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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.
The Practice of Statistics, 5th Edition
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ˆ
The Sampling Distribution of p
How good is the statistic
sampling distributi on of
The Practice of Statistics, 5th Edition
pˆ as an estimate of the parameter p? The
pˆ answers this question.
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Applet:
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Applet:
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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.
There is an important connection between th e sample proportion pˆ and
the number of " successes" X in the sample.
pˆ =
count of successes in sample X
=
size of sample
n
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ˆ
The Sampling Distribution of p
In Ch. 6, we learned that the mean and standard deviation of a
binomial random variable X are
mX = np
sX = np(1 - p)
Since pˆ  X / n  (1 / n)  X , we are just multiplyin g the random variable X
by a constant (1 / n) to get the random variable pˆ . Since multiplyin g by a constant
affects the center and spread of a random variable by multiplyin g,
1
m pˆ = (np) = p
n
pˆ is an unbiased estimator or p
1
np(1 - p)
s pˆ =
np(1 - p) =
=
2
n
n
p(1 - p)
n
As sample size increases, the spread decreases.
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ˆ
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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Note:
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On Your Own:
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Ex: Going to College
A polling organization asks an SRS of 1500 first-year college students
how far away their home is. Suppose that 35% of all first-year students
attend college within 50 miles of home.
PROBLEM: Find the probability that the random sample of 1500
students will give a result within 2 percentage points of this true
value. Show your work.
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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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