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# Introduction to Sampling Distributions

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```Chapter 7
Introduction to
Sampling Distributions
Understandable Statistics
Ninth Edition
By Brase and Brase
Prepared by Yixun Shi
Bloomsburg University of Pennsylvania
Terms, Statistics & Parameters
вЂў Terms: Population, Sample, Parameter,
Statistics
7|2
Why Sample?
вЂў At times, weвЂ™d like to know something about the
population, but because our time, resources,
and efforts are limited, we can take a sample to
7|3
Types of Inference
1) Estimation: We estimate the value of a
population parameter.
2) Testing: We formulate a decision about a
population parameter.
3) Regression: We make predictions about the
value of a statistical variable.
7|4
Sampling Distributions
вЂў To evaluate the reliability of our inference, we
need to know about the probability distribution
of the statistic we are using.
вЂў Typically, we are interested in the sampling
distributions for sample means and sample
proportions.
7|5
The Central Limit Theorem (Normal)
вЂў If x is a random variable with a normal
distribution, mean = Вµ, and standard deviation =
Пѓ, then the following holds for any sample size:
7|6
The Standard Error
вЂў The standard error is just another name for the
standard deviation of the sampling distribution.
7|7
The Central Limit Theorem
(Any Distribution)
вЂў If a random variable has any distribution with
mean = Вµ and standard deviation = Пѓ, the
sampling distribution of x will approach a
normal distribution with mean = Вµ and standard
deviation = пЃі n as n increases without limit.
7|8
Sample Size Considerations
вЂў For the Central Limit Theorem (CLT) to be
applicable:
вЂ“ If the x distribution is symmetric or
reasonably symmetric, n в‰Ґ 30 should suffice.
вЂ“ If the x distribution is highly skewed or
unusual, even larger sample sizes will be
required.
вЂ“ If possible, make a graph to visualize how
the sampling distribution is behaving.
7|9
Critical Thinking
вЂў Bias вЂ“ A sample statistic is unbiased if the
mean of its sampling distribution equals the
value of the parameter being estimated.
вЂў Variability вЂ“ The spread of the sampling
distribution indicates the variability of the
statistic.
7 | 10
Sampling Distributions for Proportions pЛ† пЂЅ r
n
вЂў If np > 5 and nq > 5, then pЛ† can be
approximated by a normal variable with mean
and standard deviation пЃ­ pЛ† пЂЅ p
and
pq
пЃі pЛ† пЂЅ
n
7 | 11
Continuity Corrections
вЂў Since pЛ† is discrete, but x is continuous, we
have to make a continuity correction.
вЂў For small n, the correction is meaningful.
7 | 12
Control Charts for Proportions
вЂў Used to examine an attribute or quality of an
observation (rather than a measurement).
вЂў We select a fixed sample size, n, at fixed time
intervals, and determine the sample proportions
at each interval.
вЂў We then use the normal approximation of the
sample proportion to determine the control
limits.
7 | 13
P-Chart Example