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Binomial and Normal Test Answer Key with Scoring – AP Statistics

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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 gives the points for each part. Credit goes for the correct model, the correct parameters and a correct answer with context; a calculator command alone does not show the setup.

Standards: AP Statistics Unit 2 (2026-27) · Level: Grade 12 AP Statistics

Question 1 Binomial Setting (2 points)

Binary: each player passes or fails. Independent: players pass independently. Number: n = 12 is fixed. Success: p = 0.8 for every player. So X ~ B(12, 0.8).

Binomial with n = 12 and p = 0.8

Scoring: 1 point for the four conditions, 1 point for the distribution with parameters.

Question 2 Binomial Calculations (3 points)

(a) C(10, 4)(0.3)4(0.7)6 = 0.2001. (b) 1 − 0.710 = 1 − 0.0282 = 0.9718. (c) μ = 3, σ = √(10 × 0.3 × 0.7) = 1.449.

(a) 0.2001; (b) 0.9718; (c) μ = 3, σ ≈ 1.449

Scoring: 1 point each part.

Question 3 Shape and Tail Probability (2 points)

(a) The distribution is skewed right because p is small. (b) P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − 0.7969 = 0.2031.

(a) skewed right; (b) 0.2031

Scoring: 1 point each part.

Question 4 Normal Probability (2 points)

z = (210 − 250)/30 = −1.33 and z = (280 − 250)/30 = 1.00. The area is 0.8413 − 0.0912 = 0.7501.

210250280
Shaded area between 210 and 280.

≈ 0.7501

Scoring: 1 point for both z-scores, 1 point for the correct area.

Question 5 Percentiles and Unknown Parameters (3 points)

(a) z = 1.036, so x = 68 + 1.036(7.2) = 75.46. (b) The 20th percentile has z = −0.842. Solve −0.842 = (63 − 70)/σ: σ = 7/0.842 = 8.32.

(a) ≈ 75.5 bpm; (b) σ ≈ 8.3 bpm

Scoring: 1 point for the z-score or invNorm use in (a), 1 point for the value, 1 point for (b).

Question 6 Tails of a Normal Model (3 points)

(a) z = −1.50, area 0.0668. (b) z = 2.00, area 0.0228. (c) 0.0668 + 0.0228 = 0.0896.

(a) 0.0668; (b) 0.0228; (c) 0.0896

Scoring: 1 point each part.

Question 7 Normal Approximation (3 points)

(a) np = 60 ≥ 10 and n(1 − p) = 240 ≥ 10. (b) μ = 60, σ = √(300 × 0.2 × 0.8) = 6.928. z = (70 − 60)/6.928 = 1.44. P(X ≥ 70) ≈ 0.0745.

≈ 0.074

Scoring: 1 point for the conditions, 1 point for μ and σ, 1 point for the probability.

Question 8 Normal Then Binomial (2 points)

z = (5.00 − 5.15)/0.10 = −1.50, so p = 0.0668. P(at least one of 5) = 1 − (1 − 0.0668)5 = 0.2923.

0.0668; 0.2923

Scoring: 1 point for the rejection probability, 1 point for the at-least-one probability.

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