Items appear on the examination but only five could b® used; item 35 is subjective in nature and was not scored uni** formly, and in a great many cases part Co) was the only part of item 30 which was scored at all* Even with th® number of usable item® thus narrowed to five, a complicating factor was In evidence* Two of the items consisted of four parts each, one of them of three parts, and the remaining two of two parts each* Making use of only the sum of correct responses to parts would thus give some items more weight than others* In order to counteract this variability, each part of th© four**part items was multiplied by three, each part of th® thre@~part items by four, and each part of the two**part 3 See Table 1 Items by six# Thus each item was credited with a composite weight of* twelve, and the total possible partial score was sixty# After the partial and absolute scores were counted and paired, a correlation table was set up and the Pearson product-moistent coefficient of correlation computed* This coefficient was found to be *87^ with a probable error of #01# The writer interprets this to mean that if on© is concerned with th® total distribution of scores, then th® two methods yield approximately th® same results* However, it can hardly be said that th© two procedures will rank an Individual in the same order in the distribution* $* groups* %$m. of , m m m M mmMM&Sm When the 1935 tests were revised and th® 1930 teats assembled, all ^multiple1* questions were broken down and the components used as separate items# Consequently, since we shall later have need to male© comparisons of responses to those Items which were given both in 1935 and 1936, we shall concern ourselves with partial rather than absolute scores# The following three tables give th© total distri bution of partial scores for each of the preparation groups 4 Due to the smell number of items involved, this coefficient is not to be relied upon too heavily# 38 together with the mean (M), standard deviation (<T) and standard error of the mean5 (d^) for each distribution. 5 _ _ I E"d® M "*y n(n - 1 ) * wher© d represents th© devia tion of a given element from Its mean* K* F# Lindquist, Statistical Analysis in Educational Research, P* 51* 39 Table 9 TOTAL DISTRIBUTION! OP PARTIAL SCORES B5T PREPARATION GROUPS FORM A, 1938 §&m& 51 50 40 48 47 46 46 44 43 48 41 40 m m 37 56 33 34 33 38 31 50 89 98 27 m m 84 83 29 21 20 19 18 17 16 13 14 13 12 11 Ex pi& 2 % i 9 2 1 3 I 7 3 f e lt 9 9 18 10 18 18 8 22 27 84 24 24 40 29 40 35 37 25 23 14 23 12 ® 6 1 1 2 5 8 7 a 5 11 15 10 88 11 14 27 IS 82 24 26 21 21 89 23 24 24 17 19 1® 10 5 9 0 0 4 5 !s i 2 1 3 2 8 ® 10 7 6 7 12 IS 14 IS 17 84 20 18 19 15 10 IS 20 18 a 12 ® s 10 5 5 7 2 2 1 5 3 1 4 8 2 ^.o Table 9 {continued) FORK A, 1935 Score *1 10 7 1 9 3 7 7 S 3 1 6 1 5 I 4 1 Totals 349 % pg 497 23*33 7*43 *3® r «S 346 33*23 7,93 ,36 32*66 7.97 .43 Table 10 TOTAL DISTRXBOTIOH OF PARTIAL SCORES BX PREPARATION GROUPS FORM B, 1935 Score 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 &3j. 1 4 1 8 2 0 3 3 9 £s 1 I 1 1 S 3 I S S 1 S I 3 1 3 3 6 6 7 6 7 11 5 9 8 3 2 6 5 10 2 4 6 8 5 10 2 41 7abla 10 FORM 8* 1955 £i III 18 295 18 109 S6 18 16 18 21 10© 12 17 ©9 11 78 189 182 82 74 2 28 1 8 1 19© 204 20*51 14*34 $#87 •42 7*48 *82 * Room 21 20 19 IS 17 10 IS 14 IS 12 11 109 8 65 824 1 totals oMr «a Is. 10s4 0 14 254 1 11 128 25*78 7*44 *6© 12 Table 11 TOTAL DISTRIBTJTXOB OF PARTIAL SCORES BT PREPARATION GROUPS m m Seort 4® 48 47 46 48 44 46 42 41 40 39 83 87 86 36 34 31 38 81 30 80 c, loss £i i % i 2 1 X t 0 1 1 87 8 m m a 24 23 82 21 30 1© 8 7 18 7 4 9 8 xa a a Totals M <r «n £2 i m 17 16 15 14 18 12 11 10 © £ii «jr 7 4 @ 7 3 1 a 3 1 119 19*43 6*20 #57 2 6 2 9 S 12 7 16 6 15 17 17 22 24 19 21 22 16 17 15 12 12 8 6 1 3 1 4 4 2 1 4 6 4 2 5 4 5 6 7 4 2 3 1 2 3 2 1 2 6 4 2 1 2 2 1 326 25*08 6*70 *57 1 85 29*12 7*15 .78 It is obvious from an examination of the preceding tables that those students who entered college with two years of high school algebra to their credit mad© higher average scores on these testa than did those with only one and on© half or one year of algebra# Similarly# th© one and one half year group responded correctly, on the average, to more Items than did the one year group# The question quit© naturally arises then, ”Are these differences real or merely due to chance?” That Is, if th© tests were given again to a similar group of students, such as perhaps would he enrolled in these or similar institutions in other year®, would th© differences again he present, or might they In some eases vanish or even appear in th© reverse form? In order to answer this question, w© shall make use of the following table# . The first four columns of this table are based on th© results of the preceding tables and th© fifth column was obtained by dividing each difference in means as shown in th© third coliimn by the corresponding standard error of th© difference® in th© fourth column* 6 *Taiff.} - F M . + *ri* • F# Kenney, ”Som© Topics In cal Monthly, XLVX (February, 1959), Bo# 2 f pp# 61, 62# 44 fable 12 OTABIDXTST OF DIFFEKBBCES OBSERVED BKWfiBH PREPARATION GB0TJP5 01 T m 1935 TESTS Emm A A A B B B C e c Preparation Orotma Difference .. Jin. Means and h i and ** F*i p rg and h 5*96- *48 12*4 4*3® *56 7*8 10*34 *64 19*1 *67 9*2 3*27 #84 3.9 9*44 *78 12*1 5*05 #68 8*3 4*04 #86 4*7 9*69 *97 10*0 h and h i h i and P2 Pi and *2 H and h i h i and P2 pi end p2 *i Standard Error of Difference Critical Katlo Th© hypothesis we wish to test is that these various pairs of preparation groups are actually random samples drawn from th© same population and hence any differences found can be readily attributed to chance* Under this hypothesis* all of th© differences observed are highly significant at the 1 per cent level# Hence we may say with assurance that our preparation group® were derived from populations which have distinctly different degrees of mathematical skill In so far as this skill Is measured by the 1935 tests* to* ICCftftfc of preparation on first m a m i a M E grades* *0o the differences fust shown carry oyer to grades? That is, does th© probability of success, as measured by these grades, increase with preparation? In order to answer this question, all of the 193$ grades which were reported have been assembled in Table 13* Gtrades given at each school, and at all schools, are classified by preparation groups* H represents the number of students with a given amount of preparation in algebra, and the numbers paired with grades are percentages* Thus, In school I there were $9 students with one year of preparation in algebraf of this group, 12*5 per cent earned grades of A, 15*6 per cent grades of B, etc* The ^miscellaneous** entry consists of withdrawals, transfers, absentees, and th© like* Table IS PERCENTAGES OP STUDENTS EARNING VARIOUS GRADES IN 1935 SCHOOL I Grad© A B C £ failure misc. H 1 12*5 15*6 12*5 25*0 25*0 9*4 52 SCHOOL II Preparation Total l| 2m* « 2*8 17*8 23*4 26.2 28*0 1.9 107 7*5 16.3 27.2 24*1 22*5 2*5 16*6 25*3 24*6 24*0 $20 459 6.8 2.8 Preparation 1 21.7 10*9 17.4 10.9 21.7 17.4 46 SCHOOL III A B C D failure mi sc* N Grad© A B C D failure misc. 1* 1 9.6 18 *2 27*5 19*8 19*5 5.8 563 1 si 13*9 18*9 25*6 20 15.1 6*6 610 2 29.2 24*0 25.0 11.5 6*3 4*2 96 22.9 14.7 11.1 16.7 16.7 18 16*5 13.8 109 8.9 45 Total 13*8 19.1 26.1 19*2 15*7 1 ii 2 Total 6*9 8.6 12 *9 21*8 22.7 38*9 20.5 3*8 5.4 185 22.6 8.5 22* 4 37.5 18.0 6.1 34.7 16*8 12 *9 6 *9 1009 101 38.7 6*5 3.2 16.1 31 Preparation 2 42*7 2*8 46*8 17*0 14*9 8*5 12.7 16.5 45*0 16*5 5*1 3*8 986 47 79 22.0 10.1 Preparation Preparation 0 12,8 14*3 22*2 22.2 11*1 11.1 ALL SCHOOLS 6*6 12*6 21*0 84.4 15.0 31.1 8*9 SCHOOL V li Total SCHOOL IV Preparation Grad© 2 Total 7.0 13.3 43.7 19.7 12*6 3*6 412 1 9*1 13*2 32.4 19*6 17.1 5.7 828 it 2 0.6 6*9 317 Total 11*0 6.0 12*9 18.2 29.6 19.3 15*8 4.2 994 544 2366 11.6 19*0 29.1 20.1 14.2 17.8 30.3 19.7 15.6 5.5 There is some tendency for high grades to be associated with increased preparation* but It Is not uniform* In general* we must agree with the research discussed in Chapter II* and conclude that there Is very little relation* ship between the two variables* CHAPTER IV ANALYSES OF THE 1930 PEE*TESTS *1# An&lveef bv the Cooperative Test Service. As has already been suggested* the 1930 teats were given to a population very much like that used for the 1930 program* Four of the five schools administering the revised forms had also cooperated th© previous year* end the fifth school was quite similar to one of those withdrawing from the group* Th© new tests were slightly shorter and easier than the first set* a fact which was known {and intended) by the test committee* and which is illustrated by th© diat tributions in Table 10* This table was constructed by the Cooperative Test Service and Includes all of the papers written* Other analyses made by the Cooperative Test Service reveal. In general* the same things as were found for th® 1930 tests * Correlations of test scores and first semester grades at each institution are given In Table 14 below* -19 Table 14 C0RRELATI0H8 BBTWEEH 1930 TEST SCORES AED FIRST SEMESTER GRADES IE COLLEGE MATHEMATICS, AT EACH INSTITOTIOH Sigma Fom I X A B 120 141 XX A B III XT Sohool Seeres EL 1*007 1*138 0.84 6*50 *65 *70 461 449 1*324 1*730 8*40 7*45 .51 *53 A B 023 143 1*144 1.194 4*20 3*62 *56 .51 A B S3 87 1*011 1*700 8.47 *57 .69 tE. (BUS A ©emp&rison of fables 8 ana 14 reveals that 193S and 1930 seta possess about the same predictive power. Table 10 DISTRIBUTIOHS OF SCORES FOR THE 1930 TESTS Frequencies Seore 50 49 48 47 46 45 44 43 40 41 40 39 38 87 36 Fens 4 0 5 3 10 11 0 10 18 14 25 30 36 29 40 45 Fern 1 1 1 2 8 6 12 20 7 18 15 20 17 24 50 Table 1© (eon.timed) §§m& 38 34 33 32 ©1 m 89 28 2? 26 2S 24 28 22 21 20 19 18 IT 16 10 14 18 12 11 10 9 8 7 6 5 4 3 t Totale Frequencies Fo ot B SmLA 44 m m 40 36 44 48 30 36 33 m 34 28 31 29 28 22 19 14 14 20 10 14 © © 4 S 3 20 29 30 a 41 39 42 40 36 02 m rf Of 36 63 34 46 3© 37 36 28 20 18 21 20 14 9 12 10 4 4 2 1 1 1 0 2 6 1 1015 1014 51 i&* M^ovtps» M w®m£§Mm Th© corresponding distribution® for each of th© preparation groups and Pg* together with the arithmetic mean# standard deviation#, and standard error of the mean of eaeh distribution# are as follows* Table 16 DISTRIBUTIONS OF SCORES FOR FORMS A AHI) B m PREPARATION © R O W S Px # $ % y and Pg ore SO 49 48 47 46 4§ 44 43 49 41 40 39 38 37 36 35 34 33 39 31 30 29 28 87 26 25 24 23 22 21 l a l ix M I 14 3 1 11 75 22 13f7 10 14 16 17 18 21 50 22 553 86 23 5 119 24 20 31 88 16 21 7 18 14 14 18 18 12 12 11 16 IS 11 5a Xi 5a 63 86 14 10 80 14 14 11 15 10 15 12S 9 4o5 a 6 6 213 1 Cl9S6> t e J Total 112 99 79 16 13 25 95 30 29 37 42 41 35 46 40 54 40 45 88 55 31 34 33 26 88 27 Ex M i i is 1 634 65 4 8 11 17 14 12 17 11 20 2s4 55 7 11 10 16 15 16 23 19 19 22 17 25 14 18 28 21 25 Ez 121 64 10 10 1284 136 11 13 109 IS 11 13 IS 9 6 8 53 S 31 Tet< 112 74 11 15 6 16 13 19 16 23 23 29 28 38 40 37 58 39 34 48 53 33 SO 33 46 52 Table 16 (continued) A SfifiEft £i Sat 20 is 11 19 10 10 11 S 8 2 9 S 10 7 s 2 2 2 2 1 2 IS 17 IS IS 14 13 12 11 10 9 £t O 7 6 5 4 3 2 Tofcala M ff* 11 s 6 2 2 Q P Total 2 1 1 1 1 1 20 20 13 14 8 19 14 14 18 24 IS IS 10 11 10 8 0 1 2 1 10 13 13 13 7 7 a 11 8 7 1 2 4 4 8 4 12 8 1 5 9 3 3 K © 1 1 2 28 10 2 1 2 X 2 940 322 220 2 2 9 1 406 35 36 35 26 25 17 20 20 12 3 2 1 1 460 Total M 2 8 1 1 254 El 2k 825 953 2&8B. 3M8* 30&L4 29,075 2X0€> £€026 35*466 25*955 7*0 6*9 8*7 63 9a 7a s*i M *3 5 *46 *28 *35 •49 *00 *40 53 As for tli© 1935 tests, wo wish to determine whether or not these differences in means are significant* We make use of the following table* Table 17 COMPARISON OF MBAIS OF THE PREPARATION GROUPS P ^ , and Pg AS FOOT!) FOB PORK A, 1956, AM) FORM B, 1936 P^na A A A B B B Preparation 0rouB3 ?1 « “« and Pj and Pi and px|? **»a Pi and Pg Pg Pi£ p2 Pg Difference In Mean® 7*875 5*128 13*003 6*08© 7*340 13*426 Standard Error of Biffex*©noe Gritleal Ratio •62 *57 *68 *53 *58 *61 12*7 9*0 19*1 11*5 12*7 22*0 With critical ratios as large as these, there can be little doubt that, as was true of the 1935 tests, a elg* nlfloant increase in mean score accompanies an Increase, from one to two years, in algebra preparation* The follow ing graphs of the distributions given In Table 16 illustrate these differences very clearly* It is to be remembered that the committee construct ed the tests essentially for the one and on© half year groups* The graphs illustrate how well they accomplished their &!mj th© crests of the distributions fall near the center of th© rang®, the crest© of th© P^ distributions are somewhat lower on th© scale, and th© crest® of th© Pg distribution® are higher* For th© purposes of comparison, th® ordinate® are given in percentage® of the total distributions* seaoos XIV jo ^^©0 Made in U. S. A. -SEgjggpP No. 5780E— 20 Squares Scores Form per In c h a to M i saa:oog xiV J° CJ.U8Q 5 6 13, IfAamCT difficulty of. tha 3S5§. £SEM.* Another ch&r&e taris tic of Table 16 which should he observed is that the mean score for Form A Is higher than the mean score for Form B for every comparable mroms* This is a strong indication that Form B is th© more difficult of the two# {Indeed, one gets this impression by simply ex amining the distribution®#} We do not know that the two groups of students who wrote Forms A and B respectively differed essentially in mathematical ability; neither do we know that they were alike* Since no effort was made to pair any particular type of student with either test form, it is reasonable to assume that no appreciable difference® existed# Under such a hypothesis we compare th© test forms for difficulty in the ©am© manner a© w© compared th© preparation groups for ability# Table 18 was constructed in exactly th© same manner a® Table 17 except that, of course, the differences in means refer to mean® found for th© two forms* A fifth column has been added which gives the probability of the occurrence of such a difference when th© groups compared are actually random samples of th© same population* Table 13 COMPARISON OP ARITHMETIC MEANS POUND POE POEM A, 19Be, AMD POEM B» 1936 VIftarenoe la Qroup Mmmna (MA-*B) *1 *lh ps Totals Critical Probability .Ratio .. of Occurrence) 2,m *65 3*5 4*060 *49 8*3 1*848 *64 2*9 3*480 *41 8*3 B m Form A m standard Error of Difference *0005 0 *0038 0 th© probability that th© two-year group finds difficult as Form B is only about four on©** thousandths* which w© consider too small to be of consequence, and hence * unless marked differences in ability exist between the two groups under consideration, Form, A f» definitely easier than Form B* 14. Mean scores made * £ aagh Afiy&SL&iMSfi. following question now arisess «» are th© differences found between th© means of th© preparation groups characteristic only of th© entire population* or are they present at indi vidual institutions as well? Due to the small number of individuals Involved in some of the sub-groups * it is im possible to give a definite answer to this question, but the following data Indicates that, for the most part, th© differences hold for Individual schools* 58 Table 19 MEAH SCORES OH TEE 1936 FIB*TESTS FOE THE PREPARATION GROUPS *1* F9 AT m m COOPEBATIHG INSTITUTION School X Frepaa*©tior%, Group I Mean £ Mean H hi 3 33 36*00 34*82 5 39 20*00 30*31 P2 96 36*15 100 34,94 *X hi ps 194 24V 37 20*05 28*04 38*00 813 247 26 18*52 24*67 29*32 III «R»aA*4i» h hi ?8 31 101 15 26*38 30*92 54*60 73 42 11 20*07 26,91 32*00 XV P1 hi *8 20 24 36 33*35 33*00 33*00 19 37 30 28*63 27*81 30*71 V pl hi 6 62 40 39*00 33*05 Ml 00*00 29*90 55*29 II P2 30*43 41 51 The one-year groups at School X* and perhaps th< two-year groups at School III and th® one-year groups at School V, are so small as to give quite unstable results It is to b© noted that th® larg© groups at School II reveal quite substantial differences In mean scores between the preparation groups * particularly between groups Pi and ^lj-* Relatively ©mall numbers Involved at School XV could readily account for the fact that no differences are found between groups* 59 That the char ac ter is tie of differences between preparation group® is a function of the m o u n t of prepara tion in mathematics» aa well as preparation in algebra alone, may be seen by am examination of fables 20 and 21* Sot® that the information blank on the first page of each test3, asks for the number of semesters of preparation in each of the mathematics courses usually taught in high school* In computing the composite prepa ration of each student, the following procedure was observedt algebra preparation was accepted as given by the students on these blanks* plane geometry preparation was considered as constituting on# year, and solid geometry and trigonometry were counted as on# half year each* Th® composite preparation was then the sum of these parts# It is this preparation which appears under th# heading 19Total Preparation in Years11 In th# tables* 1 See pages 68, 69, 70, 71, 7S, 73, 106, 107, 108, and 109, so Table SO tSBAH SCORES FOR FORM A, 1980. AT EACH COOPERATIBG IHSTITUTIOH FOR EACH HALF TEAR OF TOTAL FREPAKATIQB IE MATHEMATICS 1 §£lttsl i it E 3? M H ix 8 M 19*5 i 28 H in M H I? M 1 as I v M I §t ISO IS *01 1 56 84 32*58 SO 152 27*75 25*05 it m £ 33*12 77 36*83 75 30*58 22 34 24 26*21 62 30 37 31*05 15 32*8 0 37*33 7 32*46 14 31 80 53*28 13 35*62 16 35*06 4 32*5 36 52*04 41 54*81 13 38*15 14 41 Table SI MEAH SCORES FOR FORM B, 1986. AT EACH COOPERATING IKSTITUTIOS FOE EACH HALF TEAR OF TOTAL FREPARATIOH IE MATHEMATICS 1 1 H M II I III K M XV M H M V * M WMM & 10 27e8 18 30*78 28 32*79 87 84*93 150 67 23*85 88*81 74 27*16 18 81*07 64 18*80 26 23*62 80 m 7 26*57 7 34*86 14 28*14 10 28*37 54 27*59 16 29*8 10 36*5 5 25*8 10 29*11 41 32 14 53*64 88 86*04 1 27 176 17*50 3 14*33 1 0 1 82 1 25 at si & 1 ox In order that no misunderstanding may aria© from tli® last three tables* let it be clearly understood that the differences cited have not been represented as sianlfleant* The facts have been presented simply as evidence la support of the belief that* had enough students been Involved in each of the various subgroups* the same significant differences would appear as were found for the populations represented in fable 16* 62 CHAPTER r ANALYSES OP THE ITEMS OF THE 1935 FEE-TESTS IS* M&lirms made by the Cooperative Teat Service* Th© Items used In th© 1935 tests were studied by the Co operative Test Service according to the following plan: of Form A* a random one third was selected from each of th® highest and lowest three hundred papers* Since there were nearly fifteen hundred papers* this sampling constituted a random selection from th© highest and lowest fifth* In th® cases of Forms B and C* th© highest and lowest one hundred paper® were examined* In computing Item difficulties, the percentage of correct answer® to an item was determined for each fifth mentioned In the preceding paragraph and the two percentage® averaged* Difficulty indices derived by this method h&v© been found by the Cooperative Test Service to correlate very closely with difficulty Indices calculated by using a random sample of the whole population* Th© item valldltes were computed by th© Coopera tive Test Service by finding the percentage of correct answers to each Item for the highest and lowest fifth (or some random fraction of th© highest and lowest fifth)* These percentages wen© plotted on coordinates In th© units used and the validities computed In terms of the distances of th© resulting points from the diagonal y * x* 83 Validities of 2 or more are considered by Ben B* Wood* the director of th© Cooperative Test Service, to be satisfactory when calculated in this manner# Difficulty and validity indices for each item In th© 1935 tests are listed In th© Appendix# Heeoll that some of the items on Fart XX of each test are subjective In nature# These were not scored by the Cooperative Test Service and hence no Indices are available# Distributions of difficulty and validity indices for each test are given In Table 22* G4: Table 22 DI3TBXBUTIGBS OF VALIDITY AID DIFFICULTY IKDICBS FOOTD FOB THE 1955 TESTS PjfficttltT Form B 5b Ss s 95-100 90—94 85-89 80-84 75-79 70-74 @5-69 60-04 55-5® BO-54 45-4® 40-44 55-59 50-54 SB-9® SO- 24 15-1® 10-14 5-9 0-4 1 5 S 5 5 5 1 5 4 @ 5 5 5 5 1 5 S 1 1 S 1 4 5 1 2 5 0 1 4 5 4 5 1 S 2 1 1 1 1 2 4 5 5 4 2 1 1 5 S 5 « 5 5 5 5 1 1 Validity M i m 14 IS 11 10 9 8 7 0 6 4 5 Form A Form B Form 1 2 6 5 4 6 5 9 6 8 4 5 1 4 5 5 7 7 10 2 1 5 7 8 9 10 4 7 2n view of th© criterion used by the Cooperative Test Service, these Items are ell valid* Form B does not contain enough easy items, but for the other two forms dif ficulty indices are spread quite uniformly over the entire range* Considering the relatively small number of items in the tests, the distributions of these indices are very satisfactory* 16* Selection off samples* In order to make a detailed analysis of the items of the pre-tests, it was necessary to make use of samples* Th© procedure which was used in drawing these samples is as follows* all papers not falling into on© of th© thro© preparation groups F ^ , and Pg were separated out* Those remaining were then divided into subgroups according to test form and prepara tion* Thus there were nine such subgroups In 1935 and six in 1936* Papers in each subgroup were arranged by institu tions and then alphabetically within ©ach institution group* One sample was drawn from ©ach of th© subgroups and the procedure used in building a sample is aa followss if fifty papers were to be selected from on© hundred, alter nate papers would be taken? if fifty were to be selected from two hundred, every fourth paper would b® taken, et cetera* The net result was, then, a representative, or stratified, sample each of whose components was a random sample from some institution* "The sl&es of samples chosen by the above method are as follows* 66 Table 23 NUMBER OR PAPERS IN EACH 1935 SAMPLE Preparation Form A B C fi ?Ur 50 50 50 100 50 50 50 50 50 Tattle 24 NUMBER OP PAPERS IN EACH 1936 SAMPLE Sam A B *i 100 100 % 100 100 1O0 100 It is to b@ observed that approximately the same number of papers were used to each of th® two years % also that the 1935 sample eonsI®ting of students with on® and one half years of preparation in algebra, and who took Form A, Is larger than the others chosen for that year® The reasons for this are that more students took Form A than th® other two combined, and because the group with on® and one half years of preparation to algebra, being the most typical, Is naturally th® center of Interest# 17# Difficulty Indices for preparation aromas♦ One of the thing® w© are interested In la determining th© effect of preparation in mathematics upon student responses S7 to problems* to view of the manner In which the subject matter covered by the tests was selected, this must, for the most part, be confined to preparation In algebra# It has been demonstrated In Chapters III and IV that Increased preparation results in correspondingly increased mean scores on the tests* dividual Items? How does it affect the difficulty of the In-* We could expect that in some oases a con** cept would be mastered to a sufficient extent by those with one year, or one and one half years, of preparation that further study would not cause an appreciable improvement# In other eases, of course, we should expect additional prepa ration to carry additional achievement with it* Th® following tables give th© difficulty of each item for each preparation group as computed from the 1955 samples* The data bear out th© expectations voiced In the preceding paragraph# In a very few Instances the Indices decrease as th® preparation Increases* these differences However, non© of is large enough to b© significant* A column entitled ^tot&X** is Included In each table, Its entries being merely th© averages of the corre sponding Indices found In th® columns listed for preparation groups* It Is exhibited In order to call attention to the closeness with which It approximates th© column of Coopera tive Test Service difficulties which were computed for a sample of th© entire population* Two observations should be made in connection C O O P E R A T IV EM A T H E M A T IC ST E S TF O RC O L L E G ES T U D E N T S QS Pre-Test for First Year Students Experimental Form A - t 9 J 5 Please print: Name ____________ - _________ Last Date First Class_________________________ Age Middle Date of Birth Yrs. Mos. College or University______________________________ Classification (check one): Liberal A r ts____ City Sex_________ M . or F . Engineering___ Pre-Professional In the following spaces Indicate the number of years that you have studied each of the following subjects (one semester = 1/2 year; one quarter = 1/3 year; one summer session = 1/4 year) Algebra Plane Geometry Solid Geometry Trigonometry In High School leneral Directions: Do not turn this page until the examiner tells you to do so. This examina tion consists of two parts and requires 45 minutes of working time. There Is a time limit for sach part. You are not expected to answer all the questions In either part in the time limit, lowever, if you should finish Part I before the time limit Is up, go on to Part II. No questions nay be asked after the examination has begun. Use the blank spaces at the left of the page for four calculations, and write your answers on the lines at the right of the page. Part Minutes I 25 II 20 Total Score 45 Copyright, 1935, by The Cooperative Test Service. Printed in U. S. A. All rights reserved. G £' -3 - Part I Directions: In all answers in simplest any square roots to (Time: 25 minutes) £ x questions of this part, carry out the Indicated operations and give the algebraic orarithmetical form and without parentheses* Do not reduce decimal fractions • Write the answers in the spaces to the right of the page. x ^ 1. 2 i 5 _ 3 * 6 3 2. 2 + 1 _ 4 3 6 3(-6) - 4(-5) = 3. a) c . . . . b) 2 + 0 ~ ............... y c) 0 * 3 = ............... 6 d) 0 ^ 4 4. Add ........... 3 - 5y/2 0 . . Sum = 2 + V? 5. 2v/5 • 3 y /7 = 1 ................ 6. Add x - 4y + 1 3x + 4y - 6 4x + 3y 7. Add .06ra + .03n - 5.Op - .30m + «46n + 2.4p ----------- Sum = 8. 3x ■- [2 - (5x + 2)] = ... 7 9. (4r - St) + 3(t - t») = 10. Expand and collect terms: (2a2 - 7a - 9) . (5a - 1) = r ■jo 11. ab ^ I + I ) = a b / 12. Write with a single exponent: a • a2 • a^ = ........... 1 13. Solve for x: 3x - 5 = 8x + 10 </ f! . -jr = 14. Solve for W: P |w - W = 3 O . . W = ___________________________ (J 15. Solve for y: 2y _ 2y + 1 _ 1 3 5 ~ 3 16- Solve for T: T _ MT - G L — -—- .t = 17. Expand and collect terms: (n + l)(n2 + l)(n2 - 1) Go on to the next page* i! - 3 Reduce to lowest terms: q2 - 9 "■-*^ 3 3 q2 - 8q + 15 Reduce to lowest terms: c2 - 5c - 10 c2 - 5c (c - 2)^ * 3 3 c - 2 Perform by long division: (2x3 - 9x2 + llx - 3) 4.{2x - 3) Simplify: (x - y)(y - z)(z - x) (x - y )(x - z) , ^ (r~ Simplify: 6ab + 12b 2 3a2 + 6ab ____________________________ i/- Lf Express as a single fraction: 2m + m +1 1 m -2 3 3 ^ ^ ....... Express as a single fraction: 3 2xy _ 4y - x 4x2y^ \y/a. + x + Vx) If -y • (VaT + x - \/x) j___________________________ Va2 - x2 A/a2 - x 2 + — 7 ' Vsa2 - x2 a) Compute b) 8I - 5 Express by use of a fractional exponent:v a ------------------------ a, / — Change to simplest radical form: ^ , 3 * Solve for a (the value of b is not required): 3a + 7b « 7 5a + 3b = 29 . a = ---------------------------Solve for S: C - (1 - S2 ) - S2 . S = ____________________________ Solve for x: 3x + 7 = _ x + 6 x - 3 One root of the equation 2r2 + 3r = 2 is r - -2. Find the other ............. If t = -*04 A.B.(vy , and (A + B)r2 F * 144, f = 81, r = 2, A = B = 120, find the exact numerical value of t . . . t - Number right, Score 2. ^ r ' -> ^ Part II Directions: 34. 4 (Time: 20 minutes) Write the answers to the following questions in the spaces to the right of the page. Write formulas for each of the following statements, using the given letters: a) The volume V of a rectangular solid is the product of its length x, breadth y, and height ........ ............................................................. . b) The volume V of a sphere is foxir-thirds the product of tt and the cube of the radius ............................................................... c) The volume V of a right circular cone is one-third the product of tt, the altitude h, and the square of the radius r of the base.................. d) 35. The perimeter p of a regular polygon is the product of the number n of its sides by the length s of one side.................................. Construct a formula for the quantity desired In each of the following cases. Define by picture or words any letters you introduce for dimensions. a) The volume of a rectangular box with square ends. Volume =_ b) The product of the sum and difference of two numbers. Product =_ c) The cost of paving a street in the shape of the shaded part of Fig. 1. The street Is of uniform width and lies on two sides of a square block.............. Cost =_ yy I Pig, 1 36. If the radius of a. sphere is doubled, by what number is the volume multiplied? 37. If m books cost n dollars: a) What will one book c o s t ? ................... b) 38. What will x books cost? ................... If W and w are weights balanced at distances A, a from a fulcrum at F, the law of the lever can be expressed by the equation, WA = wa. A a) If w = 20, W = 40, a = 10, find A ........... w~| Ekl p b) Express W In terms of A, w, a. c) Fig. 2 39. - I f A = | a 3* find - ..................... w One number Is four times another and the sum of the numbers is 240. a) Write this statement in the form of an equation in one unknown. b) Write this statement as two simultaneous equations In two unknowns. c ) Find the numbers. 40. Fig. 3 d ................................................. The sum of three numbers is 150. The first number is twice the second and twenty more than the third. Find the third number................................ _ 41. In Fig. 3, BC is parallel to ED, AB = 7, BE = 5, £ ED = 6. Find B C ................................ 42. By definition a radian Is "an angle which, if Its vertex is placed at the center of a circle, intercepts an arc equal to the radius." How many radians are there In an angle at the center of a circle of radius 4 Inches which intercepts an arc of 12 inches? ................................................ 43. The base of a pyramid is the face of a cube. The altitude of the pyramid is equal to the edge of the cube. Find the volume of the cube If the sum of the volumes of the two solids is 8 cubic inches..................................... 44. Consider the operation: Multiply a number by 2 and subtract 4. Perform this operation on an unknown number x. Perform the operation on the answer. Repeat. Find a number such that the result of the third operation is 4. . .- Number right, Score COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS Pre-Test for First Year Students Experimental Form B Please print: -Name _________ Date Last Class________________________ First Middle A g e _____________ Date of Birth Yrs. Mos. -College or University_____________________________ C i t y ____________________________ Sex_________ M. or F. .Classification (check one): Engineering___ Liberal Arts____ Pre-Professional In the following spaces indicate the number of years that you have studied each of the following subjects (one semester = 1/2 year; one quarter — 1/3 year; one summer session = 1/4 year) Algebra Plane Ge ome try Solid Geometry Trigonometry In High School -General Directions: Do not turn this page until the examiner tellsyou to do so. This examina;ion consists of two parts and requires 45 minutes of working time. There is a time limit for sach part. You are not expected to answer all the questions in either part in the time limit, lowever, If you should finish Part I before the time limit is up, go on to Part II. No questions --lay be asked after the examination has begun. Use the blank spaces at the left of the page for mur calculations, and write your answers on the lines at the right of the page. Part Minutes I 25 II 20 Total Score 45 Copyright, 1935, by The Cooperative Test Service. Printed In U. S. A. All rights reserved. - 2 Part I (Time: 25 minutes) Directions: In all questions of this part, carry out the indicated operations and give the answers in simplest algebraic or arithmetical form and without parentheses * Do not reduce any square roots to decimal fractions. Write the answers in the spaces to the right of the page. 1. Divide 8 by — 7 2. Add | and | 3. Add 2 - /® . and 4. Multiply 7/3 _______________________ 5 + 4/3 by . . . . 4\ / 2 ........... 5. Find the value of 3 - 2(-5) + 4(-3).. ........... 6. Remove the parentheses and combine like terms: 5x - [Zx - y - (2x-y+3) - 6y] 7. Perform the Indicated operation: (3L - 7K) + (L + 2K) ......... 8. Write with a single exponent: 5 JZ 3 2 *2 - 2 ................. 9. Solve for x; 4 - 2x = x + 1 1 .......... 10. Solve for u* ■^u + u 11. — =2 u - 2 u = Solve for y (thevalue of x Is not required): 6x + 5y = 22 ............ y 4x - y = 6 _ = _ 12. Perform the indicated multiplication: (t3 - t2 )(t2 - t - 1) . . . . - 13. Divide x® - x^ + 2x - 2 by x - 1. 14. Simplify: (a - l)3 (a - 2) (a - 2)(a + 2)(a - l)2 15. Compute the value of 9x ~ ^y Wh en 4y + 3x x = 2, y = 3 ...................... 16. Add: 4a^ + 5ac^ — 3c® 8a2 - 7ac2 ______ 2ac2 + 5c3 17. Add: . Sum =- -Olx2 + .4 x - 1 3.00x2 - .02x + .03 Sum =_ 18. Simplify: (7 + 2)(y + 3)(-- i-— --- ^ — ) . \y + 2 y + 3' 19. Simplify: 1 1_____ /a + x - Vx “ Va + x + y/x Go on to the next page* - 3 Simplify: V(x+y)2 + 2(x-y)(x+y) + (x-y)2___________________________ Add: x 2x x + 4 x + 5 Reduce to lowest terms: 2S2 -o16S + 24 S^ - 36 ... Write in simplest radical form: 3^5 /aa ub Simplify: — x--+ Simplify: a) x ■ x5 ■ x 7 ( X 5 )2 Solve for x: b2 + (c - x)2 = a2 + x2 ..• x =Solve for x: •04x — #475 = #005 — #2x Solve for t: ... x ^ ~ 3 = 2s t + s t =. Find the positive root of the equation: 3m2 + m = 2 ............. Find the value o f : Number right, Score - 4- Part II Directions: 31. 32. (Time: 20 minutes) Write the answers to the following questions in the spaces to the right of the page, Write a formula for the following statement, using the given letters: The lateral surface S of a right circular cylinder is equal to the product of tt, its altitude h, and the diameter D of the base. . . . -------------------- - Construct a formula for: a) the entire area of the surface of a rectangular box, b) the quotient of the sum of two numbers divided by their difference. . c) the cost of a sidewalk around a square block; the walk Is of uniform width and there is no space between the walk and the curb........... 33. If a train runs M miles in 5 hours, how many miles will It run in K hours at the same rate of speed? ...................................... 34. If W and w are weights balanced at distances A, a from a fulcrum at P, the law of the lever can be expressed by the equation, WA = wa. a) If W = 40, w = 20, a = 10, find A. fwl~ b) Express a In terms of W, A, and w 7\ F c) If the ratio of A to a Is 4, find the ratio of W to w ............. Fig, 35. The sum of ‘two numbers is 25. Their difference is 17. Write this statement: a) as an equation in one unknown .................................. b) as two simultaneous equations c) Find the numbers............. 36. Let a, b, c be the sides of a triangle. The perimeter p is the If p = 100, a + b = ^c, sum of the lengths of thd sides. b + c = 3a, find a 37. In Fig. 2, the angle at B is a right angle and DE is parallel to B C . If AD = 8, DE = 6, and BC = 9, find DB. DB =. A Fig. 2 38. In the tetrahedron OABC (Fig. 3), OA = AB, OC = OB, angles OAB, AOC, and BOC are right angles. Find CB If OA = 4. CB =__ Fig. 3. 39. A parallelogram may be formed by drawing two equilateral triangles on a given base with their vertices on opposite sides of the base. What Is the perimeter, in inches, of the parallelogram If the length of the given base is 4 inches? ................................................. 40. Let y be a number obtained from x by the following operation: multiply x by 2 and subtract 4. Let F be the result of applying this operation to y. Find x so that when the operation is applied to F the result will be 1 2 ................................................................. 41. The geometric mean G of two numbers x and y Is given by the formula G = Vxy. If the geometric mean of two numbers is 6 and one of the numbers is 9, what is the other number? ............................ 42. Find an Isosceles triangle in which the altitude equals the base and each is numerically equal to the area................ Altitude = Number right, Score COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS Pre-Test for First Year Students Experimental Form C Please print: Na m e _______ Date Last Class________________________ First Middle A g e _____________ Date of Birth Yrs. Mos. College or University_____________________________ City Classification (check one): Engineering___ Liberal A rts____ Sex -________ M. or F . Pre-Professional In the following spaces indicate the number of years that you have studied each of the following subjects (one semester = 1/2 year; one quarter = 1/3 year; one summer session = 1/4 year) Algebra Plane Geometry Solid Geometry Trigonometry In High School General Directions: Do not turn this page until the examiner tells you to do so. This examina tion consists of two parts and requires 45 minutes of working time. There is a time limit for each part. You are not expected to answer all the questions in either part in the time limit. However, if you should finish Part I before the time limit Is up, go on to Part II. No questions may be asked after the examination has begun. Use the blank spaces at the left of the page for your calculations, and write your answers on the lines at the right of the page. Part Minutes I 25 II 20 Total Score 45 Copyright, 1935, by The Cooperative Test Service. Printed in U. S. A. All rights reserved. - 2 Part I (Time: 25 minutes) Directions; In all questions of this part, carry out the indicated operations and give the answers in simplest algebraic or arithmetical form and without parentheses. Do not reduce any square roots to decimal fractions. Write the answers in the spaces to the right of the page. 1 9 . 18 14 * 21 2. 1 5 + 3 _ = ............. ............................................ 4 3. 2\/3 - 3V2 ........... stun 4* 6V2 * 2^5 = ............... 5. (6 - 2) - 3(2 - 5) = . . . . 7. (6u + 5v) + (4u - 6v) = ♦ II l— I 10 i 0 02 1 1 ---- 1 1 o to 6. • . » . _2 • r „3 • r _4 = _ 8. r ............................. 9. Solve for x: 5x - 24 = 6 . , . 1 > !.■«■ 11■ II1 ...... ■' x 10. Solve for k: 2k - I k = 1 8 5 .................... k s= 11. Solve for n (the value of m is not required): 5m - 2n = 7 2m + n - 1 0 ........................ n 12. Simplify: (u + v)2 - (u - v)2 uv 13. If a = 16 and x = 9, find Va + x ................. - 14. If (a + b)2 = a2 + 2ab + b2 , find the corresponding expression for (M + 2N)2 . . I 3(1 - 1) * ............. n ' It o o + + w to 15. a) . ---------------------------- ll o| 'o 4 16. P - q2 - 4 6p + 5 + 3q2 q2 + 3 17. lg H - 2-| K 7 8» - 2 §« Difference Go on to the next page. (3x2 - 7x + 6)(2x - 1) = '6x2 + llxy - 10y2)-i- (2x + 5y)=. Solve for T: TL_j_Q = M s . . T = Solve for x: *15*-=.11 = (x - l)2. . X If x = y/2 and (6 - V^)x “ 3V2y + 2 = 0, find the value of y. x + 9 _ x x + 5 x . . . y +5 _ +8 y - * ( * - 1) x + y \/l215 = The expression a + b is a factor of a3 + b ^ . Find the corresponding factor of 8R3 + S3 ................... 1 - t _ t + 1 t - 1 V*64a2 + (\/8a)2 + \/2a • -\/8a = (xs - ys) (x® + y s ) = • • • Solve for x and y: — x + — y 3 - I = 12 = 1 y =Solve for x: 2x2 = 5x + 3. . . x —. Number right, Score - Part II Directions: Write the answers to the following questions In the spaces to the right of the page. 33. The perimeter of a polygon is the sum of the lengths of the sides. Write a formula for the perimeter of the given polygon In each of the following cases: a) An equilateral triangle of side ............... b) A square of side s c) A rectangle of base b and altitude a ........... Pig. d) A regular octagon of side s (Fig. 1) • * 34. If x blocks of metal weigh 4 pounds, what will z blocks weigh? 35. A circular cylinder and a circular cone have equal base radii and equal altitudes. The volume of the cylinder is how many times greater than the ..................................................... volume of the cone? 36. The sum of half a number and its double exceeds the number by twenty-one. a) Write an equation equivalent to this statement ................... b ) Find the number. 37. The sum of three numbers is 18. The sum of the second and third is twice the first number. The difference between the first and third numbers is one-tbird the first. Find the second number............................. C 38. In Fig. 2 the angle at B is a right angle and DE is parallel to B C . If AD = 8, DE = 6, and BC = 9, find A C ............................. AC A Fig. 2 A' 41. - (Time: 20 minutes) 32. Express the following statement as a formula, using the given letters. The energy E of a particle is equal to half the product of its mass m and the square of its velocity .................................. 40. 4 39. The triangle ABC is Inscribed In the circle whose center is 0. AB goes through 0. DC Is perpendicular to AB 0A = 5, BC = 6. (Fig. 3) Find DC. DC Fig. 3 The average of twonumbers, x and y, is k. Write an expression thatgives the value of y in terms of x and k .................................... y = If y = 3 - 2x, x = 3 + 2t, and t = 3 - 2u, express y in terms of u In Its simplest form......................................................... y =. 42. Find a number x such that the volume of the cube of edge x Is twice the area of the square of side x ........................................... .. . 43. If a positive number is added to ten times its reciprocal the result is three times the number diminished by eight times its reciprocal. Find the number................................................................... 44. If the perimeter of a right triangle Is 70 and one of the two shorter sides Is 20, find the length of the other................. Number right, Score — 74 with column four, the ntotal11 eoluam® Sine© th© samples for Forms B and G are all th© same sis©, this column In Tables 26 and 27 consists simply of th© indices derived by consider ing th© sum of th© three preparation groups as © single sample* This la not the case, however, in Table 25* though th© Al and Pg groups contained only 50 papers each, th© best estimate of their true difficulty Indices is as found* Hence they were weighted th© same as th© P^j, group which was based on 100 papers* Table 25 ITEM DIFFICULTIES FOR EACH PREPARATION GROUP Pn# P-i, and P« m m a, lose Seam £l fn £2 i 2 5a b e a 4 5 6 76 84 70 92 94 94 56 46 86 56 60 66 46 50 86 06 56 18 10 48 54 sa 92 88 98 96 95 88 74 90 68 70 92 64 55 95 80 75 45 2® 80 78 82 94 92 98 100 94 84 86 98 66 84 92 84 78 98 86 70 54 54 70 78 7 a 9 10 11 12 15 14 15 16 17 18 Total 82 90 83 96 97 94 74 69 91 03 71 83 65 50 93 77 60 38 24 61 62 Cooperative Test Be: 70 84 78 95 93 91 69 59 86 61 62 77 55 53 87 70 52 42 35 52 53 95 Table 25 (contirraed) FORM A, 1935 M m pl» l£ ?oJal Coopers Test £>©: 19 20 21 29 23 24 25 26 27a b 28 29 m si 52 SS 30 38 SO 16 1© 1© © © 1© 14 8 0 0 4 0 0 58 59 33 45 32 24 47 22 44 47 25 20 0 2© m 6 70 7© 52 ©8 44 42 7© 40 72 7© 48 30 4 30 2© 6 53 58 35 43 31 27 43 23 44 46 27 17 1 80 14 4 42 56 36 44 34 31 47 34 45 50 35 24 3 28 18 7 54a b e 6 35a b c 36 57s b 92 76 82 SO 98 83 78 91 9© 90 90 90 95 83 83 87 98 73 81 83 *# mm 38ab c 39a b 0 40 41 42 45 44 m mm 12 54 40 54 48 20 ** Wfe 22 ©5 50 78 72 29 *» 82 66 66 80 40 21 07 54 6© 07 30 «** 56 19 25 28 2 11 mm m m 80 40 2© 5© B 16 24 04 49 74 58 34 mm mm p# '#► 40 20 14 10 2 18 m«p 59 26 22 27 4 15 51 24 17 31 4 12 * This line indicates th© separation of Farts I and : Table 26 ITEM DIFFICULT IBS FOR EACH PRJ2PARATI0H GROUP P19 P.*, and P0 FORM B f 1955 ^ xir ^ H i 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 25a b 26 27 28 29 30 51 32a b e 35 34a b c MM1a T T ano xx# Cooperative M % Tot. Teat Service 74 82 SO 38 68 52 82 80 44 40 30 66 50 68 76 86 52 20 0 8 16 20 18 8 22 16 0 14 2 4 6 72 84 70 70 76 70 80 80 68 72 48 50 64 08 80 98 76 44 10 18 Oo 32 40 38 48 22 8 26 2 6 12 70 92 76 78 80 m 94 80 80 70 04 60 00 80 82 94 70 48 16 24 36 50 42 60 42 18 30 10 20 14 70 86 67 02 75 63 85 80 64 61 47 57 05 72 79 95 66 39 11 17 30 32 36 29 43 27 11 23 a 13 11 74 77 54 58 67 62 78 76 57 50 44 55 60 62 72 S3 64 43 26 33 37 42 41 39 49 33 26 32 13 26 17 66 88 98 84 75 an «** 4* mm «N* m *3SrT® mm m mm *# >#► mm W» 28 56 28 2 42 72 66 18 46 92 70 26 39 75 55 15 34 70 52 85 This line Indicates the separation of Parts I * yy, Table 26 (continued) FORM Mm. 35a b a fit *** 57 48 6 8 38 39 40 41 42 22 0 4 4 36 2 **■ 70 14 24 12 36 4 16 4 B9 1935 £2 m m> 68 18 22 14 60 4 IB 4 Total Cooperative Teat Service « 4m 4m 32 13 18 9 36 3 13 58 22 25 14 38 4 18 9 4 Table 87 ITEM DIFFICULTIES: FOB EACH FBEPAHATIGH CROUP Pf * FORM 0, 1935 Item F1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15a 64 88 30 42 62 64 84 86 92 58 6 18 66 70 78 88 74 64 68 52 64 32 b c 16 17 18 19 20 76 96 58 60 72 §6 94 94 100 78 36 26 82 94 100 90 92 80 76 64 74 48 74 94 70 84 6© 7© 90 9© 92 82 60 54 90 94 98 94 84 82 7© 86 7© 76 and P© Total Cooperative Test Service 71 93 53 ©2 ©7 ©6 89 92 ©5 71 34 83 79 86 92 81 83 75 75 67 71 52 75 84 50 65 ©5 58 82 90 95 71 43 39 71 7© 87 88 77 68 70 60 ©8 46 78 battle 27 (continued) POEM C, 1956 Item 21 22 25 24 26 26 27 28 29 30 31 32 33a b e a 34 35 56a b 37 38 39 40 41 42 43 44 Cooperative Test Service Pm-«a P ** fx| P r2 20 6 10 2 4 80 4 2 12 0 8 52 20 26 4 6 40 10 16 24 12 14 68 60 26 32 6 12 56 20 42 42 26 28 44 17 23 4 7 42 11 20 20 13 17 40 24 31 3 11 47 14 29 35 17 27 84 98 SB 82 98 44 52 @6 82 81 70 84 28 39 me 52 19 37 10 35 15 16 15 3 65 81 81 75 77 32 38 49 22 55 12 39 23 27 19 6 46 68 68 56 64 12 18 «# 40 14 14 2 20 2 8 4 0 86 88 72 92 88 43 ■m 52 18 40 10 38 16 12 10 2 84 24 58 18 4© 26 28 30 8 *otal This line indicates the separation of Parts I and XI* * Illustrative of th© differences between preparation, groups found In the preceding chapter are the following dis tributions of difficulty indices* It will b© observed that there are a great number of items which were very difficult for th© on© year groups* Difficulties are spread somewhat uniformly over th© entire rang© for th© one and one half year groups, while the Items proved to b© rather easy for the twoyear groups* More than half of th© Items in Form A disclose difficulties of 73 or more for the two-year group* Table m DISTRIBUTION OF DIFFICULTY INDICKS FOR EACH PREPARATION GROUP FORM A* 1935 Index 95-100 90-94 85-89 80—84 75-79 70-74 65-69 60—64 55-59 50—54 45-49 40-44 35-39 30-34 25-29 20-24 15—19 10-14 5-9 0— 4 p2 4 2 3 2 1 2 1 8 2 S 1 3 2 3 7 4 3 6 5 4 2 3 3 4 2 2 4 1 4 1 2 6 4 2 1 1 2 S 7 2 7 6 3 4 2 1 5 1 8 3 1 2 1 SO Table m DISTRIBUTION OF DIFFICULTY INDICES FOE EACH PREPARATION GROUP FORM B, 1935 .Siggat 95-100 90-94 85-89 80*84 75*79 70*74 05*09 60-64 55-59 50-54 45-49 40—44 35-59 50-54 85—89 80—84 15-19 10-14 0—4 *L h k 1 3 1 1 3 2 4 1 2 1 1 5 3 3 1 0 8 £a 1 i 4 1 4 3 6 3 1 1 6 2 4 2 3 2 3 3 1 1 2 4 3 2 3 2 2 3 1 1 2 2 4 3 2 Table 30 DISTRIBUTION OF DIFFICULTY INDICES FOR EACH PREPARATION GROUP FORM C, 1935 M m 95-100 90—94 85-89 80-84 75-79 70-74 65—69 60—64 55-59 50-54 45-49 40—44 85-39 80-84 25-29 20-24 15—19 10-14 5-9 0— 4 £l i 3 1 1 2 4 6 2 1 1 2 3 2 2 5 4 9 M 5 0 2 2 2 4 1 2 2 2 2 2 2 3 2 3 0 1 2 3 7 2 6 4 2 1 3 2 2 1 3 2 5 2 1 1 2 31 We shall return, now to Tables 25, 26, and 27, and compare the difficulty indices given for each of the prepara tion groups* Almost every item yields different indices for each of the three groups and the question which arises is, wWhich of these differences are too large to be explained by sampling theory?n In order to answer this question, we must have some measure of the error involved in each of the percentages given* I®# ^he standard error nuted from & stratified sample» & percentage as com** The standard error of a percentage, or a proportion, as computed from a random 1 sample is given by the formula where P is the proportion of correct responses, 0, « i-P (or Q $ 100* P If P is given in percentage rather than decimal form), and H is the number of elements comprising the sample* If a stratified sample is used, In which the elements selected from a given stratum are chosen at ran dom, the standard error is given by the formula where c"L? I s the variance of the percentage of correct responses for each stratum taken about the percentage P * 0* Udny Tula and M, 0, Kendall, An Introduction to the Theory of Statistics, p« 351* ©f the entire sample* The following derivation of the formula is, except for minor changes, the same as that given by Bowley (4)* I»et a sample contain ng, **.♦ , elements In t strata, and let the numbers which have a certain characteristic (number of successes on a given item) be Pl»X* Pgn£* •*« , ®tn t bet H * ln thea€? strata* + Eg * * • » + & £ Thua * # Pj Pln l * PSn 2 * *** * Pfcn t * (^) + % How ( ^ ) + »'*» + Pfc (-jp) ^ £§— • „ and hence^ t - 1 * 1, 2, ... , t g ZZ(w*-) ^m t-1 t s ijt H nipiql O) g If w a* ©x35! ♦ + **• + where the x *a are Independent variables and the c fa are arbitrary — ^ *w « « constants, then tfw = ^ a <r£ * J. F* Kenney, 11Some Topics In Mathematical Statistics,ft The American Mathemetic al Monthly* XLVI (February, 1939), Ho* 2, pp» 81, 62* 83 Tbaa H8 ffpz a »iPl<l - Pi> t t “ lPl * / “ lPi i n mi m i l t-l i=l -Jfe ^ / o Ki*pi *• p) _ *> hp t= l * where W ft *• N 0~ * P <T* i J Bi (Pf * *>S * t= I ^Pherefore or ^ ^ s ~~ r {8) Formula (8) demon®tratea that the error la less when a stratified aample Is used than It is when a random sample Is used (unless 0“"? *s 0)* However* Its form Is tr not as convenient for our purposes as la that of (X)* It Is to b© observed that (1) may b© written as *** & t *i«i (3) where t± Is the frequency of correct responses In stratum 1 * 34 Xf, now* P is in pereent&g© form rather than decimal form,(3) hecomes <rp* = -3gr / " n^ClOO Pj XIOO- 100 Pj) = , M t ® W tSl U l fi<i . U ) Heeall* now, that some of our samples contained 50 papers and the rest 100 papers# For the samples of 50, formula (4) becomes •*‘ s and for the samples of 4 ]UiL t °"p = ] T fiqi These are the two formulas we have used to calculate the standard errors of the various difficulty indices# These measures will be found in Tables 60 to 64 in the Appendix# 19# Comparison of the difficulty indices found for preparation groups# In order to compare the difficulty indices given for each of the preparation groups, it was necessary to compute the standard error of the difference « between two percentages# The formula used was <T(diff#} 2 3 yj <T2 + , where <Tt and <F£ are the X# F* Kenney, f,Som© Topics In Mathematical Statistics,*1 The American Mathema11oal Monthly* XLVI (February, 1939), So# 2, p # 62# standard errors of the percentages Involved# Using this formula, the standard error of the difference was calculated for each Item on each test and for all ways of pairing the preparation groups* Thus there are three such calculations for each Item, on© involving groups involving groups and P ^ , a second and Pg, and a third involving groups P*k and P«g# After the above calculations were made, each observed difference In percentage was divided by the cor** responding standard error of the difference# All quotients of 2*698 (the equivalent of 4 probable errors) or more were considered as significant# All of the differences In percentages, standard errors of these differences, and their corresponding ratios will be found in the Appendix In Tables 65 to 79# Those ratios which were large enough to be classified as ^signlfleant® are listed In the following three tables* 36 Table Si ITEMS OF FORM A, 1935, WHICH YIELD SIGNIFICANT DIFFERENCES BETWEEN PREPARATION GROUPS h m& Item pi& p2 Item 5a 4 5 pi «“a ps Item 5a 4 5 a 0 9 10 %0 11 11 14 16 16 14 16 16 17 18 10 W 10 20 22 22 26 26 27a b 28 20 51 52 26 27a b 28 58a b 300 40 20 81 22 25 24 26 20 27a b 08 00 51 50 37a « 38b 300 # This line indicates the separation of Farts I and 11# Table m ITEMS OF FORM B, 1935* WHICH YIELD SIGBXFICAKT DIFFERENCES BBTTOEK FREPiVHATXOH GROUPS P1 glj| Item ^1-j F2 Item gl Item 3 4 5 4 9 10 10 11 19 23 24 25a __________ 31 34b c 15 19 23 24 25a b 29__________29 31 34a 34a b c 39 » * This line indicates the separation of Farts I and 11# Table 33 ITEMS OP FORM C, 1935, WHICH YIELD SIGNIFICANT DIFFERENCES BETWEEN PREPARATION GROUPS Pi Hi Item Hi ps Item 3 4 11 11 12 14 15 a IB 20 21 28 28 30 and i Item 3 4 10 11 12 13 14 15a 18 20 21 22 23 2© 28 29 30 31 32 33a G d 33d 35 38 41 43 34 35 38 3© 40 41 42 43 * This line indicates th© separation of Parts I and XI* 20m Bff.ee t. of preparation upon toowled^e of the concepts and processes covered by the 1935 pro-teats# He- eall that each teat was written in two parts and that a time limit was placed on each part* The consequence of the restricted time element was that the number of omissions Increased very noticeably toward th© end of each part of each test* In an analysis of difficulty these later items should not b© considered * th© factor which determines success is probably speed rather than ability* It has thus been found necessary to eliminate from cons!deration Items 28*33 and 40*44 of Form A $ Items 20*30 and 36*42 of Form B t and items 27*31 and 30*44 of Form 0# The section which follows represents an effort to assemble the concepts and processes cowered by th© pro* tests In four overlapping groups* In one group are found the concepts and processes which were well*knowm by all groups of students* This Is a rather loos© term* but has been defined (arbitrarily) to mean that difficulty Indices for all three preparation groups were 70 or more* A second group consists of the concepts and processes which were found to be difficult for all three preparation groups and is defined so as to include only th© items for which no index exceeded 50* fourth groups In the third and are listed those areas which were materially strengthened by a study of third and fourth semester algebra respectively* In assembling these four groups* no attempt was made to differentiate between results secured for the separate examinations* All three forms were considered to gether and an effort was made to determine general con clusions* In other words* If all Items covering a certain concept or process In Form A were found to be easy, but similar Items In one of th© other forms were not, then it was not considered possible to general1b © ment was regarded as necessary* - complete agree In no case, however strong the evidence might appear, has a conclusion been drawn from a single observation* Using the procedure outlined above, th© following conclusions were derived from Tables 25, 26, 27, 31, 32, and 33* A* Concepts and processes which were well-known. 1* Elementary operations of arithmetic* By this expression Is meant addition, subtraction, multiplication, and division of positive Integers and fractions* Intro duction of parentheses and negative Inte gers Increased the difficulty, but only slightly* S* The principle of adding exponents when a series of terms with Identical bases are multiplied together* Items illustrating this principle were very elementary, consisting of no coef ficient® other than 1 and of only one base* It I® interesting to not© that th© Item S5 ,22,2« was more difficult than those con sisting of literal bases* 3* Column addition of algebraic binomials and trinomials with integral coefficients* There were three Items testing for this prooeaa and in one caae the Px index wee 64* All others were 80 or above* This particular exercise did not list similar terms in the same column, and Its signifi cance is Illustrated by th® fact that all three indices were lower than any other index in this group of exercises* 4# Removal of simple parentheses followed by addition of similar terms, when plus signs preceded each parenthesis* Insertion of a number before th© parenthe ses added to the difficulty for th© group, and removal of confound parentheses was considerably more difficult* Concepts and processes which were difficult for all groups* 1* Addition and subtraction of algebraic fractions* 92 2 * Operations with fractions containing ir rational denominators# 3 * Reduction of irrational terms to simpler form# This Is difficult for all types of radicands exhibited, whether they be alge braic or arithmetic# 4# Writing an equality of two products in th© form of a proportion and then manipulating the terms • This Is illustrated by Item 38(c) of Form A and item 34(c) of Form B* give the lever formula WA « wa th© ratio of A to a# Both and also The student Is re quired to find W/w# 0 * Areas materially strengthened by a study of third semester algebra* 1* Addition, subtraction, and multiplication of Irrational expressions* Th© agreement is not perfect but In eight cases out of nine significant differences appeared* For the ninth Item a significant difference was found between groups and Pg* 2# Solution of all but th© very simplest of th© equations Involving numerical fractions# 93 3* Solution of simultaneous equations. Strictly speaking, we have no right to list this process since only one observa tion is usable* However, there are two Items in the set we have agreed to disregard because of omissions, and all three yield significant differences*. Hence the concept Is Included# 4# Under®tending of positive fractional ex ponents « D# Areas materially strengthened by a study of fourth semester algebra# 1 # Understanding of positive fractional ex ponents * An examination of Tables 51, 32, and 33 will reveal the fact that the ^ did the groups scored significantly higher than groups on almost all of the items* Since this can be largely attributed to increased general knowledge and mathematical power, rather than to an Improvement In specific areas (other than those listed under G and D above) no detailed discussion of the processes Involved has been In cluded * 2 1 # Comparison of difficulties of related Items * Before concluding this section, mention should be mad© of certain groups of items pertaining to th© same general 94 process and which were constructed for the purpose of com parison* Bom© of the®© groups, with the accompanying dif ficulty Indices, are listed below* The results are as would be expected* Index Form A, item ©s Adds . Form A, item 7s Adds *0©m + *Q3n - 5*0p + «46n + 2*4p x * 4y + 1 Bx ♦ 4y « © 4x + 5y Form B„ item 1©* Add; Form B, item 17t Addt .01*® + «4x - 1 8,00*® - ,02x ♦ .03 Fona C, item Solve for x: 9s 4a® + 5ae® - So5 8a2 - 7ac2 2ae2 + So® 5x - 24 * © Form 0 $ item 1 0 s P1 Plf P2 8© 90 $6 68 86 98 94 52 76 70 92 100 92 58 82 98 Solve for ks 2k- Ik Z IS 72 Index P1 Form A , item13* Item14* Item15s 36 75 70 Solve for ys ®Z S Fora A f item 18 1 ♦ 1 a 1 5 3 Item 18* 16 45 54 10 28 34 34 73 78 30 59 70 Solve for T* L _ MT - Q 1’ Fora A 66 80 86 Solve for W; |w - W ; 8 Form A * P2 Solve for xs 3x « 5 Z 8x * 10 Form A , Pli Reduce to lowest terms* q2 - 9 q2 - 3q ♦ Fora A , Item 19s Reduce to lowest terms* e® - 3e - 10 ± o® - 6o (c- 2)® e— 2 96 Ha© writer has grouped another trio of items in order to compare methods of testing* Index f i l l i Ie Form A, item 57% Form B, item 55: Form C, item 54: If m books cost a dollars: la) what will on® book coat? 54 (b) what will x books cost? 46 65 82 50 60 If a train runs M miles in 5 hours, how many miles will It run in K h o w s at th® asm® rate of speed? If X blocks of metal weigh 4 pounds, what will Z blocks x weigh? 4® *B 44 Apparently an item is made easier by breaking it up Into parts* 2 2 * The predictIt® power of individual 1 terns» Correlation coefficients were given in Chapter XII to show th© relationship between the number of items solved cor* reetly on each 1935 form and first semester grades* NXA students found similar correlations between first semester grades and th© sum of correct solutions to items 2, 4, 5, 9, 15, 17, 20, 23, 29, and 52 of Form A*^ this comparison was The purpose of to determine on© of th© effects of using a short test, and th© papers used were those In the ^ These Items have no particular significance except that th©y were considered as typical of all Items* 97 samples* The correlations found were: for the P1 group, a coefficient of .305 with a probable error of *08| for the a coefficient of *379 with a probable error of .O0| for the Pg group, a coefficient of *554 with a probable error of *09j and for all groups combined, a coefficient of *390 with a probable error of *04* These ar© somewhat smaller than the coefficients found for the entire test* The writer carried the above plan to Its extreme and endeavored to find th© relationship between correct scores on Individual items, as given by the samples, and first semester grades* The purpose of this study was to determine how each Item contributed to the total correla tion and to discover whether or not some Item or Items provided a key to success* Sine© the samples were rather small for a study of this type, first semester grades were divided Into only two groups* Grades of A, B, and C were labeled Rsatis factory” and all of th© others as "unsatlafactory*n Faired with these grades in a two by two table were correct and Incorrect responses to each Item* With a four-fold table thus constructed for each item of each form, w© were In a position to determine the relationship between th© two variables (that is, between success or failure on the item and satisfactory or unsatis factory first semester grades) by calculating tetr&ehoric correlation coefficients#5 These coefficients were determined by reading directly from a set of charts pre pared by Cheslre, Safflr, and Thurston©,5 and are listea In th© following tables* at Th© validity of this procedure rests upon th© assumption that th© two variables are essentially con tinuous and would b© found to be normally distributed If It war© possible to classify them more exactly Into finer groupings# Henry B* Garrett, Statistics ip PsyeholoCT lyjfl. » P* 571* ^ Leon© Cheslre, Milton Baffin, and M* L# Thurston©, Oomoutinse h^egramg for the Tetrachoric Correlation Coefficient* 99 TahlQ 34 CORRELATIONS (TETRACHORXC) BETWE3B CORRECT SOLUTIONS OF INDIVIDUAL ITEMS IN FORM A* 1935* AND SATISFACTORY <A,B.C) FIRST SEMESTER GRADES FOR EACH PREPARATION GROUP Item % 1 2 *45 3a h e d 4 5 © 7 8 9 10 3LiJL 12 13 14 15 1© 17 18 19 80 81 22 83 24 85 86 87a hb 28 29 50 31 32 33 34a b and XX* -*11 *34 *08 -*15 *29 *17 *14 -*02 *17 #41 *4© *27 *26 a* *21 *52 ps ♦46 *80 *55 .S3 *44 *25 ♦06 •40 #26 -*11 *22 *02 *37 #37 *12 #27 *54 ,31 •30 *31 — #34 •S3 .31 -.11 .31 .57 «* ,40 *47 ,49 *38 *11 ,02 <37 *32 *47 ,53 *31 *2 © *5© #53 ,33 *14 *14 *33 *08 - *42 * *» m w .58 *40 *31 *26 — #18 -*02 -.27 .43 *40 *65 .10 ,15 ,19 #17 ,51 *55 #3© *51 *58 #5© *33 •41 ♦22 ,20 — .09 #47 *57 ,33 #54 ♦15 .25 *37 *64 *24 *37 *50 m •nr *66 *40 * This line Indicates the separation of Parts I ICO Table 34 (continued) P o m A, 1935 P JL 34e a 35a b c 56 57a b 58a b © 59a b o 40 41 42 m 44 *JL #55 -*07 .37 .60 * ■m *** *44 *25 *3© *36 *52 *44 *70 *43 *29 *45 *74 .IB ** - *93 *26 *02 *11 *80 #11 .31 *56 **# mm *3® -.06 *13 *15 ** * .09 *41 .37 *09 — .04 *35 mm .34 -.41 .03 .87 Mft ■iw Table 35 CORRELATIONS (TETRACHORIC} BETvVESH CORRECT SOLUTIONS OP INDIVIDUAL ITEMS IN FORM B, 1935, AND SATISFACTORY (A,3,C) FIRST SEMESTER OEADES FOR EACH PREPARATION GROUP IJffla 1 2 3 4 5 6 7 8 9 10 11 12 13 h .50 *00 *38 *22 .28 *31 *50 .10 *48 *46 -.37 .04 *04 M *31 *46 #60 .09 -.03 — .01 *48 .21 *09 *44 .11 *38 .10 fk -.06 .23 -.17 .13 .36 m4^* ,24 -*17 -.07 .00 .19 .24 Table 35 (continued) Form B $ 1935 Item 11 £li fk 14 15 10 17 IS 19 80 81 88 23 24 25a b 20 27 28 29 50 51 32a b 0 33 34a b e 56a b c 36 57 38 39 40 41 42 .05 -.18 #44 *39 *09 «* *14 -.12 .21 .34 — .04 w *41 •ia «* *20 *26 *14 *45 «* *45 — *08 .50 *36 *31 *36 .02 — *04 — *09 -.13 *20 -.07 +m •m mm .13 .41 .10 .14 .61 *■* mm .18 — *10 — *08 *33 .24 *39 .09 *24 - *18 *24 *08 -*11 .46 .32 *57 <*• mm ** mm ■P» mm *54 *55 *70 - mr mm *21 -*43 ** *16 *» *» *13 *09 *24 *36 - - -.06 .21 nm * *22 *22 .15 .14 .03 m* -.13 - *13 .05 *47 .33 *24 .06 1.00 .37 1.00 # This line Indicate© the ©eparation of Fart© I Table 36 CORRELATIONS (TETRAOHORIC) BETWEEN CORRECT SOLUTIONS OF INDIVIDUAL ITEMS IK FORM C, 1935, AND SATISFACTORY {A*B,C) FIRST SEMESTER GRADES FOB EACH PREPARATION GROUP 21m i 2 3 4 S 6 7 3 9 10 11 12 13 14 23a b 0 16 1? IB 19 20 91 22 23 24 25 26 27 28 29 30 31 32 35a b c a -.13 -.25 .21 **#10 *32 *40 *42 .11 *35 .32 *M .67 .4? *21 .58 *25 .53 .40 *28 *12 *63 *54 *72 *►#21 *44 4** m *21 a* ** a* • *31 .54 #42 *25 *73 fit g **.51 *14 - .03 *46 .17 *►.29 - .11 *17 .35 .25 — .36 .51 •m *53 **.24 .32 *15 *56 *23 *27 .49 *62 .23 -** -.21 **#26 -*#13 *17 .14 #22 *04 *24 •45 #02 #44 *47 *10 *38 *20 *13 •m ~*1I *41 *71 *49 #40 # 44 <w .44 .10 *23 -.15 .60 .37 #03 .23 *60 *51 -.02 *• #28 #04 *10 #39 .39 #25 *31 *44 .72 .41 «•» & This line Indicates the separation of Parts I and II* ±32 Table 36 (continued) Form C, 1935 Item 34 35 56a m m b 39 40 41 42 43 fi *52 .10 *36 *50 *11 -*11 *53 ** *35 A./L TCT fli f® *29 -.17 *03 .57 .57 .28 *32 *59 .15 - *21 *35 *40 .30 *42 ♦03 - .45 *45 — *08 •so *43 *13 ** Blank spaces In the tables are of two kinds* The first Is due to the subjective items for which no data was secured* The second arose in the following manners due to the small number of students involved, there were many times very small numbers or zeros In some of th© compartmerta* When this occurred, the coefficient could not, in most cases, be read from the charts* On two occasions, it was read with correct but misleading results* Th© two cases are th© ones Involving Items 40 and 42 of Form B with the group Pg, and the coefficients were read a® 1*0* Th© cor rectness lies In th© fact that every student who solved the problem correctly earned a satisfactory grade % the mislead ing factor is that th©r© w©re only two such students* Probably th© most striking fact about these tables Is that Items which yield unreasonably high coefficient for 104 on© preparation group seldom do for the others* A very good example of this is part b of Item 38 in Form A* Th© correlation coefficient for the one and one half year group is *74, for the one year group It is *32, and for the two year group it is -*04* Thor© are many Items which have rather high coef ficients for one or two groups, but very few have for all groups# However, items 22 and 34b In Fora A and item 33b in Form 0 are very satisfactory in this respect* It Is to be observed that th© peculiarity noted above does not necessarily mean that the Items are at fault* It certainly requires more mathematical ability for a stu dent with only one year of preparation In algebra to master, say quadratic equations, than It does for one with more preparation in the subject# In addition, success as measured by grades is rendered somewhat Ineomparable by registration In courses with different content* This ap parent discrepancy may mean, then, that concepts whose mastery is indicative of success in first year college math ematics are not the same at all for students with different degrees of preparation* The implication is that separate pre-tests for each of the preparation groups are desirable* 135 GHAFTEK VI AHALTSES OF THE ITEMS OF THE 1936 PRE-TESTS ^5# 1933 items, which war© repeated in the 1936 are-tests# Recall that the purpose in constructing three experimental test forma in 1935 was to form a sufficient number of items that, on the basis of Information secured concerning them, enough satisfactory exercises would evolve that two permanent tests coitld be assembled. The 1935 Items which were repeated In the 1936 tests are given in the following tables* iXi® to gaps in difficulty which were found In the distribution of experi mental Items, it was necessary, In constructing the 1936 forms, to Interpolate a few new items of unknown validity and difficulty and hope that they would meet the require ments# All of these Items proved to possess satisfactory validity but very few of them yielded difficulties com?** measurete with, the location of the Item In the test# Item 50 of Form A was particularly Ill-placed, since, unfortu nately, it can b© solved by Inspection# ■f v(I COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS Pre-Test for First Year Students Experimental Form A - 1936 Please print; Name________________________________________________________________________ Date Last First Middle Class_______________________ A g e ___________________ Date of Birth Yrs. Mos. College or University________________________________ Classification (check one): Liberal Arts________ City_______________________ Sex__________ M . or F . Engineering_________ Pre-Professional In the following spaces indicate the number of years that you have studied each of the following subjects (one semester = l/2 year; one quarter = l/3 year; one summer session = l/4 year): Solid Ge ometry Plane Geometry Algebra Trigonometry In High School General Directions: Do not turn this page until the examiner tells you to do so. This examina tion requires 45 minutes of working time. You are not expected to answer all the questions In the time limit. No questions may be asked after the examination has begun. Use the blank spaces at the left of the page for your calculations, and write your answers on the lines at the right of the page. Score Percentile Right Wrong Omitted Copyright, 1936, by The Cooperative Test Service. Printed In U. S. A. All rights reserved. -2 Directions: In all questions, carry out the Indicated operations and give the answers in simplest algebraic or arithmetical form and without parentheses. Do not reduce any square roots to decimal fractions. Write the answers in the spaces to the right of the page. 1. Multiply: 0*3 2. Simplify; 4 7. 4 = . . . . 3. Simplify: r2 • r5 • r5 = 4. Add the fractions: — + i 3 4 5. Write a formula for the following statement, using the given letters: The volume V of a right circular cone is one-third the product of ir, the altitude b, and the square of the radius r of the base. . ___ 6. Find the value of: .............. 3 (_6) - 4(-5) 7. Remove parentheses and collect like terms: (3L - 7K) + (L + 2K) « . . . . 8. If (a + b)2 = a2 + 2ab + b2 , find the corresponding expression for (M + 2N) . . . 9. The perimeter of a polygon Is the sum of the lengths of the sides. Write a formula for the perimeter p of a rectangle of base b and altitude a. . . 10. If y = j , find the value of k when y = 34 and x = 2. k = 11. Divide 8 by 24 .............. 7 12. Compute the value of ~ 4y + 3x when x = 2, y = 3. . . . . . . 13. If a = 16 and x = 9, find the exact numerical value of ya + x ........................ 14. Solve for x: 3x - 5 = 8x + 10 x = 15. If W and w are weights balanced at distances A and a from a fulcrum, the law of the lever can be expressed by WA = wa.*If W = 40, w ss 20, a = 10, find A, A =_ 16. Add: 3 - 5y£T 17. Express the following statement as a formula, using the given letters: The energy E of a particle is equal to half the product of its mass m and the square of its velocity v. Go on to the next page. - 3 18. Write £ 5 in decimal form. • . .------------------ ja_v> 19. Multiply: (3x2 - 7x + 6 ) (2x - 1) . •----------------------- 20. Divide x3 - x2 + 2x - 2 by x - 1 .......................... ....................... 21. Multiply 7y5" by 4V§~- • 22. Simplify: 5c - (2c - 3 ) J ........ ... 23. What is the interest on $600 for three months at the rate of 5% per annum? Interest = $ 24. The area of a triangle is 24 square units and the base is 8 units. Find the altitude of the triangle. Altitude = ___ 25. In an arithmetic progression the difference between any two adjacent terms Is constant. Find the fourth and fifth terms of the arithmetic progression whose first three terms are -3, 1, 5. Fourth term = ___ Fifth term = __ 26. If the length of the side of a square Is s, how long is the d i a g o n a l ? ................. . — 27. Simplify: ab(I + . — 28. One number Is four times another and the sum of the numbers is 240. Find the numbers. . . .__ 29. Simplify: v $ (x°)^ = . 30. Simplify: (Va + x + Vx) (Va + x - Vx) - 31. Solve for T: TP t 9- = O m . T =_ 2 32. Compute the value of 8 . . ._ 33. Solve forn (the value of not required): 5m - 2n = 7 2m + n = 10 34. Simplify: — S— “ 16s + S2 - 36 35. Solve for y: 3 m is n =_ 24 . gy. + 1 = i 5 3 y = ------------------Go on to the next page. - 36. Multiply: (x& - y^)(x^ + y^-) 38. If x blocks of metal weigh y pounds, what will z blocks weigh? .......... 39. Simplify: 2x2 » 5x + 3. X s, 41. If y = 3 - 2x, x = 3 + 2t, and t = 3 - 2u, express y In terms of u In its simplest form. y = 42. The sum of three numbers Is 18. The sum of the second and third Is twice the first number. The difference between the first and third numbers is one-third the first. Find the second number........... .......... 43. The geometric mean G of two numbers x and y is given by the formula G = y/xy. If the geometric mean of two numbers is 6 and one of the numbers is 9, what Is the other number? ............................. 44. In the figure, BC Is parallel to ED, AB = 7, BE = 5, ED = 6. Find B C . BC = A 45. Consider the operation: Multiply a number by 2 and subtract from 4. Perform this operation on an unknown number x. Perform the operation on the answer. Find a number such that the result of the final operation is 4. 46. Solve for t: « = 2s. t + s t = 47. Simplify the following radical: V^2l5 ............................ . 48. If the perimeter of a right triangle is 70, and one of the two shorter sides is 20, find the length of the other................................ 49. The base of a pyramid is the face of a cube. The altitude of the pyramid is equal to the edge of the cube. Find the volume of the cube if the sum of the volumes of the two solids is 8 cubic inches. (The volume of a pyramid equals one-third the area of the base multiplied by the altitude.)_ 50. Two men are 42 miles apart and walk toward each other at the rate of 3 and 4 miles an hour respectively. After how long a time do they meet? A33274-6 - .---------------- 37. Find the value of:-------------------- 40. Solve for x: 4 Number right, Score —-------------- — ■i\ ■V.Q _jLo O' COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS Pre-Test for First Year Students Experimental Form B - 1936 Please print: Name------------------------------------ !____________________________________Date Last First Middle Class_______________________ A g e ___________________ Date of Birth Yrs. Mos. College or University________________________________ Classification (check one): Liberal Arts________ City_______________________ Sex__________ M. or F. Engineering Pre-Professional In the following spaces indicate the number of years that you have studied each of the following subjects (one. semester = l/2 year; one quarter = l/3 year; one summer session = l/4 year): Solid Geometry Plane Ge ome try Algebra Trigonometry In High School General Directions: Do not turn this- page until the examiner tells you to do so. This examina tion requires 45 minutes of working time. You are not expected to answer all the questions in the time limit. No questions may be asked after the examination has begun. Use the blank spaces at the left of the page for your calculations, and write your answers on the lines at the right of the page. Score Percentile Right Wrong Omitted Copyright, 1936, by The Cooperative Test Service. Printed In U. S. A. All rights reserved. - 2 Directions: In all questions, carry out the indicated operations and give the answers in simplest algebraic or arithmetical form and without parentheses. Do not reduce any square roots to decimal fractions. Write the answers in the spaces t_o the right of the page. 1. Multiply:5 * 0 .................... 2. Find the value of:— ............ ............... 3. Find the value of: 4. Simplify: 5. Add: _ _ 81 * ^ ....... o a • a^ • a^ . i + £ 5 4 . . .. 6. The perimeter of a polygon is the sum of the lengths of the sides. Write a formula for the perimeter p of a square of side s .......... 7. Remove parentheses and collect like terms: (4r - 5t) + 3(t - r)._ 8. Write a formula for the following statement, using the given letters: The lateral surface S of a right circular cylinder is equal to the product of it, its altitude h, and the diameter d of the base........ 9. Divide: — -r — 14 21 10. The altitude of a triangle is 2 ft. less than the base b. Express the area of the triangle in terms of b. (The area of a triangle is one-half the product of the base by the altitude.) Area = — 11. If W and w are weights balanced at distances A and a from a fulcrum, the law of the lever can be ex pressed by the equation WA - wa. If w = 20, W = 40, A = 10, find a. a = 12. Write the formula: The volume v of a sphere is four-thirds the product of it and the cube of the radius r. __ 13. Solve for k: 2k - o = 18. k = 14. Find the value of: 3 - 2(-5) + 4(-3) 15. Multiply 6y/2 • 2\/5. 16. An electric bill varies directly as the number of kilowatt hours consumed, If 45 kilowatt hours cost $3, what will 60 kilowatt hours cost? $_ 17. If a = 4 and b = 3, find the exact numerical value of 5 y/e^ + b® . . . __ 18. The sides of the smaller triangle in the drawing are 3, 4, and 5. In the larger similar triangle the shortest side is 5. What are the other sides? (In two similar triangles any two of the sides of one are proportional to the two corresponding sides of the second.) b =___ Go on to the next page. - 3 19 * Divide: (6x2 + llxy - 10y2 ) 20. Simplify: (2x + 5y) . 3x - [2 - (5x + 2 ) ] __________________ . 21. The sum of two numbers is 25. Their difference is 17. Find the numbers............................. 22. Multiply and collect terms: (2a2 - 7a - 9) • (5a - 1 ) .......... 23. Which of the following statements are true? (Answer by number.) (1) 5 + _L_ = 5 + x, (x + 0)j X 1 (2) 5 (3) 4 3 • 2 3 = 4; (4) (x - l)2 = (1 - x) 2 24. Write a formula equivalent to the following statement: A Fahrenheit thermometer reading, F, is 32 de grees greater than nine-fifths the corresponding Centigrade thermometer reading, C ......... . 25. Add: 5 + 4V§ 2 - V3 26. How much money must be placed at 3% simple interest for one year to earn $12? ................... 27. Add: .Olx2 + ,4x - 1 3.00x2 - ,02x + .03 28. Perform the indicated multiplication: (t3 - t2 )(t2 - t - 1). 29. Simplify: Sum =. . --- 3— zJt--q2 _ 8q + 15 30. Express by use of a fractional exponent: 31. a3 ............... The expression a + b is a factor of a3 + b 3 . Find the corresponding factor of QR3 + S3 . 32. If lemons cost A cents a dozen, how many dozen can be bought for D dollars? ........................ 33. Solve for y (the value of x Is not required): 6x + 5y = 22 4x - y = 6 y = 34. The average of two numbers, x and y, Is k. Write an expression that gives the value of y in terms of x and k . y = 35. Simplify: (x +- b)2 ' \ - x2 . b 36. Solve for T: L = ~ G . T T = Go on to the next page. - 37. If a train runs M miles in 5 hours, how many miles will it run in K hours at the same rate of speed? 38. Simplify: 7a2 - x2 (v42 - x2 + 39. Simplify: | i | .- - ......... 40. Find the positive root of the equation: 3m2 + m = 2. m = 41. If x * and (6 y + 2 =* 0, find the value of y. y =. 42. If the radius of a sphere is doubled, by what number Is the volume multiplied? .............. 43. Let a, b, c be the sides of a triangle. The perimeter p is the sum of the lengths of the sides. If p = 100, a + b =-|c, b + c = 3a, find a. a =- 44. If a positive number is added to ten times Its reciprocal the result is three times the number diminished by eight times Its reciprocal. Find the number. - 45. In the tetrahedron OABC, 0A = AB, 0C = OB, angles OAB, AOC, and BOG are right angles. Find CB If OA = 4. CB =. B 46, If £ = § and - = 1 , find 47, if* fc = *04 AB(y!P ~ y/t) and (A + B) r2 F = 144, f = 81, r = 2, A = B = 120, find the exact numerical value of t. t = 48. Find an isosceles triangle in which the altitude equals the base and each is numerically equal to the area Altitude = _ 49. Let y be a number obtained from x by the following operation: Multiply x by 2 and subtract 4. Let F be the result of applying this operation to y. Find x so that when the operation is applied to F the result will be 12............................. x = 50. ( Z _ J u n f . 2*-_ - i) \x + y / \ x - y / B33274-6 = , Number right, Score 4 - 110 Table 37 ITEMS OF FORM A, 1936, WHICH WERE TAKES FROM THE 1935 TESTS U m i5 a64 a97 ii 12 ia 14 la 16 t e l Form B 5c 8 2 34o 5a 7 14 33c 1 15 15 13 58a 4 34a 32 18 17 19 20 21 13 4 22 27 28 29 30 31 32 33 34 35 36 37 39 40 41 42 43 44 45 4® 47 48 49 Form 6 11 300 25a 25 20 27a 11 22 15 29 25b 84 51 41 37 41 41 44 28 25 44 43 Table 38 ITEMS OP FORM B, 1936, WHICH WERE TAKEN PROM THE 1935 TESTS Item 1 9 4 5 Form A 1035 Item Pom B Form C So 3d 12 6 7 8 9 11 12 13 14 15 17 19 9 31 1 38a 34b 34a 10 S 4 13 19 8 21 22 35c 10 23 27 28 3 17 12 29 ia 30 31 33 34 55 36 37 38 39 40 41 42 43 44 45 47 48 49 50 27b 26 11 40 24 16 33 26 23 29 22 36 36 43 38 33 42 40 24 112 24* Cooperative Tea t Service Analyses* The dif ficulties and valid!ties of the 1956 items, as ©omputed by the Cooperative Test Service, have been placed in the Appendix* The following table gives the distributions of these indices* The distributions of difficulty indices corraborate the statement mad© in Chapter IT that the 1956 tests were somewhat easier than the 1956 forms* It is Important to repeat at this time that it was the intention of the com mittee that this phenomenon should be true, and it Is interesting to not© that it was accomplished In spite of the fact that two entirely different populations were tested* The significance of this fact will b© revealed in Chapter VIZ* 113 Table 39 DISTRIBtTTIOHS OP DIPPICULTY AND VALIDITY IHDICES POBKD FOR THE 1936 TESTS difficulty IMS* 95-100 90-94 85-8© 80-84 75-70 70-74 65-69 60-64 55-59 50-54 45-49 40-44 35-39 30-34 25-29 20-24 15-19 10-14 5—9 0-4 £ orm A Form B 1 9 1 1 3 S 1 5 5 3 4 3 0 1 3 2 3 2 7 1 S 7 6 5 S 1 4 5 0 1 3 2 3 2 2 1 Validity Index 14 11 10 9 B 7 6 6 4 5 1 Form A 1 2 4 5 4 11 10 3 6 5 1 Pom B 2 6 7 9 10 B 3 4 1 114 Observe that one item in each form falls to meet the requirement of 2 set up for satisfactory validity* These are item 1 of Form A and item 2 of Form B* Bach was given in 193d and the indices reported then were 6 and 4 respectively* That we need not be too concerned with a discrepancy of this type is pointed out by B* F* Lindquist {17} from whom we quotes #**An index of discrimination* therefore should be used with particular caution for evaluating items In a general achievement test where the criterion em ployed is the total score on the test Itself* It may be of considerable value* in such situations* in identifying Items that contain structural or technical imperfection®* but It is a dangerous basis upon which to eliminate an Item if the content of that item 1® acceptable In term© of other logical considerations * Regardless of the criterion employed* the indices of discrimination computed for the Items of a test on the basis of the performance of a given group of pupils will determine the relative effectiveness of the Items only for that &roun of pupils or for other simi lar groups * "The same item In a test may show a low index of discrimination for one group of pupils and a high index for another* depending upon the nature of their Instructional background* ** 25* Item discrimination within preparation groups* The above quotation suggest© the possibility that the power of an Item to discriminate between students grouped accord ing to ability should be listed for each preparation group* For this purpose* Table® 40 and 41 have been assembled* The manner In which they were constructed Is as follows. X and for all samples com Fox* each 1956 aample * bined* th© papers were arranged In descending order accord ing to total score* Bach such distribution was then divided into approximate thirds (approximate In order that no two papers with the same score would fall In different groups)* and the percentage of correct responses to an item for each of these thirds was computed* The three groups wax'© labeled tthlgh*” ^medium**1 and nlowH respectively* and the corresponding percentages computed for each of these groups are listed In the tables under the letters H ## M mt and L t H represents th© number of papers in each of the groups* For an Item to discriminate between different levels of achievement* then* the percentages should deereas© from column H to column M to column L - the more abrupt th© decrease* the higher the discriminating power of the Item* ^ Th© 1935 samples were considered too small to subject to this analysis* 116 Table 40 PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN EACH APPROXI MATE THIRD OF THE PREPARATION GROUP AND TOTAL DISTRIBUTIONS FORM A f 1936 N Z M m i 2 3 4 5 0 7 S 9 10 11 10 13 14 15 16 17 18 19 SO SI 00 S3 24 85 86 87 08 09 30 81 38 33 34 35 36 57 38 H. P 2i M* L# 34 32 34 97 91 100 100 79 88 94 100 74 91 88 85 79 08 88 74 94 97 74 02 76 @5 97 60 85 21 79 79 91 56 50 63 50 59 47 50 50 38 94 63 94 91 75 78 91 81 56 63 63 78 78 73 84 50 69 97 81 70 08 36 81 34 44 0 41 59 60 9 9 28 22 22 9 28 00 16 74 15 94 SB 29 SB 71 62 20 59 59 50 41 32 47 09 24 80 55 47 15 30 50 18 06 3 15 21 26 0 0 3 9 15 9 30 6 5 H. h k M* L« H. li M* D# 34 34 32 33 39 97 91 97 79 94 97 94 04 100 91 100 91 94 100 94 88 91 68 91 94 88 85 94 97 80 85 91 85 85 88 97 79 91 100 100 100 SB 82 91 79 88 47 68 68 91 91 97 68 88 85 56 26 85 71 100 91 97 94 91 62 94 60 91 71 79 32 82 56 68 35 94 08 59 44 56 15 91 59 94 94 69 81 72 69 31 69 66 50 47 59 59 34 53 97 59 53 44 38 81 44 66 0 22 69 53 41 22 63 25 31 3 08 25 3 97 97 94 82 100 95 100 100 97 87 97 100 100 95 94 95 85 90 100 70 91 85 85 95 85 92 100 07 01 85 97 97 94 85 97 100 @8 92 91 09 94 79 85 72 97 85 100 79 94 85 79 36 9 4 92 97 90 97 95 100 79 94 69 100 97 67 74 67 82 79 59 94 77 85 04 67 46 Total H* M* L# 28 92 117 01 89 68 93 93 79 75 89 82 75 61 79 82 80 79 79 80 71 93 61 68 57 48 64 57 75 29 54 79 04 46 36 82 43 21 32 04 43 14 97 93 98 98 95 99 97 95 08 93 90 91 87 97 89 96 93 99 89 87 88 75 95 93 92 61 91 98 96 93 88 99 75 72 74 92 70 59 87 40 92 90 53 71 76 68 41 62 62 60 57 52 63 37 44 90 60 58 27 42 67 32 40 4 25 45 44 10 4 26 15 21 7 23 15 7 02 81 97 97 80 89 95 91 70 81 82 88 81 86 81 81 87 98 77 07 02 01 85 62 82 23 88 84 87 59 54 74 45 48 30 63 43 24 117 Table 40 {c©n tinu®&) Form A $ 1936 fx H. 39 40 41 4® 43 44 4$ 46 47 43 49 50 26 47 21 9 53 9 6 © 3 0 3 29 L, ' 3 3 0 0 0 0 0 0 9 3 0 0 0 0 0 0 0 0 0 0 0 0 9 12 fii H. M» L« 5© 9 9 ©5 18 19 50 15 16 2© 21 © 68 56 34 47 26 © 26 0 3 21 © 0 © 24 12 18 0 0 21 3 3 ©5 18 22 Total !e , m# 79 94 82 55 94 70 39 42 58 39 30 52 3© 67 49 10 77 46 21 13 18 13 5 3© II. L# 21 39 18 7 43 14 7 0 11 0 0 25 63 76 ©1 32 86 55 29 29 37 25 22 51 17 4 38 3 23 2 14 0 50 12 18 3 7 0 5 0 a 1 i 0 i 1 22 14 Table 41 P1BGBWTAGB OF CORRECT RESPONSES- TO EACH ITEM IN EACH APPKOXIMATE THIRD OF THE PREPARATION GKO HP AMD TOTAL DISTRIBUTIONS FORM B* 1956 »* i 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1© H H. K« L# 28 44 2© 96 96 96 89 96 ©8 8© 93 71 ©1 8© 96 75 86 64 89 95 09 89 82 93 45 70 70 75 20 68 55 57 ©8 23 91 82 82 93 71 71 14 4© 39 71 4 32 32 32 54 10 71 M M, L. H* fs M4 28 40 32 31 41 9© 82 100 100 9© 89 S9 93 89 71 82 100 89 89 ©8 9© 95 90 95 90 90 85 80 98 S3 45 75 83 80 83 68 90 88 88 97 6© 75 44 ©9 75 47 13 59 53 59 ©9 38 81 Total L. H. M. 1* 28 94 113 93 9© 88 96 89 93 73 82 90 79 41 BO 81 75 82 56 93 89 85 92 72 80 34 ©3 63 ©1 12 54 4© 51 62 26 82 100 100 9© 94 90 82 100 98 9© 100 95 82 97 95 89 97 88 64 100 85 93 97 95 89 90 90 ©8 84 59 52 97 90 82 94 95 ©8 90 93 79 100 93 79 100 73 54 97 95 100 99 91 98 97 96 91 93 96 89 7© 90 97 93 95 82 96 118 Table 41 (continued) Fona B. 1936 Item H, 17 IB 19 90 21 22 93 24 25 26 27 28 29 30 81 32 m 34 55 86 37 38 39 40 41 42 45 44 45 46 47 48 49 50 86 57 86 ©6 79 75 18 64 71 89 71 64 75 71 71 21 46 43 57 25 50 14 11 25 14 14 11 7 7 29 11 7 0 0 fi M* 64 39 86 64 82 73 7 36 43 SO 52 64 52 36 55 9 25 7 16 2 16 0 5 2 2 0 0 2 7 25 2 0 0 0 29 7 59 36 57 39 0 7 25 71 25 21 32 18 7 0 7 0 0 0 11 0 0 0 0 0 0 0 4 14 0 0 0 0 03 93 89 75 96 86 36 82 06 100 82 89 m 82 79 32 79 71 86 34 75 57 32 57 18 36 11 21 14 29 7 11 4 11 IU l*w 86 68 83 78 85 75 8 58 88 7© 68 83 60 73 48 5 33 13 23 8 28 8 47 44 75 44 83 38 © 16 50 ©6 54 47 19 56 53 0 IS 3 9 0 9 0 0 0 3 0 0 8 0 19 0 a 13 5 0 5 8 3 25 0 8 0 3 6 0 0 fs, M* ,L* 97 94 87 90 100 94 48 90 90 100 87 100 90 100 97 55 90 74 87 71 87 94 55 81 71 71 65 77 81 74 52 39 26 45 95 85 83 76 95 73 37 6© 78 98 78 73 85 95 S3 27 71 54 76 49 ©8 46 22 ©8 44 27 15 15 15 29 10 5 0 2 ©2 64 82 71 82 75 32 50 86 64 71 54 64 82 57 14 46 21 32 18 21 11 11 32 11 11 4 18 21 21 0 4 0 4 Total H. M* U 94 88 88 82 100 85 38 82 88 99 85 07 91 94 88 37 85 68 84 60 77 ©7 40 73 48 45 32 38 31 48 23 18 10 19 87 65 85 77 81 75 19 55 80 81 67 65 67 73 58 13 37 22 35 15 52 10 9 19 9 7 4 10 9 22 3 5 0 2 44 30 ©7 46 68 49 4 20 38 69 37 40 31 35 39 3 IS 3 © 0 13 O 1 0 1 0 0 2 3 18 1 2 0 0 Two characteristics of these tables are of con~ siderabl© importance# First of all, the 11total® column, the one given for test items when only on© is available, is inadequate to describe the discrimination within the preparation groups# In some oases, of course, where the percentage differences are about the same for all groups, the general index is satisfactory, but where wide divergence between groups Is found, Its reliability is considerably diminished# The second characteristic is that Items which discriminate for one group do not necessarily do so for another# The earlier and easier Items do not discriminate between members of the often the group, and the later Items are so difficult for all members of the group that no differentiation takes place# Th® implication of thes© two features is Hi at not only should indices be given for each group, but separate tests themselves are desirable# Further research should b© mad© in connection with this point# 26# Difficulty indices for v reparation groups# Corresponding to Tables 25, 26, and 27, are the following two which give the difficulty Indices, as derived from the samples, for each 198© preparation group# Attention should be called again to column four, whos® Indices ar© the averages of th© corresponding ones given in th© first three columns, and which Is presented solely for the purpose of exhibiting th© degree to which it approximates th© column of indices derived by the Cooperative Test Service* The same remarks can be mad© of Tables 42 and 43 as were mad® of those involving th© 1935 items# In a few cases th© index listed for a particular group exceeds that given for another group with mor® preparation, but all these differences are small enough to b© entirely accounted for by chance fluctuations# For most of the items there is a definite, though not necessarily signifi cant, Increase In difficulty accompanying an increase in preparation* Distributions of the difficulty indices for each preparation group are listed in Tables 44 and 45# Table 4£ 12.1 ITEM DIFFICULTIES FOR EACH PREPARATION GROUP pl- *1* , and Pg FORM A* 1930 Cooperative Total Ei Test Service Sfctt !i hk 88 l 93 95 92 92 2 56 78 82 72 74 96 3 95 96 96 95 4 95 98 94 95 90 87 5 61 88 79 66 6 77 91 92 87 82 85 89 95 7 90 84 81 a 84 91 85 78 52 9 64 ©7 84 47 10 71 85 79 82 72 70 80 85 11 78 75 71 12 81 88 80 75 m 72 89 13 71 70 14 65 79 93 79 73 yy 71 78 15 85 78 16 71 53 93 72 68 62 17 82 84 76 68 18 92 97 99 96 77 69 88 75 75 69 19 75 20 59 70 73 68 21 41 80 70 60 83 22 58 59 69 52 S3 77 89 23 83 78 80 63 38 70 24 ©3 52 70 SO 72 85 85 26 8 48 29 80 31 87 45 82 61 63 54 86 89 75 28 76 54 87 29 60 76 73 ©2 77 51 30 ©5 55 22 50 48 21 68 00 31 07 03 32 94 32 75 47 27 65 45 46 33 47 39 32 52 57 34 38 45 35 57 81 36 61 59 79 36 64 59 46 45 66 26 45 37 24 43 29 25 38 19 28 28 4© 39 26 11 39 43 16 68 40 34 28 30 7 27 41 51 20 42 15 3 IS 24 42 49 73 22 58 43 34 25 27 46 3 44 15 23 11 10 45 1 18 10 19 2 9 46 21 15 47 29 1 14 14 8 0 6 18 48 10 7 9 49 1 12 28 17 29 50 38 31 ±32 Table 45 item Mm 1 2 3 4 5 6 7 8 © 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 28 27 28 29 SO 31 32 33 34 55 36 37 38 59 40 41 42 43 44 45 46 47 48 49 50 DIFFICULTIES FOB EACH I5REPARATION GROUP ^1* ^1 P0BM B, 1936 !i 92 89 92 81 8© 43 68 88 73 87 63 60 55 69 30 85 60 35 73 69 74 64 8 56 46 80 50 52 53 41 46 10 26 IS 23 3 24 4 5 3 5 4 3 3 6 23 4 2 0 0 ..fir 93 87 97 85 07 73 79 89 75 42 72 78 76 ©0 SB 89 75 67 82 65 81 65 15 51 78 79 61 65 57 71 08 11 40 26 36 18 35 19 12 21 8 10 5 11 5 23 2 a 1 4 a* 99 89 98 93 94 84 92 94 84 59 90 87 a© 91 76 97 92 82 84 79 93 80 39 69 84 89 79 76 81 93 80 32 70 51 67 47 61 51 32 62 43 36 27 35 31 41 20 15 S 10 Total 95 88 96 80 90 67 80 84 77 43 75 75 73 80 55 90 70 61 80 09 83 70 21 52 69 83 63 64 64 68 61 18 45 31 42 24 40 25 16 30 19 17 12 16 14 29 9 8 3 7 ^2 Cooperative Tost Service 95 87 91 85 87 57 72 67 74 31 73 65 64 72 50 84 @8 55 73 56 77 60 24 45 62 83 54 54 55 66 57 19 49 55 44 15 29 33 20 36 27 25 15 IB 18 36 15 10 4 10 Table 44 DISTRIBUTION OR DIFFICULTY INDICES FOR EACH PREPARATION GROUP FORM A, 1936 S M m 95*100 90*94 85*89 80*84 75*79 70*74 65*69 60*64 55*59 50*54 45*49 40*44 55*30 30*54 25-99 20*24 15*19 10*14 0*9 0*4 !i lii i 2 2 1 2 A tSU 5 3 2 5 1 1 2 2 2 4 3 X 2 7 2 3 5 6 0 3 1 5 2 1 1 1 1 3 4 0 1 2 3 0 £2 5 5 8 8 4 1 4 1 1 2 3 1 I 0 1 2 2 1 0 0 Table 45 DISTRIBUTION OF DIFFICULTY INDICES FOR EACH PREPARATION GROUP POEM B» 1936 Index 95*100 90*94 85*89 80*84 75*79 70*74 05*69 60*64 55*59 50*54 45*49 40* 44 35*59 30*54 25*29 20*24 15*19 XU 4 0*4 Ek 0 2 5 2 0 3 3 5 1 3 2 2 2 1 2 3 1 1 6 8 ^8 M 1 1 5 3 0 4 4 1 3 1 0 2 2 0 1 2 3 t 3 3 9 4 8 4 1 2 2 1 2 1 2 3 3 1 1 2 ? 0 These distributions reveal differences between preparation groups in the same manner as was noted for the 1955 tests* That is, indices increase in si£© quite generally as th© preparation is Increased, giving the im pression that th© entire distribution has been forced upward* Th© distributions also reveal th© fact that the 195© tests were relatively easy, perhaps too easy* FYen for th© one*year groups, there are tw©nty*five items in Form A and twenty* two in Foma 8 whose indices are 50 or more# 27* Comparison of difficulty indices found for preparation groups» We seek, now, to determine those items for which the differences between preparation groups are significant, and th© procedure used is identical to that employed for the 1955 items* Following the same pattern as before, the calculations have been placed in th© Appendix and th® significant differences found are listed In th© next two tables * Table 46 , ITEMS OB' FORM A, 1934 WHICH YIELD SIGNIFICANT DIFFERENCES BETWEEN PREPARATION GROUPS P1 and Item Plj 8113 f2 Item P1 a»d pg Item 2 2 5 B 6 9 e 9 12 16 17 21 13 14 16 21 24 .25 26 27 28 29 30 31 32 33 54 32 35 56 57 39 40 41 42 43 44 45 47 49 37 58 39 40 41 43 44 47 48 13 14 18 17 21 22 24 25 26 27 2B 29 30 31 32 33 34 53 36 57 38 59 40 41 42 43 44 45 46 47 48 49 50 JJ36 Table 47 ITEMS OF FORM B* 193© # WHICH YIELD SIGHIFICAHT DIFFERENCES BBTWEEH PREPARATIOK GROUPS \ and % Item F1 F2 Item Item 7 © 7 8 IX 11 6 8 10 12 12 15 IS 14 15 16 17 18 20 21 15 15 17 IB 22 23 24 25 27 30 58 40 29 30 51 32 33 54 35 3© 37 58 39 40 41 42 43 44 45 4© 47 50 23 84 85 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 30 88* B|T t of preparation upon knowledge of the fe&n_Q.SP.tfi. and processes covered by the 1956 p r o t e s t s * Hslng Tables 42, 45, 46, and 47, we shall now classify the 1938 Items In the ©am© manner as was described for the 1955 items* The groups are as followss A* Concepts and processes which were well~known* 1* Elementary operations of arithmetic* The agreement here la not perfect, but It is very nearly so and hence Is included In the list* Of the ten items covering these operations, only on® failed to meet the criterion of a difficulty Index of 70 for 9well-known*w The group yielded an Index of only 54 for Item 2 of Form A* The process involved was a simplification 2* The principle of adding exponents when a series of Items with Identical bases are m l tipi led together* 3* Substitution of numbers for letters In simple algebraic expressions followed by simplification* When extraction of roots was added to these processes, difficulties were Increased* A cursory examination of the responses revealed on© common error ( \/9 ♦ 16~»3 ♦ 4) 128 made by those with one year of preparation* 4* Simple written problems commonly found in arithmetic* In order to illustrate the type of exerelse referred to, the three items ar© listed below* Item 2 3 » Form A ; What is the Interest on $600 for three months at the rate of 5 p©r cent per annum? Item IQ, Form B s An electric bill varies directly as the number of kilowatt hours consumed* If 45 kilowatt hours cost $3, what will 60 kilowatt hours cost? Item 2 6 , Form B s How much money must b© placed at'^per cent simple interest for one year to earn #12? B* Concepts and processes which were difficult* 1* Subtraction of algebraic fractions* C* Areas materially strengthened by a study of third semester algebra* 1* Addition and multiplication of irrational expressions* Mo subtraction was involved in 1936# 2* Solution of unfamiliar problems testing the power to follow definitions or direc tions* 3* Simple written problems involving linear equations In two or three unknowns* 4* Operations involving fractional exponents* 5* Solution of quadratic equations* 129 B* Areas materially strengthened by a study of fourth semester algebra* 1# Extraction of square roots# 2* Multiplication of irrational expressions* 3# Subtraction of algebraic fractions. 4. Evaluation of monomials with integral bases and fractional exponents. 5. Hqp re sen tation of literal relationships such as the follow!ngi a* If lemons cost A cents a dosen, how many dosen can be bought for D dollars? h« Th© average of two numbers, x and y, is k. Write an expression that gives the value of y in terms of x and k* c* If a train runs M miles in 3 hours, how many miles will it run in & hours at th© same rat® of speed? 6* Solution of quadratic equations. 7# Multiplication of algebraic expressions Involving fractions* It is to be remembered that no process was listed In the above groups, or the corresponding groups Involving the 1935 Items, unless more than on© item covered th© process. Ther© were two such items omitted from th© 1935 groups and also from thO 1936 groups, but which exhibited the same characteristic In both years. That Is, these two Items will b© classified as were the others since two observa tions were mad© for different years. S' rs f, JLo-U Th© following item falls Into th© difficult groups 11If x - {2 and (6 - /s)x - 5 ^ 2 y ♦ 2 » 0, find th© value of y,n 3# 7 For th© other item, ”simplify significant difference was found between the P^ and groups# a 131 CHAPTER VII THE DIFFICULTY OF AK ITEM FROM YEAR TO YEAH 29# Comparison of 1935 and 1936 difficulty Indices, Th© purpose of this section is to investi gate the difficulty indices reported for those Items which were given In 1935 and repeated In 1936# groups of items will he ©onaidereds Three those which were repeated in Identical form; thos© which were repeated in almost Identical form, differing only In the direc tions Involved or through some simple change of number; and those which do not fall into the other two groups but which Involved th® same process or concept# These three groups of corresponding pairs of Items, with difficulty Indices for each, are listed In Tables 48, 49, and 50, respectively# The apparent stability of th© difficulty index for a given item Is quite noticeable# That this agreement should, and does, become less marked as th© items become less alike is evident# Table 48 COMPARISON 0F DIFFXGULTIK,s of ;ETE&S Oilr m ib loss AMD REPEATED IB IDENTICAL FORM IB 1956 Item £l Pl& *2 1* 1935 1936 0*14 A-8 70 91 94 84 94 91 88 85 76 78 8* 1935 1936 B~1 A-ll 74 70 72 80 70 as 78 78 74 75 3. 1935 1936 B-15 A—12 76 71 80 SI 82 88 79 SO 72 75 4* 1935 1936 A**13 A—14 66 @3 80 79 8© 93 77 79 70 73 5* 1935 1936 JW4 A*16 56 53 83 71 84 93 74 72 ©9 68 6* 1935 1936 C<»$2 A-17 46 62 68 82 84 84 68 7© ©5 ©8 n 1936 1936 B—13 A* 20 50 59 64 75 80 75 65 70 80 ©8 8« 1935 1936 B**.4 A-2X 38 41 70 60 78 78 82 60 58 53 9* 1935 1936 B~25a A-29 82 60 48 82 60 87 43 7© 49 73 10* 1935 1936 C-2Q A-31 32 21 48 @0 7© 68 52 50 4© 48 11 • 1935 1936 0-11 A-83 6 27 36 46 60 63 34 45 45 47 • 01 H Cooperative Test Service Year 1935 1936 A—15 A-36 16 21 45 3© 54 57 38 38 42 45 13* 1935 1936 0—31 A— 40 8 16 14 , 34 28 88 17 39 27 43 14. 1935 1936 C—41 A— 41 2 7 16 27 26 51 15 28 23 30 Total 13: Table 48 (continued) Cooperative Test Service Year Item fl is# 1935 1936 0-37 A—42 14 3 18 18 24 24 19 15 22 20 16* 1935 1956 B-41 A-43 4 28 1© 53 18 73 13 49 IB 42 17* 1935 1956 A-41 A*»44 14 3 25 27 2© 46 22 23 17 34 18 * 1955 1956 B-28 A*#46 2 2 2 9 10 19 5 10 13 18 19* 1935 1936 C*44 A~48 0 0 2 6 © 18 3 8 6 14 20 * 1935 1936 B~3! B**8 @6 ©8 88 89 98 94 S4 84 75 87 21* 1955 1936 G~10 B~18 58 55 72 7© 82 88 71 73 71 64 22* 1936 1956 B*5 B-14 68 ©9 76 80 so 91 75 80 67 72 23* 1955 1936 B~17 B~27 52 50 76 61 70 79 8© 63 54 54 24* 1935 1936 B-12 ■Q i x >«*O*5Co 5@ 52 56 ©5 60 76 57 64 53 54 25* 1935 1936 A~27b B~30 14 41 47 71 76 93 46 60 50 86 26* 1935 1936 C-26 B~31 30 46 40 58 56 80 42 61 47 57 27* 1955 1936 B~ll B~33 30 26 48 40 64 70 47 45 44 49 28* 1935 1936 C~40 B-54 20 15 38 2© 48 51 35 31 39 35 29* 1936 1936 B-24 B~35 8 23 38 36 42 87 29 42 39 44 1'jgT !e Total 4^4 fable 48 {continued} Cooperative Test Service Xf** Item Ei Eli Eg 30* 1935 1936 iW16 B-36 10 8 28 18 34 47 24 24 35 15 31* 1935 1936 B~53 B~37 28 24 42 55 46 61 39 40 34 29 32* 1935 1938 B*S9 B~40 4 8 6 81 28 62 13 30 85 36 33* 1935 1936 0-22 B- 41 8 5 20 8 26 43 17 19 24 27 34* 1935 1936 A-36 8-42 12 4 22 10 28 36 21 17 24 25 35* 1935 1936 B-86 B-43 6 3 14 5 18 27 13 12 22 16 36* 1935 1936 C-43 B-44 4 5 10 11 30 35 15 16 19 18 57* 1935 1956 B~38 IB'*"45 2 8 12 5 14 31 9 14 14 18 38* 1935 1986 A-53 B-47 0 4 6 8 6 20 4 9 7 15 39* 1935 1936 3*42 B~48 4 2 4 8 4 15 4 3 9 16 40* 1985 1936 8-40 B-49 0 0 4 1 4 8 3 3 4 4 %1rM. Table 49 GGMPABISOB OF DIFFICULTIES OF ITEMS GIVEN IN 1955 AND REPEATED IN ALMOST IDENTICAL FORM IN 193 a Cooperative Teat Service fiMwr Item PI Eli 1* 1935 193© A-3 c A«*l 94 88 96 93 100 95 97 92 93 92 2# 1935 1936 A-2 A-4 84 95 92 94 94 98 90 95 84 90 3* 1938 1936 A~34c A***5 82 ©1 78 87 90 88 83 79 81 60 4* 1938 193© A-3a A«*6 70 77 88 91 92 92 83 87 78 82 5# 1938 1936 B-7 A~7 82 85 80 89 94 95 85 90 78 84 6* 1985 1930 C«,33c A*9 56 52 72 ©4 82 84 70 67 75 47 7* 1935 1930 C-13 A«»13 66 66 82 72 90 89 79 76 71 71 8§ 1955 193© B**54a A-38 a A-15 56 54 71 72 78 78 92 66 85 73 66 78 70 74 77 9* 1955 1956 0*18 A«*19 52 69 ©4 75 8© 82 07 75 60 69 10* 1935 1936 O**0 A~22 @4 51 5© 58 76 69 65 59 58 52 11 * 1955 193© A~ll A~27 3© 45 55 61 78 82 56 03 83 54 18. 1985 195© A-39 c A~28 40 54 5© 86 80 09 59 76 51 75 13* 1935 193© A~25 A-30 0 22 47 65 76 77 43 55 47 51 14# 1955 193© A«*27a A-32 1© 32 44 75 72 94 44 67 45 63 £& Tots Tatola 49 (eont limed) Cooperative hk £s Tot 32 32 57 44 52 32 47 42 39 12 39 24 64 42 79 20 61 35 59 A-37 16 26 22 43 42 00 27 45 33 46 1935 1936 A-*24 A-59 16 11 24 26 42 46 27 28 31 28 1935 1936 C-23 A* 47 4 1 6 14 12 29 7 15 21 1935 1936 A-43 A-49 2 2 9 12 4 7 4 1 10 1935 A*»3d C~lSc 1936 B~£ 94 74 89 95 92 87 94 84 89 94 85 8© 91 77 87 22 * 1935 1936 A-12 1W*4 86 81 95 85 98 93 95 00 87 85 23* 1933 1936 0-2 B-5 88 88 96 87 94 94 93 90 84 87 24* 1935 1936 C-33b 0 68 43 86 73 88 84 ©1 67 81 57 23* 1935 1936 A-9 B-? 66 68 92 79 92 92 83 80 77 72 26# 1935 1936 C-l B-9 64 73 70 73 74 84 71 77 75 74 27* 1935 1936 A-34b B-12 76 00 83 78 90 87 83 75 73 65 28* 1935 1936 G-4 8-15 42 30 60 58 84 70 62 55 65 50 29* 1935 1936 0-19 64 73 74 82 70 84 71 80 63 73 15. 16. 17. 18. 19 * 20, 21 , Itiam £i 1936 1936 B-22 A-34 @0 1935 1936 C-29 1935 1936 A-36 B~25t> B—19 a Test Service 11 127 Table 49 (continued) Cooperative Teat Service Year £t®» !i 50. 1955 1936 A—8 8-20 60 62 70 65 84 79 71 69 62 66 31. 1935 1936 B~35e B-21 48 74 70 81 68 93 62 83 58 77 32* 1938 1936 A-10 B-22 46 64 64 6$ 84 80 65 70 55 60 33. 1935 1936 B—3 B-25 50 46 76 78 76 84 67 69 54 62 34. 1935 1936 A—18 B-29 34 53 73 57 78 81 62 64 53 55 55# 1935 1936 A-26 B-38 6 4 22 19 40 51 23 25 34 33 36* 1935 1936 0-23 B-39 10 5 26 12 32 32 23 16 31 20 37* 1933 1936 G-24 3-50 2 0 4 4 6 16 4 7 10 £2 3 138 Table 50 COMPARISON OP DIFFICULTIES OF ITEMS GIVES IK 1956 WHICH INVOLVED THE SAME PROCESS BUT WERE NOT IDENTICAL TO 1935 ITEMS e w « M «w iin eM m iiiiiii i m u <M*w»ilWei m uw.i■■ w m m m w w rn im rriir nnii*i— w im m ■ » m w im i uooperstxve Test Service XS££ £fe®» P 11 hk ■£» IS 1935 ■A**12 B~8 0*8 A*3 86 80 86 96 95 80 94 95 98 80 90 96 93 80 92 96 87 70 90 95 1938 B—S C~2 A***4 82 88 93 84 96 94 92 94 98 86 93 95 rjfty 84 90 3e 1935 1936 B«*5 A**6 08 77 76 91 80 92 75 87 67 82 4* 1933 1936 A*0 0*7 A* 7 60 84 88 92 94 89 92 90 95 83 89 90 77 82 84 5. 1935 1936 8*9 A* 14 44 08 68 79 80 93 64 79 57 73 6. 1935 1936 8-3 A-10 50 88 76 71 76 93 67 72 54 68 7. 1935 1936 A-34'to A* 34c A~17 76 82 62 85 78 82 90 90 84 83 83 70 73 81 ©8 8. 1935 1936 A* 10 A* 19 46 69 64 75 84 82 85 75 55 69 9. 1935 1936 A-20 A-20 38 59 59 75 76 75 58 70 56 68 10. 1935 A* 3 0*4 A* 21 46 42 41 74 60 60 80 84 78 69 62 60 59 65 53 A*8 8*8 A-22 00 52 51 70 70 58 84 86 69 71 63 59 62 52 52 A—29 B~ll A* 33 0 80 27 20 48 46 30 64 63 17 47 45 24 44 47 1* 1936 2* 1935 1936 11. 1935 1936 IS. 1935 1936 Total 139 Table 50 (continued) P Cooperative Test Service Tear Item fl fli JSL 13* 1933 1936 JW18 4*3 4 34 52 73 S7 78 52 ©2 47 53 39 14* 1933 1936 A* 3Tb A* 38 46 19 50 25 66 43 54 29 49 24 13. 1935 1936 A* 5c 0*1 94 92 96 93 100 99 97 95 93 95 16* 1935 1936 B~S 0*8 B~4 SO 86 81 80 94 85 80 96 93 80 92 86 7© 90 85 17. 1935 1936 A* 2 B~5 84 88 92 87 94 94 90 90 84 87 IS. 1933 1936 e~33& B*6 m 43 86 73 92 84 82 07 81 57 1©. 1935 1936 A* 54a B*8 92 68 98 89 96 94 95 84 92 ©7 20. 1935 1936 A*1 B~9 76 73 88 73 82 84 82 77 78 74 21. 1935 1936 A* 38a B*34a B*ll 54 50 63 78 72 72 66 92 90 6© 73 75 74 70 73 22. 1935 1936 0*32 B-12 40 60 08 78 84 87 66 75 65 65 23. 1935 1936 A* 3a B~14 70 69 88 80 92 91 83 80 78 72 24* 1935 1936 A* 5 B-*4 0*15 46 38 30 74 70 58 86 78 70 69 62 55 59 58 50 23. 1935 1036 B-0 BU20 52 62 70 65 60 79 63 69 52 ©6 26. 1933 B~!2 0*18 B~22 56 52 64 56 64 65 60 86 80 57 67 70 53 60 60 1936 Total 140 TaBle 30 (continued) 27* 28* 29* Year Item F JL fli 1935 1836 A—4 0-3 B-25 56 30 46 1935 1936 A«*7 B-27 56 50 1935 A*10 C-18 B~28 1936 30. 1935 1936 31* 1935 1936 Cooperative Teat Service B 2 Total 83 58 78 84 70 84 74 53 ©9 69 50 62 68 61 6© 79 63 ©3 61 54 46 52 52 64 64 84 ©5 76 65 67 ©4 55 60 54 A~22 B**22 B**80 16 20 55 45 32 57 68 44 81 43 32 ©4 44 42 55 A**20 CML1 B-53 0 6 26 20 30 36 40 60 70 17 34 45 24 43 49 86 Keeall that* because of a great number of omissions* it was deemed necessary in the previous section to eliminate some items from conaideration* If this policy were to be followed here, the following pairs of items would be elimi nated from Table 48s 56, 37, 38, 39, 40* 13* 14* 16* 17, 18, 19, 28, 32, 35, Similarly th® pairs 16, 19, 20, and 37 would disappear from Table 49, and item A-29 would be dropped from the two pairs labeled 12 and 31 in Table 50# The effect of this would be to strengthen considerably th© relationship mentioned# However, it would mean a loss of one third of th© identical pairs and hone© all items are h©pt as given# In compiling distributions of difficulty indices, groups of five units each were used# vals, Using the same inter- the Pearson product-moment coefficient of correla tion was computed for the paired difficulties as given in each table by th© Cooperative Test Service* The coeffi cient found for th© identical items was #924 with a probable error of #016$ for the !falmost identical*1 Items it was *885 with a probable error of #024$ and for th© items Involving the same process, th© coefficient was found to b© #761 with a probable ©rror of *042* Ho corre lation study has been mad© for the Indices derived for the preparation groups* These Intervals are undoubtedly too small to tab:© Car© of all chance fluctuations, and hone© th© correlation coefficients are probably too small* Befor® generalising we should make on© point clear# The Committee, in constructing the 1936 tests, mad© a deliberate effort to place the items in a different setting from that employed in 1935# At the same time, how ever, theconventional policy of arranging items in th© order of difficulty was roughly followed# In terns of these facts, then, there is strong evidence to support the theory that the difficulty of an Item of a pre-test for students of first year college mathematics, as computed for a large heterogeneous population, la Invariant with respect to its location on a test, with respect to th© year in which it Is given, and with respect to the specific population to whom It Is given, provided that population is similar with respect to preparation to the one for which the Index was derived# ■i* W l -V#— hi A school freshmen* *1* nlr.nl|lw ( I# * # ! * # W W M M w w M M )# **— — found for ........................... rrtwn#** * ! . 30# Comparison of difficulty indices In order to substantiate a belief that the Invariance described above Is not confined to the pre-tests nor to college students, th© writer mad© on© further Investigation* Th© basis for this study is furnished by th© 1932, 1934, and 1936 Iowa Every-Pupil Tests o in ninth year algebra* Difficulty Indices for the Items ® These tests constituted, in the years mentioned, on© part of th© annual Iowa &v©ry-Pupil Testing Program* This program Is carried on each spring under the direction of Professor E* F# Lindquist of the State University of Iowa, and extends to all high schools in the state who wish to participate# 143 of th© 1932 test* reported for the entire population of 11*572* were presented fey Professor Lindquist at a con ference of teachers of mathematics which was helc3 at the University of Iowa In October* 1935* Indices for th© 1934 and 1936 tests were taken from two unpublished master’s theses % which were directed by Professor Lindquist* Th© 1934 study Is based on a sample of 1000 papers and the 1936 study on a sample of 500 papers* The writer carefully examined the three examina tions mentioned and selected all of th© items which were enough alike to be considered as testing for the same process* (Since schools participating in the program have access to tests of former years* It is very seldom that an item Is repeated*) Each test is divided Into three sec tions which are labeled ‘’Fundamental Processes*tf Algebraic Representations and R e l a t i o n s a n d "¥©rfeal Problems*n The Items are classified her© In the same manner, and the number in parenthesis immediately following an item is Its difficulty index* In most cases where noticeable differences occur* It is easy to see th© reason for th© difference* S Fridolf Lun&hofom* "Achievement in th© 1934 EveryPupil Test In Algebra*" Thesis (M*A#) Iowa* 1934* Alfred A* Kaskadden* "A Critical Analysis of th© 1956 Iowa Bvery-Pupll Test In Algebra*" Thesis (M*A«) Iowa, 1936* 144 PffiTt 1* l t 1934, Item 2t 1936, Item 2t 2 * 1932, Fundamental Process©® If a £ - 2 , find the value of (-3 &)^* (48) If x » (5 7 ) 2, what Is the value of Sx'H Item28 Subtract a2 - 6 a * 7 from 2 a2- 3 * (50) item 4s Subtract 2 a2 * a- 3 from 4 a2 - 2 a- 1 # (67) 3# 1934, ItemIs Divide 4a2x® by ax2 , (62) 1936, Item9s Divide 6n4x6 by 3n2x2* (57) 1934, 4# 1934, 1936, 5, 1932, 1936, 6 # 1934, 1934, Solve for x In terms of a and b, item5* a « & * Item5s Solve for Rx K s Item6s Solve for m s * 2n » k * Item1 4 •m » ax- b# j| » (59) (60) (59) Solve for x in terns of a, b, and m* (57) Item61 If 3A - 2B, what is the ratio of A to B? (7) Item 19s Solve the equation G * ax ratio l/x* (18) 7, 1932, Items: 1934, Item 22: Solve for r: ^j| - 4 /®"* V *J^ for 1h© . What is the value of H? (5) ( 5) ±4n> 1934, item 9* Write in factored form the low®at common denominator that can be used in adding .x and ay + xy (35) Find the algebraic expression that is the product of th© highest powers of all factors of the terms (14) 6 ©xS , 3a^x and 4asxs # 1936, item 105 What is the lowest common denominator that should be used In adding three fractions whoa© denominators are (57) Sax, 8 &S , and 12x3? * 1932, item 10: 1934, item 11: If 3x - 2y s 11 and 5x+3y m 12 what is the numerical value of x? (44) What is the value of x found by solving simultaneously these equations? (36) x ~ 2y z 10 1936, item 19: 10# 1932, item Hs and 2x ~ y « 1 1 * Solve for the unknowns: 2a •* 3b 3 13 ♦ 2b « 4 2 m 7 Writ© the expression x ~ 2 ~ r ^'r+~*> (30) (19) as on© fraction having x + 2 as its denominator* 1934, Item ID: Simplify* ( 9) 1936, item 15s Reduce the following mixed expres sion to on© fraction in its simplest forms 2 _ Sns x + n - —---=»- ♦ (14) x -n XI# 1932, item 15* If R * —— write a formula for 0 in terms of E, r, and R# 1934, Item 24s X, m M fe-r, B, # Solve for t In terms of L, M, and g# 1936, Item 21* S a ^ (1 2 ) (12) 1 ® * Writ© a formula for ® Wg in terms of S and ( 5) For what negative value of y will th© expression 5y^+5y**2 he equal to a©rot (25) 1934, Item 16* What are the two roots of the equation 2 n 2 « 5n * 3f (2 2 ) 1936, item 22* Find the two roots of the equation 2 b2 - 4b S 18* (26) 13# 1954, Item 23* Multiply ( 9) 1936, Item 10s Simplify 12# 1932, item 19* 2+ & n V? by - 1] • / 4n+ 6 . — ™— 4x - 9 • (17) 147 ?jRjrt II * Algebraic Representations and Halations 14* 1932 , 1 tern 1936, item 15* 1934, item 7s 1934, Item 11* 16. 1934, item 10* Write a trinomial whose factors are (x ~ 0ft) and (2x 4- a)* (5 4 ) What trinomial has 2x + y of Its two equal factors? (3 5 ) as on© Write the quotient obtained by dividing a + 1 by a ^ * 1 * 2 (26) What Is the ratio of x - 9 5x~ 9 ? (16) Write th© fraction which is equal to jj but which has a numerator (12 ) Of 3$. 1936, item 12* 17, 1934, Item 22* 1936, item 18* Hi© expression — j^g may be changed into a fraction whose denominator is What Is the numerator of this fraction? (28) For what value of m is the expres sion equal to 3 * (22) For what value of x is th© expres* (20 ) * 18* 1934, item 24* 1936, Item 14* 3 equal to zero? Suppose H varies inversely as R and that H Is 6 when H is 2 * What is the value of N when R Is 3? A varies Inversely as 16 when B is 3* What A when B is 8 ? B, is ( 3) and A is( 3 ) the valueof 148 Problems If the average speed of a train should be Increased 4 miles an hour it would require 30 minutes less time to travel a distance of 180 miles* What is the average speed of the train? ( 1) 1934, item 10* If the average rate of a certain train should b© Increased by 4 miles per hour. It would require 30 minutes less time to travel a distance of 180 miles* What Is th© average rat© of the train? ( 2) 1934, item 9* A man worked for a number of days and earned #63* If he had received 75 cents more per day, he would have earned the same amount in 2 days less. How many days did he work? ( 1) 20* 1934, Item 3s The sum of three numbers is 150* The first number is twice th© second, and 20 more than th© third* What is the third number? ( 9) 1936, Item 2* Three numbers are so related that the first Is equal to twice the second, while th© second is equal to three times the third. What is the largest of th© three numbers If their sum Is 45? (24) 19* 1932, item 10* Only one pair of th© entire set of exercises can b© classified as M e n t l c a l ^ that pair being th© first two items of th© group numbered 19, and the Indices reported are practically identical* The 1934 Item was the last one on the examination so its index perhaps doesn11 mean much, but there were five Items following Item 10 in 1932 and three of them yielded Indices of 10 or more, on© being as high as 28* It is quite likely, then, that the true index of th© item is between 0 and 6* There are several pairs of nalmost identical” Items and, with on© exception, the agreement of their In dices is very close* These pairs ares those items numbered 5, the last two Items of group 8, th© first two of group 9, th© last two of group 12, the pair labeled 18, and th© pairs one and three and two and three of group 19* Th© two whose Indices differ markedly are those found In group 8* Th© unusual terminology of the 1934 Item is un questionably responsible for a large part of this difference* In general, th© paired items taken from the Every-Pupil Tests behave very much as do those taken from th© pro*tests* Th© evidence Is strong that th© invariant quality of th© difficulty index is characteristic of mathematics tests at both Instructional levels* BIBLIOGRAPHY 151 1* Allen, Arthur A*, “Studies of the Retention of Algebra by High School Graduates* I Algebraic Representa tions, II Formal Skills, III Verbal Problems•" Master1® thesis, State University of Iowa, Iowa City, August, 1939# 2* Atchison, R* H#, "Variability from School to School In Difficulty of Items in the 1933 Iowa EveryPupil Test in General Science," Master’s thesis. State University of Iowa, Iowa City, August, 1934# 3# Bergen, M# C#, "Achievement in College Algebra Com pared with the Humber of Semesters of Preparation in High School," School Science and Mathematics. XXXVIII (October,^1938), 763-765# ' ~~ 4# Bowley, A# L#, Element® of Statistics# 4th editions Bond on* P • "STJ King and Son, Ltd*, 1920, 5# Branham, Arthur K#, "Knowledge of High School Algebra Retained by High School Seniors#" Master’s thesis, State University of Iowa, Iowa City, August, 1938 6# Brumbaugh, Aaron J,, "The Uses of Tests and Examina tions in the Selection, Guidance, and Retention of Students#" Institute for Administrative Officers of H i g h e r lnstitu tion a , Proceedings-. VIII, 1936, pp# 176-189# x/t * Burch, Robert L#, "Studies of Retention of Algebra ^ by High School Graduates* I Algebraic Representa tion#" Master’s thesis, State University of Iowa, Iowa City, August, 1939# 8# Chesire, Leone, Milton Saffir, and L* L# Thurstone, Computing Diagram® for the Tetrachorlo Correlation ffoefficTeiat» The University of Chicago, 1933* 9# Crawford, A# B# and Paul S* Burnham, "Entrance Exami nations and College Achievement," School and Society. XXXVI (September 10 and 17, 1932), 344352 and 378-384# 10# Douglass, Harl R#, "Permanence of Retention of Learning In Secondary School Mathematics," The Mathematics Teacher, XXIX (October, 1936), 287,288* 11# Douglass, Harl R# and Jessie H# Michaelson, "The Rela tion of High School Marks and of Other Factors to College Marks In Mathematics," School Review5 XLIV (October, 1936), 615-619# 12* Garrett, Henry E*, Statistics in Psychology and EducaM o n * 2nd editions Hew York * Longmans, Green and Company, 1937* IS# Gilkey, Royal, 11The Helation of Success in Certain Subjects in High School to Success in the Same Subjects in College,ff School Review* XXXVII (October, 1929), 576-588* 14# Hammond, H* P # and George B* Stoddard, A Study of Placement Examinations, University""of Iowa Studies in Education, Vol* XV, Ho# 7* Xowa Citys Uni versity of Iowa, 1929* 15* Hansen, George W*, 11An Analytical Study of the Iowa Placement Examinations#1* Master*s thesis, State University of Iowa, Iowa-City, August, 1928# 16# Hart, William L*, "Some Debatable Facts and Some Sug gestions, ” The Mathematics Teacher * XXXI (Decem ber, 1938), 355-362* 17* Hawkes, H* E*, E* P* Lindquist, and C* R* Mann, The Construction and Use of Achievement Examinations* Hew Yorks Houghton, Mifflin Company, 1936* 18* Kaskadden, Alfred A#, ”A Critical Analysis of the 1936 Iowa Every-Pupli Test in Algebra**1 Master1s thesis, State University of Iowa, Iowa City, August, 1936* 19* Kenney, J* P*, "Some Topics in Mathematical Statistics,” The American Mathematical Monthly % XL VI (February, 19391, Ho* 2, 59-74* 20* Lindquist, E* F*, "Cooperative Achievement Testing,” Journal of Educational Research, XXVIII (March, 1935), Ho* 7, 511-520* 21# Lindquist, E* P*, Statistical Analysisin Educational Research* Hew York: Houghton, Mifflin Company, 1940* 22* Lundholm, Frldolf, "Achievement In the 1934 Every-Pupil Test In Algebra•” Master*s thesis, State Uni versity of Iowa, Iowa City, August, 1934* 23* Miller, Laurence William, An Experimental Study of the Iowa Placement Examinations * University of Iowa Studies in Education, Vol* V, Ho* 6* Iowa City: University of Iowa, 1930* 153 24* Olson, Owen L #, "Studies of Retention of Algebra by High School Graduatest I Algebraic Representa tion, IX Formal Skills, III Verbal Problems*" Master1s thesis, State University of Iowa, Iowa City, August, 1939. 25* Hemmers, H* H. 4 Study of Freshmen Placement Testa at Purdue university. 1926-29* Purdue University Bulletin, XXIX,No* 13, Lafayette, Indiana, 1929* 26* "Report of the Committee on Tests," The American Ma them a 11ca! Monthly. XL VII (May, 194QT, Ho* 5* 27* Robb, Eva V., "A Study of the Variability In Item Dif ficulty from School to School for the 1933 Iowa Every-Pupil Teat in Ninth Grade Algebra." Master's thesis, State University of Iowa, Iowa City, August, 1934* 28* Schoonmaker, Ha&el E*, "The Value of the Hotz Algebra Scales In Sectioning College Classes In Freshman Mathematics." School Science and Mathematics XVIII (November, 1928),880-884* 29* Segel, David, Prediction of Success in College* Bulletin 1934, Ho* 15* Washington, D* G*: United States Department of the Interior, 1934* 50* Stoddard, George D., Iowa Placement Examinations* University of Iowa Studies in Education, Vol. Ill, Ho* 2* Iowa City: University of Iowa, 1925. 31* Stoddard, Georg© D., "Iowa placement Examinations," School and Society* XXIV (August 14, 1926), Ho. 607, 212-216* 32* Tippett, L. H* C*, Random Sampling Numbers * Tracts for Computers» S o * XV, Cambridge University Press, 1927* 33* Turner, Nura, "A Study of Certain Pre-Tests for First Year College Mathematics*" Master's thesis, State University of Iowa, Iowa City, August, 1936* 34. Yule, G* Udny and M* G. Kendall, An Introduction to the Theory of Statistics* 11th edition® Charles Griffin and Company, 1937. London: its4 APPENDIX Table 51 DIFFICULTY AND VALIDITY INDICES FOR EACH ITEM OF FORM A, 1935, AS GIVEN BY THE COOPERATIVE TEST SERVICE Item 1 2 5a b c d 4 5 8 7 8 9 10 11 12 15 14 15 16 17 18 19 30 21 22 23 24 25 26 27a b 28 29 30 31 32 33 34a b c d 35a b c Difficulty Validity 78 84 78 95 93 91 69 59 86 61 62 77 55 55 87 70 52 42 35 52 53 42 56 56 44 34 51 47 34 45 50 35 24 3 28 18 7 92 73 81 83 3 6 6 4 6 4 8 7 3 3 5 5 5 10 7 8 8 9 11 7 7 8 6 9 11 11 11 12 12 10 10 4 9 4 11 9 6 3 4 5 6 mm mm - 1^6 Table 61 (continued) Item 66 37a b 58a b c 59a b e 40 41 42 45 44 Difficulty Validity 24 64 49 74 68 64 8 5 6 4 •7 4* m *» m 81 24 17 51 4 12 6 4 8 4 5 4 Table 52 AMD VALIDITY IHDIOES FOR EACH FORM B* 1956 Item Difficulty 1 8 5 4 5 0 7 a 9 10 11 12 15 14 16 16 17 18 19 20 '74 64 58 67 52 78 76 57 50 44 53 60 62 72 85 54 45 26 53 Validity 4 5 10 7 6 6 5 5 -8 8 5 7 10 5 5 4 7 6 10 12 157 Table 52 (continued) Item Difficulty 21 22 23 24 25a b 26 27 28 29 30 81 32a % a 35 54a b 0 58a b 0 m 57 38 39 40 41 42 37 42 41 39 49 58 26 32 15 25 17 78 Validity 8 12 12 3 11 10 9 3 9 9 5 «•» ■m 34 70 52 25 ** m 58 22 25 14 86 4 IS 9 7 8 7 7 ** 9 8 7 6 5 5 6 6 Table S3 DIFFICULTY AMD VALIDITY INDICES BOR EACH ITEM OF FORM C, 1938 Item 1 2 3 4 5 6 Difficulty Validity 75 84 50 65 68 58 82 4 4 6 5 7 7 4 Table 555 (continued) Item Difficulty a 9 10 11 12 13 14 15a b 6 16 J7 IB 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33a b e & 34 35 56a b 37 38 39 40 41 42 43 44 90 95 71 43 39 71 76 87 88 77 68 70 •60 68 46 40 24 31 3 11 47 14 29 Validity 81 ai 78 77 6 4 6 8 10 7 9 8 7 4 5 4 6 6 14 9 a 9 4 6 6 8 9 9 8 8 6 7 9 8 7 82 8 85 17 27 65 38 m 49 22 85 12 59 23 27 19 6 5 m 7 5 6 7 11 10 10 7 6 Table 54 DIFFICULTY AMD VALIDITY INDICES FOR EACH 1936 ITEM AS GIVES BY THE COOPERATIVE TEST SERVICE Form A* 1936 Item Difficulty 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 92 74 95 90 66 82 84 78 47 72 75 75 71 73 77 68 68 77 69 68 53 52 78 63 70 30 54 75 73 51 48 63 47 39 45 59 46 24 28 43 Validity 1 7 4 6 4 5 6 7 3 6 7 4 3 6 6 9 4 10 7 5 7 4 4 6 6 11 7 10 6 11 14 10 7 5 8 9 Y 9 Y 8 Form Difficulty 95 87 91 85 87 57 72 67 74 31 73 65 64 72 50 84 68 55 73 66 77 60 24 45 62 83 54 54 55 66 57 19 49 35 44 15 29 33 20 36 1936 Validity 5 1 3 4 4 9 6 5 3 8 7 7 7 3 6 5 5 8 4 6 6 5 7 6 7 6 7 5 8 7 6 7 7 9 10 8 6 9 8 10 •]* ‘Table 84 (continued) Form A m 1956 Item Difficulty 41 43 43 44 45 46 47 48 49 50 50 20 42 34 15 18 21 14 10 28 Validity 10 6 8 6 7 9 9 8 7 3 Form B, 1936 Difficulty Validity 27 25 18 18 18 36 15 16 4 10 9 9 8 9 5 3 8 6 5 6 * f; ii *>-•* M Tabl® 55 NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W), AND OMISSIONS (0), ARISINQ FROM EACH SAMPLE, FOR EACH ITEM OF FORM A, 1935 1 Year 1& Years E Years Item E W 0 R mtm* W 0 H W 0 1 2 3* b e d 4 5 6 7 8 9 10 11 12 13 14 18 16 17 18 19 20 21 22 23 24 23 26 27a b 28 29 30 31 32 33 34a 38 42 38 46 47 47 28 28 48 28 30 33 23 18 43 53 18 8 8 24 17 15 19 10 8 8 8 3 11 7 14 4 3 2 21 21 7 21 16 14 19 26 6 11 16 20 22 19 19 18 15 21 24 20 11 10 5 6 5 6 6 4 3 1 2 2 10 7 7 1 1 1 O 0 1 1 6 0 1 4 3 8 6 1 6 16 22 23 7 14 17 16 19 18 22 31 37 42 36 40 40 44 46 45 49 48 2 2 2 3 88 92 88 98 96 95 83 74 90 68 70 92 64 55 95 80 75 45 28 60 73 58 59 33 45 32 24 47 22 44 47 25 20 0 26 16 6 98 83 78 91 11 8 11 2 3 3 15 23 9 31 27 6 30 37 5 19 21 48 43 30 20 33 20 54 37 45 38 13 15 13 10 18 15 24 5 S 10 2 15 20 8 1 0 1 0 1 2 2 3 1 1 3 2 6 8 0 1 4 7 29 10 7 9 21 13 18 23 38 40 63 43 43 57 65 76 69 79 84 0 2 2 1 41 47 46 49 50 47 42 43 49 33 42 46 42 39 49 43 35 27 17 38 39 35 38 26 34 22 21 38 20 36 38 24 15 2 15 13 3 48 45 45 45 8 3 4 1 0 3 8 3 1 16 8 4 7 9 0 7 13 17 18 11 10 12 6 19 12 23 15 3 11 3 2 10 14 20 4 6 8 2 4 4 3 1 0 0 0 0 0 0 4 0 1 0 0 1 2 1 0 2 6 15 1 1 3 6 5 4 5 14 9 19 11 10 16 21 28 31 31 39 0 1 1 2 b c a 3 8 7 4 0 0 2 0 0 46 58 41 40 ±S2 Table 58 (continued) 1 Year Item 35a b 0 36 37 a b 38a b e 39a b e 40 41 42 43 44 14 Years H W 0 ** mm e* m¥ mm- a# 4m- 6 27 23 27 24 10 26 22 22 18 9 18 18 1 5 a 17 25 22 6S 50 78 72 29 1 W 2 Years MM* «MWM 0 E w 0 #* 4m mm mm * *» ee M mm 58 32 42 16 14 34 20 3 8 6 14 57 14 41 33 33 40 20 23 9 12 14 2 16 13 0 5 5 a 14 m mm ae mm flW- m* mm ** mm m- 4m aw eft- « 4m mm ** m 20 10 7 3 1 9 13 12 3 3 2 3 17 28 35 39 47 39 56 19 85 28 2 11 25 48 59 6© ©7 80 40 20 13 10 4 8 19 33 16 © 11 9 5 14 9 2 4 7 5 16 28 30 42 36 Table 86 NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W), AND OMISSIONS (0), ARISING FROM EACH SAMPLE, FOR EACH ITEM OF FORM B, 1935 l£ Years 1 Year 2 Years ■ M A a r t M M M M i M W W M * Jin nnrm-. Jinn NTT Item E W 0 H W £ E W 0 1 2 5 4 5 6 7 8 9 10 11 37 41 25 19 34 26 41 40 22 20 15 13 9 18 17 15 24 8 8 21 17 19 0 0 7 14 1 0 1 2 7 13 16 36 42 38 35 38 35 40 40 34 36 24 13 8 12 8 12 15 9 9 15 10 18 i 0 0 7 0 0 1 1 1 4 8 55 4© 58 39 40 33 47 40 40 55 52 15 4 10 6 9 17 5 9 10 14 16 0 0 2 5 1 0 0 1 0 1 2 fable 56 (continued) 1 Year Item 1 IB 13 14 15 16 17 18 19 20 21 22 23 24 25a b 26 87 28 29 30 31 52a b c 33 34a b c 35 a b c 36 37 38 39 40 41 42 28 25 34 38 43 26 15 0 4 - h, 8 10 9 4 11 8 3 7 1 2 3 33 *•. li Years «•»# w 0 20 16 11 9 4 19 26 25 16 13 13 12 19 14 @ 4 4 2 2 8 9 #» 2 Years m W 0 & W 0 2 9 6 5 3 5 11 25 30 24 27 29 27 25 34 43 39 47 46 39 8 28 32 34 40 49 38 22 8 9 19 16 20 19 24 11 4 13 1 3 22 JLi^r 12 9 1 18 19 29 25 0 5 4 1 0 0 9 13 16 16 21 20 21 23 24 31 32 36 41 36 1 30 40 40 41 47 35 24 19 1 3 4 2 0 2 7 13 11 14 14 14 16 14 17 24 25 34 31 34 1 ■put 6 44 IS 13 10 10 5 15 15 5 13 6 8 5 a 12 18 22 25 81 30 21 9 15 5 14 7 49 «■» «» ** *» mm «* urn «■* mm mi mm mi «* am m m tm 8 7 17 19 21 56 33 9 m 14 28 14 1 28 15 19 30 23 12 11 32 m mm* «► !*» 24 3 4 1 11 0 2 2 5 6 17 8 4 6 3 4 21 41 29 41 35 44 45 44 35 7 12 6 18 2 8 2 6 2 6 9 - mm mm 4 11 15 7 3 8 4 1 11 32 23 37 29 40 38 47 23 46 35 13 y 6 7 3 13 19 29 27 18 14 11 13 6 12 17 10 11 5 9 0 - mm m 34 9 11 7 25 2 9 2 as 20 3 8 28 7 1 7 9 an* mm 7 13 20 10 5 3 5 9 28 19 33 20 45 36 42 mm mm — m 6 Table 57 NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W), AND OMISSIONS (0), ARISING FROM EACH SAMPLE, FOR EACH ITEM OF FORM C, 1935 I Tear Item E PM* 1 2 3 4 5 6 7 8 9 10 11 12 IS 14 15 a b c 16 17 18 19 20 21 22 25 24 25 26 27 28 29 30 31 32 33a b C a 32 44 15 21 51 32 42 43 46 29 3 9 33 35 39 44 37 32 34 26 52 16 10 3 5 1 2 15 2 1 6 0 4 23 34 34 28 32 W 18 6 27 14 19 14 @ 7 4 18 31 34 15 12 10 2 10 16 14 19 11 15 19 6 18 13 7 6 12 10 11 9 8 25 7 5 7 8 lit Years 0 0 0 3 15 0 4 0 0 0 3 16 7 2 3 1 4 3 2 2 5 7 19 21 41 27 56 41 29 36 39 33 41 38 4 9 11 15 10 R W 38 43 29 30 36 28 47 47 50 36 18 13 41 47 50 45 46 40 38 32 37 24 26 10 13 2 3 20 5 8 12 6 7 34 43 45 36 46 12 2 20 13 14 21 2 3 0 12 25 28 9 2 0 4 3 7 11 15 5 17 13 15 18 19 12 8 14 9 7 8 9 14 6 6 12 3 2 Years 0 R W 0 0 1 7 0 1 1 0 0 2 tp 37 47 35 42 33 38 45 48 46 41 30 27 45 47 49 47 42 41 38 43 38 38 30 13 16 3 6 28 10 21 21 13 14 42 46 44 41 48 12 3 15 5 17 11 4 2 4 9 17 21 5 3 1 3 8 9 11 7 7 8 14 27 21 28 10 8 20 9 8 5 9 8 3 5 8 2 9 0 1 0 1 1 3 1 3 8 9 11 25 19 29 35 22 31 33 31 36 34 2 1 1 2 1 0 PPM- 1 0 0 3 0 1 1 0 0 0 3 2 0 0 0 0 0 0 1 0 5 4 6 10 13 19 34 14 20 20 21 32 27 0 1 1 1 0 105 Table 67 (continued) 1 Year Item R w M M 34 35 36a b 37 38 39 40 41 42 43 44 6 9 25 24 20 7 7 1 10 1 4 2 0 12 18 14 12 18 6 O 3 7 ijfr Years *0" 19 17 «•> 18 25 29 37 22 43 46 45 43 2 Years R W 0 R W 0 14 25 25 22 11 5 22 26 22 16 6 8 mm mm 26 9 20 5 19 8 6 5 1 15 27 14 13 12 14 6 2 11 mm 9 14 16 32 19 28 38 43 38 52 12 29 9 23 13 14 15 5 mm 13 17 8 © 9 14 6 3 5 5 21 15 35 18 23 30 32 42 fable 58 NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W), AND OMISSIONS (O), ARISING FROM EACH SAMPLE, FOR EACH ITEM OF FORM A, 1936 1 Year l| Years 2 Years Item K W 0 R W 0 R W 0 1 2 5 4 5 6 7 8 9 10 11 12 13 14 15 1© 88 56 96 93 61 12 43 4 7 22 22 15 17 28 23 26 26 32 24 19 36 0 1 0 0 17 1 0 2 20 6 4 3 2 11 10 11 93 78 95 94 87 91 89 84 64 85 80 81 72 79 78 71 7 21 3 6 10 9 11 15 30 15 19 IB 28 18 18 28 0 95 82 96 98 88 92 95 91 84 82 85 88 89 93 85 93 5 16 4 2 11 8 5 8 15 18 14 12 11 6 14 7 0 2 0 0 1 0 0 1 1 0 1 0 0 1 1 0 77 85 81 52 71 70 71 6© 65 71 53 2 0 3 0 0 1 6 0 1 1 0 3 4 1 Table 58 (continued) 1 Tear lj» Tears 2 Years Item R W 0 R W 0 R W 0 17 IB 19 20 62 92 69 59 41 51 77 38 52 8 45 54 60 22 21 32 27 32 21 39 26 19 11 16 7 3 22 3 1 2 1 0 1 17 26 8 28 31 30 40 21 45 19 59 42 20 35 43 24 33 27 36 27 40 33 34 38 26 16 23 6 21 23 21 11 21 2 10 12 0 3 10 29 9 2 17 29 33 IS 26 7 35 55 35 48 32 52 21 41 47 51 58 77 74 72 76 76 *77 88 79 97 73 82 99 75 75 60 58 89 70 80 31 61 86 82 65 60 75 46 57 36 64 43 26 26 34 27 18 53 27 10 9 14 6 9 31 13 0 22 10 27 39 8 24 8 @4 32 4 15 27 25 22 31 26 42 29 43 37 40 39 34 20 13 22 32 28 21 18 8 5 5 1 3 15 13 3 3 6 12 15 7 10 3 8 15 3 23 17 22 7 14 38 34 27 39 62 34 51 58 63 65 76 83 64 84 97 82 75 78 69 83 80 85 48 82 89 87 77 68 94 63 52 57 79 66 43 46 68 51 24 73 46 25 19 29 18 12 SB 14 2 16 13 16 31 15 16 9 46 18 8 12 18 24 5 51 40 59 20 31 39 48 26 29 25 9 23 40 43 20 16 10 3 2 1 2 12 6 0 2 4 6 6 0 3 1 5 8 1 6 8 4 1 3 18 6 6 20 51 18 31 37 38 51 66 78 59 21 22 23 24 25 26 27 26 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 48 47 48 49 50 I—j JL*5 J Table 59 NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES <W), AND OMISSIONS (0), ARISING FROM EACH SAMPLE, FOR EACH ITEM OF FORM B, 1936 1 Year Item R Wmm* m 1 2 3 4 5 6 92 89 92 81 88 43 68 68 73 27 63 60 55 69 30 85 60 35 73 62 74 64 8 36 46 80 50 52 53 41 46 10 26 15 23 8 24 4 8 3 7 18 12 35 30 8 83 64 30 24 24 28 38 9 33 47 12 33 6 31 82 37 39 9 39 42 32 15 33 64 27 44 45 39 42 32 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 29 30 31 32 33 34 35 36 37 38 li Years 2 Years 0 £ W 0 R W 0 0 3 1 1 0 22 2 24 4 9 7 16 21 3 38 6 *7 18 15 5 21 5 10 27 15 11 11 6 15 44 21 26 47 41 32 53 34 64 93 87 97 85 87 73 79 89 73 42 72 78 76 80 58 89 75 67 82 65 31 65 15 51 78 79 61 65 57 71 58 11 40 26 36 18 35 19 5 10 3 13 12 20 20 7 24 54 28 19 19 19 25 9 25 24 8 33 10 32 77 34 21 10 36 29 26 18 22 72 30 42 45 44 37 35 2 3 0 2 1 7 1 4 3 4 0 3 5 1 17 2 0 9 10 2 9 3 8 15 1 11 3 8 17 11 20 17 30 52 19 38 28 46 99 89 98 93 94 84 92 94 84 59 90 87 88 91 76 97 92 82 84 79 93 80 39 69 84 89 79 76 SI 93 80 32 70 51 67 47 61 51 1 11 2 6 6 12 8 6 16 38 10 13 10 9 15 3 8 14 8 19 3 20 56 29 15 5 19 23 14 5 17 61 24 41 31 39 28 50 0 0 0 1 0 4 0 0 0 3 0 0 2 0 9 0 0 4 8 2 4 0 5 2 1 6 2 1 5 2 3 7 6 8 2 14 11 19 ±68 Table 59 (continued) 1 Year ijr Years 2 Years Iteia B W 0 a W 0 R W 0 39 40 41 42 43 44 45 46 47 48 49 50 5 8 5 4 5 3 6 23 4 2 0 0 46 15 7 28 5 5 4 8 10 6 4 16 49 77 88 68 92 92 90 69 86 92 96 84 12 21 8 10 5 11 5 23 2 8 1 4 51 17 22 26 3 3 12 2 11 1 7 16 37 62 70 64 92 86 83 75 87 91 92 80 32 62 43 36 27 35 51 41 20 15 8 16 51 15 29 36 9 9 17 7 14 6 14 16 17 23 28 28 64 56 52 52 66 79 78 68 1G9 Table 60 V npq u p * 10*000 / FORM A, 1935 Item 1 2 3a b o & 4 3 6 7 a 9 10 11 la 13 14 15 16 17 1© 19 20 21 22 23 24 25 26 27a b 28 29 30 31 52 33 34a b c a 31*064 ' 25*360 30*976 14.100 9*768 9*232 46*040 44*108 21*668 42*924 39*444 42*260 43*272 44* 924 23*436 39*592 41*308 24.988 16.176 42.272 38.828 36.048 36.732 27.936 25.064 25*596 20.396 9,544 9, 544 20*436 17.176 6.692 O O 5.852 O 0 14.176 34.372 22.900 29.900 9*913 7,146 10*046 1.913 5.753 4*653 13.933 19.013 8.966 21.086 19.986 7.113 21.346 23.746 4.553 15*446 18.32 23.833 18*713 23.48 19.146 23.613 22.866 19.886 24.133 20.453 17*086 21.113 16.153 21.533 22.753 18.466 14.553 0 17.286 12,253 5.133 1.913 13.113 16.546 7,946 28,172 10,668 14.000 3.852 O 10.740 25.504 20*304 3.852 42,804 23.212 14.376 24*072 30*376 3*852 22.596 30.000 43.472 33.204 30.000 31.508 39.244 34 «156 38.036 40.396 41.508 42 *804 32.156 35.364 35,284 33.880 36,060 34.348 10*156 38.244 36*848 13,712 7.408 17.636 17*212 17.212 Table 60 (continued) Item 35a 13 c 36 37a b 38a b o 39a b 0 40 41 42 43 44 £1 h i m «» *» 19.938 43.984 46.744 48.076 38.844 88.438 15.753 21.846 83.113 16.313 19.213 19.433 m* a* 46.260 28.936 81.640 84.436 3 *852 28*064 «* * 34.396 85.932 38.544 44.036 88*840 44.176 «* m ** 28.68 15*158 18*613 19.646 1.92 9*453 28*520 42*908 35*064 42*316 13.712 23.516 Table 61 CT| ? 4 ^ npt| FOBM 8 # 1935 Item 1 2 3 4 5 6 7 8 9 10 11 12 15 14 15 16 fi 35.472 28*596 48.132 44*448 42.976 48*696 29*376 30.976 43.584 42.744 35.796 44.000 46.428 41.376 34*464 20*316 Pl§ 39*296 22*228 32*916 39*524 33.524 37.296 31.828 28.096 41.423 37.924 48.384 46*668 40*896 39.200 27.028 3.716 I* 36.376 14.236 55.004 27.676 28.036 41.588 10.760 30.396 29.840 39.500 44.856 46.828 29.748 30.968 29 *444 9.600 tTable 61 (continued) Item 17 18 19 20 21 22 23 24 25a b 26 27 28 29 30 31 32a b G 33 54a b c 35a b c 36 37 38 39 40 41 42 47*096 52.272 O 10.780 23.544 28.464 22.448 10.780 30.064 23.900 7.228 20.848 O 3.896 7.580 40.384 m 36.096 41.584 25.524 25.184 46.584 41.296 44.000 42.868 47.984 29.924 14.516 36*856 3.2 10.784 20.856 20.784 39.272 41.728 85.784 55.708 41.472 40.500 45.856 46.272 40.640 46.804 27.340 41.516 17.064 58.140 21.748 3.732 #*- 35.604 46.152 34.272 3.896 47.316 56.628 41.656 29.256 46.456 14.836, 58*204 36.468 37.896 10.408 14.384 3.896 32.804 0 7.580 7.580 41.256 22.916 33.584 20.724 40.384 6.856 23.296 7.584 39.804 28.300 53.588 23.348 47.596 7.304 28*100 7.304 Table 62 Op* S 4 £ npq FORM C» 1936 Item 1 2 3 4 5 6 7 8 9 10 XI 12 13 14 15a b c 16 17 18 19 20 21 22 23 24 25 2© 27 28 29 30 31 32 53a b c a 34 35 36a P1 40*076 19*900 41*268 45*204 42*020 44*860 26*416 23*820 14*604 46 *496 9*636 23*820 43*91© 41*212 33*624 19*720 57*784 45*568 42* 504 49*648 44*308 36*29© 31*240 10*840 11*248 2.000 7*484 36.980 7.636 2*000 14 *604 0 13.248 46.360 42*624 42*020 43*916 44*060 19*900 29.240 mm *1* 56*000 7*600 45*900 45*600 39*500 49*100 9.400 9.900 0 59*600 45*500 35*600 25*500 11*100 0 17*100 14.400 31*400 35*500 45*500 37*500 49*900 49.500 31.100 57*400 7*600 9*400 47.100 17*500 25.500 55*500 19*500 25.000 41*900 23.500 23 *500 39.600 14.600 39.400 45.900 mm *8 37.656 11*223 41*428 24*228 43 *916 34*628 17.628 5*800 14.628 28.200 44*916 47*056 15.716 11*05© 3.800 11.228 26.516 29.056 35*656 22.228 35.228 33*916 46*428 37*85© 41.716 11.228 20.228 39*65© 30.056 44.656 44.856 33.916 39.428 25.716 11.228 20*916 28.716 7.656 46.516 49.856 mm Table 62 (continued) Item 36b 37 38 39 40 41 42 43 44 Ei h k 44*860 23*820 22*540 3*764 26*172 2*000 12*932 5*764 0 49*500 29.100 46.600 16.600 41.600 99.400 19.100 17*900 S.500 Ha 44*428 36*000 45.516 29*056 43*716 37*228 36*916 36.628 10.716 Table 63 <r/ = H [ npq FOES A, 1936 Item 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 El 10*509 21*271 3*797 6*444 21.171 17.530 12.394 14.430 22.469 20.064 20.418 19.984 21*976 21*438 19.543 25*097 20*571 7.190 19.451 22*894 21.418 21.970 17.193 19.348 M Er 6*380 15*564 4*588 5*535 10*960 7*819 9*501 12.789 21.313 12.627 14.593 14.914 19.594 16*322 16.326 19.529 14*141 980 18.280 17.106 22*335 22.268 9.378 19.329 4*674 13.197 3.659 1.924 9.832 6.797 4.600 7.879 12.449 14* 015 12.183 10.394 9.226 5.668 12.628 6.068 13.000 2.847 13.^92 18.226 17.038 20.396 13.805 15*583 • f,ryA UXm *+ Table 63 (continued) Item 25 26 27 28 29 50 31 32 35 34 35 36 37 58 39 40 41 42 45 44 45 46 47 48 49 50 23.597 S.261 22.110 22.771 22.469 14.836 11.575 17.037 17.159 20.184 13.568 22.218 17.317 10.250 3 *084 8*284 4.849 2.762 15.461 2.275 0 1.5 .5 0 .875 13.777 15.538 20.770 20.823 11.817 14.509 19.396 20.437 16.670 22.866 22.566 20.017 20.591 22.431 18.340 16.830 20.135 18.817 14.318 23.348 19.359 8.860 7.747 11.672 5.514 7.809 20.250 12.571 22.913 14.112 9.350 10.894 17.318 19.610 5.150 21*619 23.134 21.823 15.866 19.834 22.776 24.429 19.692 23.198 18.220 18.967 25.949 16.029 15.291 18.918 14.288 9.698 22.137 Table 64 C~p2 » 21 FORM B, 1936 11 1 2 3 4 5 6 7 8 9 10 11 12 15 14 15 16 17 18 19 00 21 22 25 04 25 26 27 28 29 30 31 32 33 34 36 36 37 38 39 40 41 7 b198 9b681 7 b198 14b603 10w420 22•718 20*093 20*971 18*579 17*818 22.521 22*820 24.195 20.862 17.316 12.420 21.745 21.181 19*387 23.179 18*720 22.554 6.795 19.454 23.526 14.806 25.987 23.798 24*256 00.391 23.141 8.862 14.471 9.281 15.993 5.318 17.691 2.333 4.479 S. 396 3.776 pi& 6*238 10*786 2*689 12*417 10b146 17.884 15.076 9b 451 19.418 22.966 19.678 15.551 17.687 15.543 21.44© 9.586 14.700 21.684 14.307 21.957 14.700 21.613 11.240 23.235 16.551 15.528 23.479 22.413 22.735 17.518 22*849 9.197 17.203 18.543 19.484 12.653 22.165 13.261 9 .494 13.219 6.952 f® .8 9.211 1.918 6.426 5.502 12 .953 7.009 5.486 12.79© 23.086 8.729 11.169 10.362 7.559 16.203 2.875 6.900 14.412 13.323 15.572 6.091 13.552 22.918 19.498 12.453 8*990 16.547 17.714 14.964 5.395 15.571 21.101 19.872 24.217 21.267 24.274 22.407 23.348 20.229 21.512 22.524 Table ©4 (continued) Item 42 43 44 45 4© 47 48 49 50 II 3*526 2*583 2*541 4*712 15.816 2.688 .985 0 0 fit 8.338 4.633 9.310 4.634 17*243 1.884 7.141 .934 3.084 fs 18.457 18.525 20.009 19.845 20.540 14.614 11.336 7.041 12.600 -r J i5 - t* w Tabid 65 COXFARXSOB OF OBSERVED DIFFERENCES BETWEEN GROUPS P^ and P-ji IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IB FORM A, 1935, AMD THE STANDARD ERRORS OF THE DIFFERENCES Item 1 a 3a b e d 4 3 6 7 a 9 10 11 12 13 14 15 16 17 1© 19 20 21 22 23 24 25 26 27a b 28 29 30 31 32 33 34a b o a Observed Difference 12 8 18 6 2 1 27 28 4 12 10 26 18 19 9 14 39 29 18 12 39 28 21 13 29 16 8 41 1© 28 33 17 20 0 22 16 6 6 7 4 11 5*12# (diff * ) 6*4 5*7 6*4 4*0 3*7 3*7 7*7 7*9 5*5 8*0 7*7 7*0 8*0 8*3 3*3 7*4 7*7 7*0 5*9 8*1 7*6 7*7 7*7 6*9 7*0 6*8 6*1 5*5 5*1 6*6 6*3 5.0 3*8 0 4*8 3*5 2*3 4*0 6*9 6*5 6.2 S.B.Td: 1*9 1*4 2*8 1*5 *54 *27 3*5 3*5 *73 1*5 1*3 3*7 2*3 2*3 1.7 1.9 5.1 4.1 3.1 1*6 5*1 3*6 2*7 1.9 4*1 2*4 1*3 7.5 3*1 4*5 5.2 3* 4 5.8 4*6 4*6 2*6 1.5 1*0 *63 1.8 178 Tab!a 66 (continued) Item 85a b c 36 37a b 38a b c 39a b a 40 41 42 43 44 Observed Difference m m S*E*(diff*) ■m m m 10 11 4 24 24 9 6*0 8*1 8 #4 7*6 7*6 6*9 e*» mm a* Difference §*E*(diff*) mm- 1*7 1.4 *48 3*2 3*2 1.3 «* a t 16 1 11 12 0 7 8*3 6*6 6*3 6*6 2*4 6*0 1.9 .15 1*7 1*8 0 1*2 Table 66 COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS Pji and P, IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM A, 1935* AND THE STANDARD ERRORS OF THE DIFFERENCES Item Observed Difference 1 2 3a b 0 d 4 5 6 7 8 9 10 11 12 13 6 2 4 0 4 1 1 12 8 2 14 0 20 23 3 6 S*B.(diff.) 6*2 4*2 4*9 2*4 1*9 3*9 6.3 6.3 3*6 8*0 6*6 4*6 6.7 7*4 2 *9 6*2 S*E*(d; *97 *48 *82 0 2.1 *26 *16 1.9 2.2 *25 2*1 0 3.0 3*1 1*0 .97 -f. S ^ Q m- w Table 66 (continued) item Observed Difference 14 15 16 17 18 19 20 21 22 23 24 25 26 27a b 28 5 9 6 16 5 12 17 19 23 12 18 29 18 28 29 23 10 4 4 10 0 2 7 12 1 30 31 32 33 34a b e a 35a b e 36 37a b 38a b e 39a b c 40 41 42 43 44 *» aa 4* 6 17 16 12 8 11 *» 24 21 1 8 6 5 S*E*(aif£) 7*0 8*2 7*2 7*5 7.1 7 *9 7*6 7 *6 8.0 7.9 7*7 7.5 7.2 7.6 7.5 7.4 7.0 5.2 7*5 7.0 4.3 3.1 5.5 3.8 5.0 £~£S§|§g® S»£«vdirf•/ .71 1*1 *83 8.2 *70 1*5 2.2 2.5 2.9 1.5 2*3 4.0 2.5 3*7 3*9 3.1 1.4 1.3 .55 1.4 0 .65 1*3 2.1 .20 <* ■w* m 7*1 6*9 7.9 7.8 6.9 8.0 ** 7.2 7.6 7.3 7.9 4*0 5.7 ** .85 2.5 2.0 1.5 1.2 1.4 e* 4* 3.3 2.8 *14 1.0 1.5 .88 180 Table 67 COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS P^ and Pg IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM A, 1935, AND THE STANDARD ERROR OF THE' DIFFERENCES Item I ® 3a to 6 0 4 8 € 7 8 9 10 11 18 13 14 15 16 17 19 80 21 22 23 24 25 26 27a b 28 29 30 31 32 34a b q d Observed Difference 6 10 28 0 6 0 28 40 12 10 24 26 38 42 12 80 34 38 24 28 44 40 38 32 52 28 26 70 34 56 02 40 30 4 26 26 6 4 14 8 10 S.E.{diff.) 7.7 6.0 6.7 4,2 3,1 4.8 8,5 8,0 5.1 9.3 7.9 7.5 8.2 8.7 5.2 7.9 8.4 8.3 7.0 8.5 3.4 8.7 8.4 8.1 8.1 8.2 7.9 6.5 6.7 7.5 7.1 6.5 5.9 3.2 6.6 6.1 3.7 4.6 7.2 6.3 6.9 .78 1.7 3.1 1,4 1.9 0 3.3 5.0 2.4 1,1 3.0 3.5 4.'S 4.8 2.3 2.5 4*0 4.6 3.4 3.3 5.2 4«'6 4.5 4.0 6*4 3.4 3.3 10.8 5.1 7.5 8.7 6.2 5,1 1.3 3.9 4.3 1.'6 .87 1.9 1.3 1.4 131 Table ©7 {continued) X test Difference S.B.(dlff*) Difference 8*E*{diff.) b e 56 57 a b 50a b c b e 46 41 42 45 44 16 28 20 12 20 6 2 f. 2*2 8*4 9.2 9*5 8 #2 ©•6 5*5 2.2 1*3 3.9 2*4 8*6 8*5 7*5 8*2 4*2 7*2 4.7 2.4 1*6 2*4 1.4 .28 Table 68 COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS Px and Px j IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM B* 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES 1 teia Observed Difference 1 2 3 4 5 6 2 2 03 32 8 18 2 0 24 32 18 0 14 0 4 12 8 9 10 11 12 13 14 15 1© S.E.{dlff*) 8*6 7.1 9.0 9.2 8.7 9.3 7.8 7.7 9.2 9.0 9 #1 9.5 9.3 9.0 7.8 4.9 uxx x or1 SZSTTHi .23 .28 2 .9 3.5 .92 1.9 •26 0 2.6 3.6 2.0 0 1.5 0 .51 2.4 S .S S Table 68 (continued) Item 17 18 19 SO 21 22 23 24 25a b 26 27 28 29 30 31 32a b o 33 34a b c 38a b c 36 37 38 39 40 41 42 Observed Difference 24 18 16 10 22 12 22 30 26 6 2 12 0 8 6 22 ** m 14 16 38 16 ** 22 8 16 10 14 4 12 0 S*E.(diff*) 9*1 8*6 5*1 6.0 8*4 8*4 8*2 7*3 8*8 7*3 4*7 7.6 1*8 3*8 5*3 7*8 PMtwmse SeETTWIfFr) 2*6 2.1 3*1 1*7 2*6 1*4 2*7 4*1 3*0 .82 .43 1.6 0 .53 1.1 2.8 *•* dr #* d» «*» *» 9.1 9 .1 8.7 5.8 1.5 1*8 4*4 2.8 CM dr dr 8*9 5*8 6*9 5.0 8.6 2*6 5*6 3.9 m 2*5 1.4 2.3 2*0 1*6 1*5 2.1 0 133 Table 69 COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS P-ji and Pg IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM B, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES Item 1 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 23a b 26 27 28 29 30 31 32a b c 33 34a b 0 35a b c 36 Observed Difference 2 3 0 a 4 4 14 0 12 2 16 4 16 12 2 4 6 4 0 6 2 12 10 4 12 20 10 4 8 22 2 10 *■ m mm 4 20 4 8 3 «E.(d 8.7 6.0 8.2 8.2 7.8 8*9 6.5 7.0 8.4 8.8 9.7 9.7 8.4 8*4 7.5 3.6 8.7 9.1 7.2 7.8 9.4 9.0 9.5 9.4 9.4 8.8 6.5 8.9 4.5 7.0 6.5 4.9 p 1 Difference •23 1*3 0 •98 •51 •45 £•2 0 1*4 .23 1.6 •41 1.9 1.4 .27 1.1 •69 •44 0 •77 •21 1.3 1.1 .43 1.3 2*3 1.5 *45 1.8 3.1 •31 2.0 m mm mm 9.7 7.1 8.9 8.1 4* mm m mm 2 4 9.0 7.2 .41 2.8 .45 *99 .22 .36 134 Table 69 (continued} Item 3? 38 39 40 41 42 Observed Difference S.E.(dlff.) a 2 14 O 2 0 8.2 6.6 9.4 3.8 7.2 3.9 .24 .30 1.5 0 .28 O Table 70 COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS. and Pg IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM B, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES Item 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25& b 26 27 Observed Difference 4 10 26 40 12 14 12 0 36 30 34 4 30 12 6 8 18 22 16 16 20 24 32 34 38 26 12 16 s*E*{diff•) 8*5 6*5 9*1 8*5 8.4 9*5 6*3 7*8 8*6 9*1 8.9 9*5 8«?7 8*5 8*0 5.5 9*3 8*6 5*1 6*8 8.1 8*3 8*3 7*6 8*4 8*4 5.9 7.9 Difference S.E*(dI: *47 1 *5 2.9 4*7 1*4 1*5 1.9 0 4*2 5.3 3*8 *42 3.4 1.4 *75 1*5 1.9 2*6 3*1 2*4 2*5 2*9 3 .9 4*5 4*5 3 *1 2.0 2.0 £85 3?&ble 70 (continued) s,E.(aiff.) Item Observed Difference 28 29 30 31 32a b 0 33 34a b c 35a b e 36 37 38 39 40 41 42 8 24 8 32 4*1 6*5 S.4 6.6 2*0 3*7 1*5 4*8 - to* w 18 56 42 24 * to* 20 18 14 12 28 4 14 0 «* to* to« to* 9*1 7*8 8.5 6*4 2.0 4*6 4.9 3*8 to* to* 8.8 6*8 6*9 5.2 9.0 2*7 6*0 3*9 to* 2.3 1.9 2*0 2*3 3*1 1*5 2*3 0 Table 71 COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROOTS P-j_ and Pj| IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM C, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES Item Observed Difference 1 2 3 4 5 6 7 8 9 10 11 12 8 20 IS 10 8 10 8 8 14 30 s*E*(aiff.) 8.7 5*2 9*3 9*5 9*0 9*7 6*0 5.8 3.8 9*3 7.4 Difference 1*4 1*5 3*0 1*9 1.1 .82 1*7 1*4 2.1 1*5 4.1 1LSS lable 71 (continued) Item 12 13 14 15a t> e 16 17 18 19 20 21 22 23 24 25 m 27 28 29 30 31 32 33a b c a 34 35 36a b 37 38 39 40 41 42 43 44 Observed Difference S.H.(dif£.) PM£er|R.go 8 16 24 22 2 18 16 8 12 10 16 32 14 16 2 8 10 6 14 12 18 6 22 IB IB 16 28 16 28 7*7 8*3 7*2 5*8 6.1 7*8 8*8 3*8 9*3 9*0 9*3 9*0 8*5 7*0 3.1 4*1 9*2 5*0 5*2 7*1 4* 4 8*0 9*4 3*1 8*1 9.1 7.7 7.7 8*7 m •» im 9.7 7*3 8*3 4*6 8.2 4*9 5.7 4*9 1.9 1.2 12 4 26 S IB 14 4 6 2 1*0 1*9 3.3 3.8 .35 2*5 1*8 *91 1.2 1.1 1*7 3.6 2.2 2.3 .65 .49 1.1 1.2 2.7 1*7 2*7 1.0 2.2 2.2 2.2 1*8 3.6 2.1 3.2 esq .<oO 3.1 1*8 2.2 2.9 *70 1*2 1*1 137 Table 72 COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS Pji and Pg IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM C, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES Item 1 2 5 4 5 6 7 3 9 10 11 12 13 11 15a "cb 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 53a b cs a 34 35 Observed Difference 2 2 12 24 6 20 4 2 a 10 24 28 8 0 2 4 82 0 22 2 28 a6 6 2 6 16 10 26 IS 14 14 166 2 10 4 16 6 S.2.(diff.) 8*6 4*3 9.3 8.4 9*1 9*2 5*2 4.0 5*8 9*5 7*4 7*7 6.4 4*7 1.9 5*3 07*8«4 8.4 8*2 8.5 9.2 9*8 8*3 8*9 4.3 5.4 9*3 6 *9 8*4 9.0 7*3 7 *9 8*2 5*9 6*7 8.3 4*7 9*3 9*8 ^ § " ~fdfe?^') *23 ,47 1,3 2.9 .66 2.2 ,77 ,50 2*1 1*1 3*2 3*6 1*3 0 1*1 *75 1*3 *26 0 2*7 *24 3*0 .82 *72 *67 .47 1*1 1*7 1*4 3*1 2.0 1.9 1*8 2.0 1*0 .30 1*2 .85 1.7 .61 138 Table 72 (continued) Item 36a to 37 38 39 40 41 42 43 44 Observed Difference 12 6 18 8 8 10 16 20 4 S.E*(diff*) 9*7 8*1 9*6 6*8 9*2 7*7 7*5 7*4 3*8 Difference s.S.fdiff*;) 1*2 *74 1*9 1*2 .87 1*3 2*1 2*7 1*1 Table 73 COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS Pj_ and Pg IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IK FORM C, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES Item 1 2 3 4 3 6 7 8 9 10 11 12 13 14 15a to c 16 17 18 19 20 Observed Difference 10 6 40 42 4 12 6 10 0 24 54 36 24 24 20 6 10 18 8 34 12 44 3*K*(diff*) 8*8 5*6 9*1 8*3 9*3 8*9 6*6 5*4 5*4 8*6 7*4 8*4 7*7 7*2 6*1 5*6 8*0 8*6 8*8 8.5 8*9 8*4 iiirrerei &# & •(dl: 1#1 1 *1 4* 4 5*1 *43 1*5 *91 1*9 0 2*8 7*3 4*3 5*1 5*3 3.5 1.1 1 *e 2 *1 *91 4*0 1 *3 5 *2 139 T a b le Item 21 22 23 24 25 26 27 28 29 30 31 32 33a b © & 34 35 36a b 37 38 39 40 41 42 43 44 75 ( c o n tin u e d ) Observed Difference 40 20 22 4 5 26 16 40 30 26 20 38 24 20 20 32 32 34 s,E.(diff.) 8,8 7.0 7.3 3.6 5.3 8,8 6.1 6,8 7,7 5,8 7,3 8,5 7,3 7,9 8,5 7,2 8.1 8,9 m* 24 10 44 10 26 24 20 26 0 §-£--f£vB£9o • \fixfr•/\ 4.5 2.9 3,0 1,1 1.5 3.0 2,6 5.9 3,9 4,5 2.7 4.5 3.3 2.5 3.1 4.4 4.0 3.8 *» 9*4 ^*7 8.2 5,7 8.4 6.3 7.1 6.5 3,3 2*6 1*3 5.4 2.8 3.1 3.8 2.8 4.0 1.8 190 Table 74 COMPARISON OP OBSERVED DXPFEE.BKCES BETWEEN GROUPS and IH PEHCKNTAOE OF CORRECT RESPONSES TO EACH ITEM IB FORM A, 1936, Af W THE STANDARD ERRORS OF THE DIFFERENCES Item 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 35 30 37 58 40 41 Obaerred Difference 5 22 1 1 26 14 4 5 12 14 10 10 6 14 7 18 20 7 0 16 19 7 12 52 28 23 16 32 22 43 39 43 19 25 15 25 17 6 15 18 20 S.E*(diff*) 4*1 6*1 2*9 3*5 5*7 5*0 4*7 5*2 6*6 5*7 5*9 5,9 0,4 6,1 6*0 6*5 5*9 2*9 6,1 6*3 6*6 6*7 5*2 8*2 6*3 5*1 6*6 5*9 6*1 5*9 5*7 5*8 6*3 6*5 6*8 6*5 6*3 b *3 5*0 5*3 4*9 Difference S.E.fdIff*) 1*2 3*0 .34 *29 4*6 2*8 *85 *58 1.8 2,5 1.7 1.7 ,94 2.3 1*2 2.8 3*4 2.4 .98 2.8 2*9 1*0 2*8 5*2 4.4 4*8 2*4 5*4 3*6 7*3 6.8 7*4 3*0 3*8 2.6 3*8 2*7 1.1 3*0 3.4 4.1 19:1 T&ble 74 (contlnuod) Item Observed Difference 42 43 44 45 46 47 48 49 50 15 31 24 9 7 18 6 8 14 S.E,(diff*) 4.1 6*8 4.7 3*0 3*0 3.5 2*5 2*9 5.8 Difference o *hi*{(3if f .) 3.7 4*9 5*1 3.0 2.3 3.7 2*6 2*8 2,4 Table 75 COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS P^.3, and Pg IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM A, 1930, AND THE STANDARD ERRORS OF THE DIFFERENCES Item Observed Difference 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2 4 1 4 1 1 6 7 20 5 5 7 17 14 7 22 2 2 7 0 18 11 6 S.E.(diff,) 3.3 5*4 2.9 2.7 4.6 3*8 3.8 4, 5 5.8 5*2 5,2 5,0 5.4 4,7 5*4 5.1 5,2 2,0 5.7 5,9 6,3 6,6 4,8 -uirrere* S.E* {dlj .61 .74 *34 1*5 *22 ,26 1,6 1,6 3*4 *53 .96 1.4 3*1 3.0 1.3 4*3 ♦38 1.0 1*2 0 2.9 1.7 1 *3 192 Table 75 (continued) Item 04 25 2$ 27 28 88 S0 31 m 33 34 33 3© 37 m 39 40 4% 42 43 44 45 46 47 48 40 50 Observed Difference 10 5 17 01 3 5 10 8 19 17 5 01 15 23 18 20 34 24 © 00 It 13 10 13 12 5 7 S*E*(diff.) 3*9 5*3 ©*© 5*9 4*6 3*0 6*1 6*3 4*7 6*7 6.8 6*3 6*0 6*5 6*4 6*4 6*3 ©*S 5*7 8*5 ©*© 5*0 4*8 5*5 4.4 4*2 6.5 ^flCSXSl S.E,{dt: 1.7 *94 2*6 3*6 *63 1*0 9*0 1*3 4*0 0*5 *74 3*2 2.5 3*5 2.8 3.1 5.4 3*7 1*1 3*1 2.9 2.6 2.1 2.7 2.7 *71 1*1 Table 76 COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS and Pg IS PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM A* 1936, AND THE STANDARD ERRORS OF THE DIFFERENCES Item Observe© Difference 1 9 7 96 0 8 4 8 © 3 27 15 s.K„(diff.) 3,9 5.9 2.7 2*9 5*6 4.0 Difference 1*8 4.4 0 1*7 4*8 3*1 193 Table 70 (continued) Item 7 a 9 10 11 12 IB 14 IB 16 17 18 19 20 2% 22 2B 24 25 20 27 28 29 50 31 32 33 34 50 38 37 38 39 40 41 42 43 44 4S 40 47 48 49 50 Observed Difference 10 10 32 11 15 17 23 28 14 40 22 5 13 16 37 18 6 42 S3 40 37 35 27 55 47 62 36 20 36 40 40 24 38 52 44 21 81 43 22 17 28 18 11 21 $*B.(aiff.) 4.1 4*7 5*9 8*8 5*7 5.5 5.6 8*8 5*7 5*4 8*8 3.2 5*8 6*4 6*2 6*5 5*6 5*9 6*0 5*3 6.0 5*7 5*8 k *f » O. 5*6 4*7 6*2 6*6 8*9 6.2 '6.1 5*7 8*7 5.8 5*3 4*6 8*9 5*1 4.0 4.1 4*4 3*8 3.3 6.0 Difference STKfdiff*) 2*4 2*1 8*4 1*9 8*8 3*1 4*1 6*4 2*5 7*4 3*8 1*6 2*8 2*5 6*0 8*8 1*1 7*1 5*0 7*5 6.2 6*1 4*7 9*6 8*4 13*2 5*8 3*0 6*1 0*5 0*6 4*2 0*1 9*8 8*3 4*0 8*0 8*4 5*5 4.1 6*4 4*7 3*3 3*5 194 Table 77 COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS P^ and P ^ IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN FORM B* 1936* AND THE STANDARD ERRORS OF THE DIFFERENCES Item Observed Difference % 1 2 3 4 B 6 7 5 4 1 30 a9 2 11 21 10 11 12 13 14 15 16 17 ia 19 20 21 22 23 24 25 26 27 28 29 30 31 32 35 54 35 36 87 38 59 40 9 18 21 11 28 4 15 82 8 5 7 1 7 IS 1 11 13 4 30 12 1 11 15 10 11 IS 7 13 •*b *{diff*) 3 #7 4*5 5*1 5*2 4*5 6*4 5*9 5*5 6*2 6*4 6.5 6*2 6*5 6*0 6*2 4*7 6*0 6*5 5*8 6*7 5.8 6.6 4*2 6*5 6*3 5*5 6.9 6*8 6*9 6*2 6*8 4*2 5*6 5*3 6*0 4*2 6*3 5*9 3*7 4*5 >61100 S.E.tdirn) *27 *44 1*6 •77 •22 4*7 1*9 8*8 0 2*3 1*4 2*9 3.2 1.8 4*5 *85 2*5 4*9 1*6 *43 1*2 *15 1*7 2*3 5*1 *18 1*6 1*9 *58 4*8 1*8 *24 2*5 2*1 2*2 2*4 1*7 5*8 1*9 3*0 195 Table 77 (continued) Item 41 42 43 44 45 46 47 48 49 50 Difference s. hi.(diff.) Observed. Difference 3 6 2 8 X 0 2 6 1 4 *91 1*8 *74 3*3 3*4 2.7 3*4 3*1 5*7 2*1 2*9 1*0 1*8 *32 0 *95 2*1 1*0 2*2 Table 78 COMPARISON OP OBSERVED EtPFERE/X-ES BE W E E N GROUPS Px£ and Pg IN PERCENTAGE OF CORRECT ilSSPOHSES TO EACH ITEM IN FORM B, 1936, AND THE STANDARD ERRORS OP THE DIFFEKEtfCES Item 12 3 4 a6 7 8 9 10 11 12 13 14 15 16 17 18 10 20 21 22 Observed Difference 62 18 7 11 13 5 11 17 18 9 12 11 18 8 17 15 2 14 12 15 S.E*(dlff 2*7 4.6 2*1 4.3 3*9 5*6 4*7 3*9 5*7 6*8 5*3 5*2 5*3 4 *0 6,1 3*5 4*6 6*0 5,3 6*1 4*6 5,9 *J Differs TiiTm 2*2 *44 *48 1*9 1*8 2,0 2*8 1*3 1*9 2,5 3*4 1*7 2,5 2.3 3*0 2*3 3*7 2*5 *38 2*3 2*6 2*5 -a Table 79 {continued} Item 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 £7 28 29 30 31 32 33 34 35 36 37 38 89 40 41 42 43 44 45 46 47 48 49 50 Observed lifference 41 2 4 26 11 82 27 rvr/ rv 5 t> 22 46 12 32 47 11 17 19 16 51 «.>o 38 9 29 24 28 52 84 22 4-4 36 44 89 37 47 27 54 58 52 24 52 25 18 16 15 8 16 S.lMdtff.) 6 ,0 5 *2 5*1 5*6 6 *4 5 *6 5 *8 5 *9 5*3 5*8 8*9 5*4 6 *0 5*7 6*2 5 *0 6*0 5*5 6 .2 6*0 4 .9 6 .4 6*4 6*3 5*1 6 .2 5*5 5*9 ■5*8 6 ,1 5 *4 6*3 5 .1 5*0 5 .2 5*1 4 *7 4 *6 4 .7 5*0 8*0 4*2 3 *5 2*7 3*5 Mlgerence 3 * Aii* (d 12i?V) 6*8 4 .6 b « J2 .0 5 ,0 4*8 4 .7 5 ,6 4 .2 7 .9 3 ,1 5 .9 7 .8 1 .9 2 .7 3 .8 2 .7 5 *6 5 .3 8 *5 1*8 4 ,5 3 .8 4 ,4 1 0 .2 5 ,5 4, 0 7 ,5 6 .2 7 ,2 7 ,2 5 ,8 9 .2 5 *4 1 0 .4 7 .5 6 ,3 5 .2 6 ,3 5 ,0 3 .0 3 *8 3 .7 3*0 4 *6 ±96 Table 78 (continued) Ite m Differen.ee O b s e rv e d 23 24 05 26 27 28 29 30 31 32 55 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 24 18 6 10 IS 11 24 02 02 21 30 25 31 29 26 32 00 41 55 26 22 24 20 18 18 7 7 12 C O M P A R IS O N IN FORM Ite m 1 2 3 OF OBSERVED PERCENTAGE B, 1936, OF AND O b s e rv e d t* **>*> \ tauf., Difference 5 - [ Z 5 *8 6*5 5 *4 5 *0 6*3 6*3 6*1 4 *8 6 .2 5*5 6 ,1 6 ,5 6 ,4 6 .1 6 .7 6 ,1 5 ,5 5 *9 5*4 6*2 4*8 5*4 4*9 6 *1 4*1 4*3 2 ,8 4*0 T a b le IB S .E * T IE - D iffe re n c e 7 O 6 _ . ) 4 .1 2,8 1*1 2*0 2*9 1*7 3*9 4*6 3*5 3.8 4*9 3.8 4.8 4.8 3.9 5 .2 3.6 6*9 6.5 5*0 4*6 4*4 5.3 3*0 4 .4 1*6 2*5 3.0 BETWDDN G R O U PS RESPONSES STANDARD _ 79 D li< F E K K K C S S CORRECT - T _ ERRORS TO EACH OF THE and Pg IT E M D IF F E R E N C E S S * E * (d iff* ) ) 2 .8 4*3 3*0 2 .5 0 2*0 4 12 4*6 2.6 5 6 4 .0 1*5

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