If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Sampling distribution of a sample proportion example

Here's the type of problem you might see on the AP Statistics exam where you have to use the sampling distribution of a sample proportion.

## Example: Proportions in polling results

According to the US Census Bureau's American Community Survey, 87, percent of Americans over the age of 25 have earned a high school diploma. Suppose we are going to take a random sample of 200 Americans in this age group and calculate what proportion of the sample has a high school diploma.
What is the probability that the proportion of people in the sample with a high school diploma is less than 85, percent?
Let's solve this problem by breaking it down into smaller parts.

### Part 1: Establish normality

Note: The sampling distribution of a sample proportion p, with, hat, on top is approximately normal as long as the expected number of successes and failures are both at least 10.
Question A (Part 1)
What is the expected number of people in the sample with a high school diploma?
people

Question B (Part 1)
What is the expected number of people in the sample without a high school diploma?
people

Question C (Part 1)
Is the sampling distribution of p, with, hat, on top approximately normal?

### Part 2: Find the mean and standard deviation of the sampling distribution

The sampling distribution of a sample proportion p, with, hat, on top has:
\begin{aligned} \mu_{\hat p}&=p \\\\ \sigma_{\hat p}&=\sqrt{\dfrac{p(1-p)}{n}} \end{aligned}
Note: For this standard deviation formula to be accurate, our sample size needs to be 10, percent or less of the population so we can assume independence.
Question A (Part 2)
What is the mean of the sampling distribution of p, with, hat, on top?
mu, start subscript, p, with, hat, on top, end subscript, equals

Question B (Part 2)
What is the standard deviation of the sampling distribution of p, with, hat, on top?