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```AP Statistics
7.2 Notes
Sampling Distribution of a Sample Proportion
count of successes in sample X

size of sample
n
We know that the mean and standard deviation of a binomial random variable X are  X  np and
The sample proportion of successes is closely related to X. pˆ 
1
X 1
   X , we’re just multiplying the random variable X by a constant to
n
n n
get the random variable p̂ . Recall that multiplying by a constant multiplies both the mean and the standard
deviation of the new random variable by that constant. Therefore,
1
 pˆ   np   p (confirming that p̂ is an unbiased estimator of p )
n
 X  np 1  p  . Since pˆ 
np 1  p 
p 1  p 
1
np 1  p  

(as sample size increases, spread decreases)
2
n
n
n
That takes care of center and spread. What about shape? Multiplying a random variable by a constant
doesn’t change the shape of the probability distribution. So the sampling distribution of p̂ will have the same
 pˆ 
shape as the distribution of the binomial random variable X. Therefore, whenever np and n 1  p  are at
least 10, a Normal distribution can be used to approximate the sampling distribution of p̂ .
Sampling Distribution of a Sample Proportion
Choose an SS of size n from a population of size N with proportion p of successes. Let p̂ be the sample
proportion of successes. Then:
 The mean of the sampling distribution of p̂ is  p̂  p .

The standard deviation of the sampling distribution of p̂ is  pˆ 
p 1  p 
as long as the 10%
n
1
N.
10
As n increases, the sampling distribution of p̂ becomes approximately Normal. Before you perform
condition is satisfied: n 

Normal calculations, check that the Normal condition is satisfied np  10 and n 1  p   10.
Example
About 75% of young adult Internet users (ages 18 to 29) watch online video. Suppose that a sample survey
contacts an SRS of 1000 young adult Internet users and calculates the proportion p̂ in this sample who watch
online video.
1. What is the mean of the sampling distribution of p̂ ? Explain.
2. Find the standard deviation of the sampling distribution of p̂ . Check that the 10% condition is met.
3. Is the sampling distribution of p̂ approximately Normal? Check that the Normal condition is met.
4. If the sample size were 9000 rather than 1000, how would this change the sampling distribution of p̂ ?
```
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