AP Statistics · Exam guide
AP Statistics Inference for Proportions Explained: Intervals and Tests with a Worked Example
Questions on a single proportion are a standard on the AP Statistics free-response section. Graders look for the same pieces every time: the parameter, the procedure, the conditions, the calculation, and a conclusion in context. The data below are made up for practice.
A made-up survey
A pollster takes a random sample of 400 voters from a large city. In the sample, 232 support a proposed park bond.
The confidence interval
| Step | Work |
|---|---|
| State | p is the proportion of all voters in the city who support the bond. Estimate it with 95 percent confidence. |
| Plan | One-sample z interval for a proportion. Random sample. The sample is less than 10 percent of the voters in a large city. n times p-hat is 232 and n times (1 - p-hat) is 168, both at least 10. |
| Do | p-hat = 232 / 400 = 0.58. The standard error is the square root of 0.58 times 0.42 divided by 400, about 0.0247. The margin of error is 1.96 times 0.0247, about 0.048. The interval is 0.58 plus or minus 0.048, or about 0.532 to 0.628. |
| Conclude | We are 95 percent confident that the interval from 0.532 to 0.628 captures the true proportion of all voters in the city who support the bond. |
The significance test
Is there convincing evidence that more than half of all voters support the bond?
| Step | Work |
|---|---|
| State | H0: p = 0.50. Ha: p > 0.50, where p is the proportion of all voters in the city who support the bond. |
| Plan | One-sample z test for a proportion. The same random and 10 percent conditions. Large counts: n times 0.50 is 200 and n times 0.50 is 200, both at least 10 (use the hypothesized value for a test). |
| Do | z = (0.58 - 0.50) / the square root of (0.50 times 0.50 / 400) = 0.08 / 0.025 = 3.2. The p-value is about 0.0007. |
| Conclude | The p-value of 0.0007 is less than 0.05, so we reject H0. There is convincing evidence that more than half of all voters in the city support the bond. |
Writing the conclusion
It never says what p is, and it does not link the decision to the p-value.
It uses the p-value to justify the decision and states the conclusion in context.
Common mistakes
- Checking the large-counts condition with p-hat in a test, where the hypothesized p belongs.
- Writing "the sample is random" without saying whether the problem tells you so.
- Reporting a p-value without comparing it to the significance level.
- Describing the interval as the probability that p is in it. The 95 percent describes the method.
- Leaving out the population in the conclusion.
Frequently asked questions
How do I check conditions for a proportion?
Random sample or assignment, the sample is less than 10 percent of the population, and at least 10 successes and 10 failures.
Which proportion goes in the standard error for a test?
The hypothesized value p0. For an interval, use the sample proportion.
What does the p-value mean?
It is the probability of getting a result at least as extreme as the sample if the null hypothesis were true.
What if the problem does not say the sample is random?
Say so, and note that the conclusion applies to the population only if the sample can be treated as random.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every unit in the AP Statistics course ends with a ten-problem practice set, three multiple choice and seven free response, with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Statistics course Free lessons and lesson list
Other free-response guides:
- AP Statistics FRQ Guide: How Free-Response Answers Are Scored
- AP Statistics Inference for Means Explained: The One-Sample t Test and Checking Conditions
- AP Statistics Chi-Square Test Explained: Homogeneity and Independence with a Worked Example
- AP Statistics Regression Questions Explained: Interpreting Slope, r-Squared and Residuals
AP is a registered trademark of the College Board, which is not affiliated with and does not endorse this site. This guide is our own explanation of the question types and scoring. For the official scoring guidelines, see the College Board.