Transcript Sampling Distribution of a Sample Proportion

CHAPTER 7
Sampling
Distributions
7.2
Sample Proportions
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Sample Proportions
Learning Objectives
After this section, you should be able to:
 FIND the mean and standard deviation of the sampling distribution
of a sample proportion. CHECK the 10% condition before
calculating the standard deviation of the sample proportions.
 DETERMINE if the sampling distribution of sample proportions is
approximately Normal.
 If appropriate, use a Normal distribution to CALCULATE
probabilities involving a sample proportion.
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The Sampling Distribution of p
How good is the statistic pˆ as an estimate of the parameter p? The
sampling distribution of pˆ answers this question.
Consider the approximate sampling distributions generated by a
simulation in which SRSs of Reese’s Pieces are drawn from a
population whose proportion of orange candies is either 0.45 or 0.15.
What do you notice about the shape, center, and spread of each?
http://www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=1
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ˆ
The Sampling Distribution of p
What did you notice about the shape, center, and spread of each
sampling distribution?
Shape : In some cases, the sampling distribution of pˆ can be
approximated by a Normal curve. This seems to depend on both the
sample size n and the population proportion p.
Center : The mean of the distribution is m pˆ = p. This makes sense
because the sample proportion pˆ is an unbiased estimator of p.
Spread : For a specific value of p , the standard deviation s pˆ gets
smaller as n gets larger. The value of s pˆ depends on both n and p.
 pˆ 
The Practice of Statistics, 5th Edition
p(1  p)
n
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The Sampling Distribution of p
Sampling Distribution of a Sample Proportion
Choose an SRS of size n from a population of size N with proportion p
of successes. Let pˆ be the sample proportion of successes. Then:
The mean of the sampling distribution of pˆ is m pˆ = p
The standard deviation of the sampling distribution of pˆ is
p(1- p)
s pˆ =
n
as long as the 10% condition is satisfied : n £ (1/10)N.
As n increases, the sampling distribution becomes approximately
Normal. Before you perform Normal calculations, check that the
Normal condition is satisfied: np ≥ 10 and n(1 – p) ≥ 10.
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ˆ
The Sampling Distribution of p
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• CYU on p.445
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Sample Proportions
Section Summary
In this section, we learned how to…
 FIND the mean and standard deviation of the sampling distribution of
a sample proportion. CHECK the 10% condition before calculating the
standard deviation of the sample proportions.
 DETERMINE if the sampling distribution of sample proportions is
approximately Normal.
 If appropriate, use a Normal distribution to CALCULATE probabilities
involving a sample proportion.
The Practice of Statistics, 5th Edition
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