AP Statistics Inference for the Difference of Two Proportions Test Solutions and Scoring (PDF)

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

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This two proportions test answers 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.

Each solution includes a scoring guide. Award points for the conditions, the correct standard error, the interval or z statistic, and a conclusion linked to the context.

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 test answers: worked solutions

Question 1 – Two-sample interval [4 pts]

Interval.

(a) Counts 84, 116, 66, 154 are all at least 10. (b) \(\hat{p}_X=0.42,\ \hat{p}_Y=0.30,\ SE=0.0466,\ ME=0.0914\), interval (0.029, 0.211). (c) We are 95% confident that the interval captures the true difference in the proportions of successes between populations X and Y. (d) 0 is not in the interval: convincing evidence of a difference.

(0.029, 0.211).

Scoring: 1 point for (a); 1 point for the SE and margin of error; 1 point for the interval; 1 point for the interpretation and the conclusion about 0.

Question 2 – Two-sample test [4 pts]

Two-sided p-value.

\(\hat{p}_c=\frac{150}{420}=0.3571\). \(SE=0.0468,\ z=\frac{0.12}{0.0468}=2.56\). p-value \(=0.0105\).

p-value < 0.05, so reject H0: convincing evidence that the success proportions differ between X and Y.

Scoring: 1 point for the pooled proportion, 1 point for z, 1 point for the p-value, 1 point for the conclusion in context.

Question 3 – Hypotheses [2 pts]

Randomized experiment results.

p_D = true proportion who would improve with the drug; p_P = true proportion who would improve with the placebo.

H0: p_D = p_P; Ha: p_D > p_P.

Scoring: 1 point for defining the parameters in context, 1 point for the hypotheses.

Question 4 – Sample difference [1 pt]

Drug: \(\frac{56}{80}=0.70\). Placebo: \(\frac{40}{80}=0.50\).

0.70 and 0.50.

Scoring: 1 point for both proportions.

Question 5 – Error type [1 pt]

Rejecting a true null hypothesis.

Type I error.

Scoring: 1 point for Type I.

Question 6 – Pooled or not [2 pts]

The test assumes H0 is true, which says both populations share one proportion, so combining the samples estimates it.

An interval does not assume equality, so it uses each sample proportion separately.

Scoring: 1 point for the test assumption, 1 point for the interval difference.

Question 7 – Conditions [3 pts]

Conditions.

(a) The sophomores have only 6 successes, fewer than 10, so the large counts condition fails. (b) Take larger samples.

Large counts fails; use larger samples.

Scoring: 2 points for identifying the condition and the count; 1 point for a fix.

Question 8 – Interpret the interval [3 pts]

(a) We are 95% confident that the true on-time proportion of route A is between 0.04 less and 0.12 more than that of route B. (b) 0 is in the interval.

No convincing evidence of a difference.

Scoring: 2 points for the interpretation with direction and context; 1 point for the conclusion.

Total score

Total

20 points

Key facts behind this two proportions test answers

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 answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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