AP Statistics Inference for the Difference of Two Proportions Worksheet (PDF)

AP Statistics math worksheet: Inference for the Difference of Two Proportions – Maths PDF

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This two proportions worksheet for AP Statistics covers inference for the difference of two proportions (Common Core AP STAT 6.8, AP STAT 6.9, AP STAT 6.10, AP STAT 6.11, HSS-IC.B.5), with a free PDF to print.

These 20 problems practice AP Statistics topics 6.8 to 6.11. You will check conditions for two proportions, build and interpret two-sample z intervals, run pooled z tests, and use intervals to judge claims about p1 – p2.

AP Statistics · Common Core State Standards: AP STAT 6.8, AP STAT 6.9, AP STAT 6.10, AP STAT 6.11, HSS-IC.B.5

Two proportions worksheet: exercises 1 to 10

Exercise 1 – State the parameters

Survey results for two independent random samples.

Using the survey table, define the parameters p_A and p_B and write the null and alternative hypotheses for testing whether the proportions differ.

Exercise 2 – Check conditions

An experiment randomly assigns 80 volunteers to a new app and 90 to an old app. Of the 80, 52 improve; of the 90, 45 improve. Check the conditions for a two-proportion procedure.

Exercise 3 – Difference and standard error

For the app experiment, find the sample proportions, their difference (new minus old), and the standard error.

Exercise 4 – Confidence interval

95% interval.

Construct a 95% confidence interval for p_new – p_old in the app experiment.

Exercise 5 – Interpret the interval

Interpret the interval (0.003, 0.297) in context and say whether it suggests a difference between the apps.

Exercise 6 – 90 percent interval

In random samples, 180 of 400 adults in a city and 150 of 400 adults in a neighboring city say they recycle regularly. Find a 90% confidence interval for the difference (first city minus second).

Exercise 7 – Pooled proportion

For H0: p1 = p2, with 126 successes of 300 in sample 1 and 90 of 250 in sample 2, find the pooled proportion and the pooled standard error.

Exercise 8 – z statistic and p-value

Two-sided p-value for z = 1.43: 0.1527.

Complete the test for the school surveys: find z and the two-sided p-value.

Exercise 9 – Conclusion

Using alpha = 0.05, write a conclusion for the school survey test (p-value 0.1527).

Exercise 10 – One-sided test

Right-tailed p-value.

Test whether the new app improves a higher proportion than the old app (52 of 80 compared with 45 of 90) at alpha = 0.05.

Exercises 11 and up

Exercise 11 – Interval and test agree

A 95% interval for p1 – p2 is (0.01, 0.15). A two-sided test of H0: p1 = p2 at alpha = 0.05 is run. What is the conclusion?

Exercise 12 – Two-way table

Flu study results.

In a randomized experiment, 300 volunteers got a flu vaccine and 300 got a placebo. Find the sample proportions who got the flu and a 95% interval for p_placebo – p_vaccine.

Exercise 13 – Significance in the flu study

For the flu experiment, a test of H0: p_vaccine = p_placebo versus Ha: p_vaccine < p_placebo gives z = -3.60. Find the p-value and state the conclusion.

Exercise 14 – Large counts fails

In a sample of 60 teens, 4 own a drone. In another sample of 50 adults, 9 own one. Which condition fails for a two-proportion z procedure?

Exercise 15 – Dependent samples

A teacher gives 25 students a quiz before and after a lesson and compares the proportions that passed. Why is a two-proportion z procedure inappropriate?

Exercise 16 – Margin of error comparison

For the school surveys the interval (-0.022, 0.142) contains 0. Explain what a larger sample might do.

Exercise 17 – Sample proportions

A random sample of 200 men and 250 women is asked about electric cars: 90 men and 130 women say they would consider buying one. Compute p-hat for each and the difference (women minus men).

Exercise 18 – Interval for the electric car survey

Find a 95% interval for p_women – p_men and interpret it.

Exercise 19 – Test for the electric car survey

Test H0: p_women = p_men against a two-sided alternative at alpha = 0.05 for the same data.

Exercise 20 – Multi-step

In one town, 72 of 150 voters in a random sample support a tax; in another, 60 of 160 do. Find the 99% interval for the difference and state whether the data suggest a difference.

Key facts behind this two proportions worksheet

Key facts

Parameter: p1 – p2; statistic: p-hat1 – p-hat2.
Interval: (p-hat1 – p-hat2) +/- z* sqrt(p-hat1(1 – p-hat1)/n1 + p-hat2(1 – p-hat2)/n2).
Test: pooled p-hat_c = (x1 + x2)/(n1 + n2); z = (p-hat1 – p-hat2)/sqrt(p-hat_c(1 – p-hat_c)(1/n1 + 1/n2)).
Conditions: independent random samples (or random assignment); 10% condition; at least 10 successes and 10 failures in each group.
If the interval contains 0, a difference of 0 is plausible; if it does not, there is evidence of a difference.

How to use this two proportions worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the two proportions worksheet as a PDF with the button below and keep it with your notes.

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