Inference for Proportions Test Answer Key and Scoring

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

AP Statistics Math Test Answer Keys – Mathbot, the Math Made Easy robot coach, with free printable test answer keys for AP Statistics

The scoring guide shows how points are earned. Each inference answer needs conditions, correct calculations and a conclusion in context that refers to the p-value or the interval.

Standards: AP Statistics Unit 3 (2026-27) · Level: Grade 12 AP Statistics

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Question 1 Confidence Interval (3 points)

(a) Random sample; 400 is less than 10% of a large county; 158 and 242 are at least 10. (b) p̂ = 0.395, SE = 0.02444, ME = 0.0479, interval (0.347, 0.443). (c) We are 95% confident that the true proportion of county voters who plan to vote for the bond is between 0.347 and 0.443.

(0.347, 0.443)

Scoring: 1 point conditions, 1 point interval, 1 point interpretation in context.

Question 2 Sample Size (2 points)

n ≥ (1.645/0.04)²(0.5)(0.5) = 422.82. Round up.

n = 423

Scoring: 1 point for the formula with p̂ = 0.5, 1 point for rounding up correctly.

Question 3 Interval Justification (2 points)

Program X: the interval (0.08, 0.20) does not contain 0 and is all positive, so there is convincing evidence that X improves the graduation rate. Program Y: the interval (−0.03, 0.11) contains 0, so a difference of 0 is plausible and there is not convincing evidence of a difference.

X: yes; Y: no

Scoring: 1 point each program, with the reason about 0.

Question 4 Significance Test (3 points)

(a) H₀: p = 0.50; H_a: p ≠ 0.50. Random sample, 150 is under 10% of the plays, and 150(0.5) = 75 for both counts. (b) p̂ = 0.4067, SE = √(0.25/150) = 0.0408, z = −2.286. The two-sided p-value is 2 × 0.0111 = 0.0222. (c) Since 0.0222 < 0.05, we reject H₀; there is convincing evidence that the true win proportion differs from 0.50 (it appears to be lower).

Two-sided p-value for p̂ = 0.4067.

z ≈ −2.29, p-value ≈ 0.022; reject H₀

Scoring: 1 point each part: hypotheses and conditions, statistic and p-value, conclusion in context.

Question 5 p-Value and Decision (2 points)

(a) If the true proportion of students who work part-time were 0.30, the probability of getting a sample proportion as large as the one observed, or larger, would be 0.021. (b) Since 0.021 < 0.05, reject H₀: there is convincing evidence that the true proportion is greater than 0.30.

Reject H₀

Scoring: 1 point for a correct interpretation, 1 point for the decision with context.

Question 6 Errors in Context (2 points)

Type I: deciding that the defect rate is above 2% when it is actually 2% (rejecting a true H₀), which could stop production unnecessarily. Type II: not finding convincing evidence that the rate is above 2% when it is really higher, so defective items continue to be shipped.

Type I: reject true H₀; Type II: fail to reject false H₀

Scoring: 1 point for each error described in context.

Question 7 Difference of Two Proportions (3 points)

(a) Random and independent samples; counts 52, 78, 36 and 84 are all at least 10. (b) p̂_A = 0.400, p̂_B = 0.300, difference 0.100. SE = 0.0600, ME = 0.1175. Interval (−0.018, 0.218). (c) The interval contains 0, so there is not convincing evidence of a difference.

(−0.018, 0.218); no convincing evidence

Scoring: 1 point each part.

Question 8 Power (3 points)

(a) Power is the probability of rejecting H₀ when a particular alternative value of the parameter is true (1 − β). (b) Increase the sample size, or use a larger α, or test an alternative that is farther from the null value. (c) A larger α raises the probability of a Type I error.

Power = P(reject H₀ | H_a true)