AP Statistics Inference for the Difference of Two Proportions Test with Point Values (PDF)

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

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This two proportions test 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.

This 45-minute test covers conditions, confidence intervals, and hypothesis tests for a difference of two proportions. Follow State, Plan, Do, Conclude and write every conclusion in context.

Time and points

About this test

Grade: AP Statistics
Chapter: Inference for the Difference of Two Proportions
Standards: AP STAT 6.8, AP STAT 6.9, AP STAT 6.10, AP STAT 6.11, HSS-IC.B.5
Time: 45 minutes
Total: 20 points

Two proportions test: questions

Question 1 – Two-sample interval [4 pts]

Results for two independent random samples.

Use the table (independent random samples, each under 10% of its population). (a) Check the large counts condition. (1 point) (b) Construct a 95% confidence interval for p_X – p_Y. (2 points) (c) Interpret the interval and say whether it suggests a difference. (1 point)

Question 2 – Two-sample test [4 pts]

With the same data, test H0: p_X = p_Y against Ha: p_X not equal to p_Y at alpha = 0.05. Show the pooled proportion, z, p-value, and a conclusion in context. (4 points)

Question 3 – Hypotheses [2 pts]

Randomized experiment results.

The table shows a randomized experiment. Define the parameters and write the hypotheses to test whether the drug helps a greater proportion.

Question 4 – Sample difference [1 pt]

Using the table in the previous question, find the sample proportion improved in each group.

Question 5 – Error type [1 pt]

If the drug has no effect but the test rejects H0, what type of error was made?

Question 6 – Pooled or not [2 pts]

Explain why a test of H0: p1 = p2 uses a pooled proportion while a confidence interval does not.

Question 7 – Conditions [3 pts]

Conditions.

A school surveys 40 randomly chosen sophomores (6 own an e-bike) and 50 randomly chosen seniors (14 own one). (a) Which condition fails? (2 points) (b) What could be done? (1 point)

Question 8 – Interpret the interval [3 pts]

A 95% confidence interval for p_A – p_B is (-0.04, 0.12). (a) Interpret in context where A and B are two bus routes’ on-time proportions. (2 points) (b) Is there convincing evidence of a difference? (1 point)

Key facts behind this two proportions test

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 test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Statistics chapters: AP Statistics math.

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