← Back to the Sampling Distributions test
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: AP Statistics Unit 2 (2026-27) · Level: Grade 12 AP Statistics
Question 1 Parameter and Statistic (2 points)
(a) The parameter is the mean commute time of all 1,400 students, μ. The statistic is the sample mean, x̄ = 24.5 minutes. (b) No. Different samples contain different students, so x̄ varies from sample to sample (sampling variability).
(a) μ and x̄; (b) no, because of sampling variability
Scoring: 1 point each part.
Question 2 Sampling Distribution of p̂ (3 points)
(a) 180(0.52) = 93.6 and 180(0.48) = 86.4 are at least 10, and 180 is under 10% of all voters. (b) Mean = 0.52, σ = √(0.52 × 0.48/180) = 0.0372. (c) z = (0.47 − 0.52)/0.0372 = −1.34, probability 0.0897.
(a) conditions met; (b) 0.52 and 0.0372; (c) 0.0897
Scoring: 1 point each part.
Question 3 Sample Size and Spread (2 points)
(a) Size 100: a larger sample makes the statistic vary less. (b) n = 25: √(0.5 × 0.5/25) = 0.10; n = 100: √(0.5 × 0.5/100) = 0.05.
(a) n = 100; (b) 0.10 and 0.05
Scoring: 1 point each part.
Question 4 Sample Mean from a Normal Population (3 points)
(a) Mean 125, σ = 6/√9 = 2. (b) The population is normal, so x̄ is normal for every sample size. (c) z = ±(3/2) = ±1.50, probability 0.8664.
(a) 125 and 2; (c) 0.8664
Scoring: 1 point each part.
Question 5 Central Limit Theorem Shape (2 points)
(a) With n = 4 the sampling distribution is still skewed right, similar to the population, since 4 is too small for the Central Limit Theorem. (b) With n = 50, the Central Limit Theorem applies and the distribution is approximately normal.
(a) skewed right; (b) approximately normal
Scoring: 1 point each part with an explanation.
Question 6 Difference of Proportions (3 points)
(a) Mean = 0.10. σ = √(0.6 × 0.4/100 + 0.5 × 0.5/120) = √(0.0024 + 0.002083) = 0.0670. (b) z = (0 − 0.10)/0.0670 = −1.49. Probability = 0.0677.
(a) 0.10 and 0.0670; (b) 0.0677
Scoring: 1 point for the mean, 1 point for the standard deviation, 1 point for the probability.
Question 7 Difference of Means (2 points)
Mean = 30 − 27 = 3. σ = √(52/25 + 42/36) = √(1 + 0.4444) = 1.2019.
mean 3, standard deviation ≈ 1.202
Scoring: 1 point each value.
Question 8 Bias and Variability (3 points)
(a) Method A is centered near 0.40 (mean 0.40), so it has little bias, but its estimates range from 0.22 to 0.58: high variability. (b) Method B is centered near 0.48 (mean 0.48), so it is biased high, but its estimates stay between 0.44 and 0.54: low variability. (c) Method A, because its errors average out and larger samples would shrink its spread, while bias cannot be fixed by a larger sample.
A: unbiased but variable; B: biased but consistent; choose A
Scoring: 1 point each part.
Mis à jour le