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Statistics 101
Chapter 9 : Sampling Distributions
Section 1
Sampling Distributions
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Parameter – a number that describes the
population. Fixed number, but in practice we
do not know its value because we cannot
examine the entire population.
Statistic – a number that describes a
sample. The value of a statistic is known
when we have taken a sample, but it can
change from sample to sample.
Difference between a parameter and a
statistic
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Parameter –come from population
Statistics –come from samples
Example 9.1: Making Money
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The mean income of the sample of
households contacted by the Current
Population Survey was $57,045. The
number 57,045 is a statistic because it
describes this one CPS sample. The
population that the poll wants to draw
conclusions about is all 106 million U.S.
households. The parameter of interest is the
mean income of all of these households. We
do not know the value of this parameter.
Sampling Variability
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What would happen if we took many
samples?
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Take a large number of samples from the same
population
Calculate the sample mean (x) or sample proportion p
for each sample
Make a histogram of the values of X or p
Examine the distribution displayed
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WHY WOULD WE DO THESE THINGS??
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Example 9.3 Baggage Check!
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http://statweb.calpoly.edu/chance/applets/app
lets.html
Simulation is powerful for studying chance. It
is faster to use table B then to actually draw
repeated SRS’s
The distribution of the sample proportion from SRSs of size 100 drawn from
population with population proportion p = 0.7. The histogram shows the
results of drawing 1000 SRSs.
Sampling Distribution
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The distribution of values taken by the
statistic in all possible samples of the same
size from the same population.
The population used to construct the random digits table (Table B) can be
described by the probability distribution.
Values of mean in all possible samples of two random digits.
The sampling distribution of x for samples of size n=2.
Bias of a statistic
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A statistic used to estimate a parameter is
unbiased if the mean of its sampling
distribution is equal to the true value of the
parameter being estimated.
Variability of a Statistic
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The spread is determined by the sampling
design and the size of the sample. Larger
samples give smaller spread
The spread of the sampling distribution is
approximately the same for any population
size as long as the population is much larger
than the sample (10 times or greater.)
Summary
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Parameters describe the population
Statistics describe the sample
A statistic from a probability sample or
randomized experiment has a sampling
distribution
Bias and variability come in when a statistic is
used as an estimator of a parameter.
Properly chosen statistics do not suffer from
bias or variability.
Bias and variability – a visual
Exercises
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2, 8, 9abcde, 10, 17