This ap statistics proportions worksheet gives 24 problems on inference for proportions, ordered from easy to hard, with a separate answer key.
These 24 problems practice inference for proportions from the AP Statistics course: checking conditions, building and interpreting one-proportion and two-proportion intervals, choosing a sample size, running z tests, interpreting P-values, and describing Type I errors, Type II errors and power. Problems move from short skill checks to full four-step procedures.
AP Statistics Proportions Worksheet: problems 1–5
Problem 1: Large Counts check
In a random sample of 45 customers at a farmers market, 9 paid with cash. Check whether the Large Counts condition is met for a confidence interval for the proportion of all customers who pay with cash.
Problem 2: Standard error of a proportion
In a random sample of 150 Little League parents, 90 support adding a pitch clock. Calculate the standard error of \(\hat{p}\).
Problem 3: Critical values
Find the critical value \(z^{*}\) for a 90%, a 95% and a 99% confidence interval for a proportion. Round to three decimal places.
Problem 4: One-proportion z interval
A random sample of 250 youth players in a regional league includes 85 who play a second sport. Assume the random and 10% conditions are met. Check Large Counts, then construct a 95% confidence interval for the proportion of all players in the league who play a second sport.
Problem 5: Interpreting an interval
The 95% confidence interval for the proportion of players in a regional league who play a second sport is \((0.281, 0.399)\). Interpret this interval in context.
AP Statistics Proportions Worksheet: problems 6–10
Problem 6: Meaning of the confidence level
Priya writes: “There is a 95% chance that the true proportion of players who play a second sport is between 0.281 and 0.399.” Identify the error in this statement and explain what 95% confidence actually describes.
Problem 7: Sample proportion and margin of error
A confidence interval for a proportion is \((0.412, 0.508)\). Find the sample proportion \(\hat{p}\) and the margin of error.
Problem 8: Effect of confidence level
In a random sample of 400 shoppers at a mall, 140 used a coupon. Construct a 90% and a 99% confidence interval for the proportion of all shoppers who use a coupon. Explain why the intervals differ in width.
Problem 9: Sample size without a prior estimate
A polling group wants a 95% confidence interval for a proportion with a margin of error of at most 0.03. No prior estimate of the proportion exists. Find the smallest sample size that guarantees this.
Problem 10: Sample size with a prior estimate
An earlier study found that about 30% of teenagers volunteer each month. Find the smallest sample size needed for a 99% confidence interval with a margin of error of at most 0.02, using the earlier estimate.
AP Statistics Proportions Worksheet: problems 11–15
Problem 11: Writing hypotheses
A battery company states that 8% of its batteries are defective. Ethan suspects the true proportion is greater than 8%. State the null and alternative hypotheses and define the parameter.
Problem 12: Right-tailed z test
A bakery claims that at most 10% of its muffins are underweight. Sofia suspects the rate is higher. In a random sample of 120 muffins, 18 are underweight. Perform a one-proportion z test: find the standardized test statistic and the P-value.
Problem 13: Two-sided z test
A school reports that 40% of its students ride the bus. Jamal thinks the percentage has changed. In a random sample of 180 students, 63 ride the bus. Find the test statistic and the P-value for a two-sided test of \(H_0: p=0.40\).
Problem 14: Interpreting a P-value
A test of \(H_0: p=0.10\) versus \(H_a: p>0.10\) for the share of underweight muffins, based on a sample with \(\hat{p}=0.150\), gave a P-value of 0.034. Interpret this P-value in context.
Problem 15: Drawing a conclusion
The muffin test gave a P-value of 0.034. Use a significance level of \(\alpha=0.05\) to state a conclusion in context. Then decide whether the conclusion would change at \(\alpha=0.01\).
AP Statistics Proportions Worksheet: problems 16–20
Problem 16: Interval for a difference
In a random sample of 300 residents of Town A, 156 support a new park levy. In an independent random sample of 250 residents of Town B, 105 support it. Assume all conditions are met. Construct a 95% confidence interval for \(p_A-p_B\).
Problem 17: Justifying a claim with an interval
A 95% confidence interval for \(p_1-p_2\), the difference in the proportions of two groups who prefer online shopping, is \((-0.031, 0.117)\). Does the interval provide convincing evidence that the proportion is higher in group 1? Justify the answer.
Problem 18: Two-proportion z test
A company randomly assigns 200 users to App A and 200 users to App B. In App A, 130 users finish the sign-up; in App B, 112 do. Test whether App A has a higher completion rate than App B. Find the pooled proportion, the test statistic, and the P-value.
Problem 19: Describing a Type I error
A battery company claims that 8% of its batteries are defective. An inspector tests \(H_0: p=0.08\) versus \(H_a: p>0.08\). Describe a Type I error in context and state its probability if the significance level is 0.05.
Problem 20: Describing a Type II error
For the same battery test of \(H_0: p=0.08\) versus \(H_a: p>0.08\), describe a Type II error in context and one possible consequence.
AP Statistics Proportions Worksheet: problems 21–24
Problem 21: Power and Type II error
A test has power 0.72 against the alternative \(p=0.60\). Find the probability of a Type II error for that alternative, and list two ways to increase the power without changing the alternative value.
Problem 22: Computing power
A test of \(H_0: p=0.50\) versus \(H_a: p>0.50\) uses \(n=100\) and \(\alpha=0.05\). Find the power of the test when the true proportion is \(p=0.60\). Use the critical value \(z^{*}=1.645\).
Problem 23: Four-step significance test
A school newspaper claims that 70% of the seniors in a large state plan to attend college. Noah thinks the true percentage is lower. In a random sample of 220 seniors from the state (far more than 2,200 seniors), 143 plan to attend college. Carry out a significance test at the 0.05 level.
Problem 24: Interval and test agree
A 95% confidence interval for a population proportion is \((0.43, 0.57)\). Use the interval to decide what a two-sided significance test at \(\alpha=0.05\) would conclude for (a) \(H_0: p=0.60\) and (b) \(H_0: p=0.50\).
How to use this ap statistics proportions worksheet
Print this ap statistics proportions worksheet as a PDF, solve every problem on paper, then compare with the answer key.
See every chapter on the AP Statistics math page.