AP Statistics Sampling Distributions and the Central Limit Theorem Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Sampling Distributions and the Central Limit Theorem – Maths PDF

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This sampling distributions test answers for AP Statistics covers sampling distributions and the central limit theorem (Common Core AP STAT 5.1, AP STAT 5.2, AP STAT 5.3, AP STAT 5.4, AP STAT 5.5, AP STAT 5.6, AP STAT 5.7, HSS-IC.A.1, HSS-IC.A.2), with a free PDF to print.

Each solution includes a scoring guide. Give credit for correct parameters, a sketch or setup, a correct z-score, and a conclusion in context.

AP Statistics · Common Core State Standards: AP STAT 5.1, AP STAT 5.2, AP STAT 5.3, AP STAT 5.4, AP STAT 5.5, AP STAT 5.6, AP STAT 5.7, HSS-IC.A.1, HSS-IC.A.2

Sampling distributions test answers: worked solutions

Question 1 – Proportion probability [4 pts]

Sampling distribution of p-hat.

(a) Mean 0.30; sd \(\sqrt{0.3(0.7)/100}=0.0458\). (b) np = 30 and n(1-p) = 70, both at least 10 (and the sample is under 10% of the population). (c) \(z=1.09\), \(P(Z > 1.09)=0.1379\).

0.30 and 0.0458; conditions met; 0.1379.

Scoring: 1 point for each of the mean and sd, 1 point for conditions, 1 point for the probability.

Question 2 – Sample mean [3 pts]

(a) \(\mu_{\overline{x}}=4.2,\ \sigma_{\overline{x}}=\frac{1.5}{6}=0.25\); n = 36 is at least 30, so by the CLT the shape is approximately normal.

(b) \(z=\frac{4.6-4.2}{0.25}=1.6\), \(P(Z > 1.6)=0.0548\).

Approximately normal, mean 4.2, sd 0.25; 0.0548.

Scoring: 1 point for the mean and sd, 1 point for CLT and shape, 1 point for the probability.

Question 3 – Effect of n [2 pts]

\(\sqrt{0.25/100}=0.05\) and \(\sqrt{0.25/400}=0.025\).

0.05 and 0.025: the spread is cut in half when n is multiplied by 4.

Scoring: 1 point for each standard deviation (with the halving statement required for full credit on the second).

Question 4 – Bias and variability [2 pts]

The first is centered at the parameter but spread widely; the second is narrow but centered at the wrong value.

Unbiased is usually preferred because its center is correct, and its variability can be reduced by a larger sample, but bias cannot.

Scoring: 1 point for each correct description and reason (2 points total).

Question 5 – Difference of proportions [3 pts]

Mean: \(0.5-0.4=0.1\). SD: \(\sqrt{\frac{0.5(0.5)}{80}+\frac{0.4(0.6)}{100}}=\sqrt{0.003125+0.0024}=0.0743\).

Mean 0.1; standard deviation 0.0743.

Scoring: 1 point for the mean, 1 point for variances, 1 point for the sd.

Question 6 – Poll probability [2 pts]

Sampling distribution, n = 100.

\(\sigma=\sqrt{0.5(0.5)/100}=0.05\). \(z=\frac{0.42-0.5}{0.05}=-1.60\). \(P(Z < -1.60)=0.0548\).

0.0548.

Scoring: 1 point for the sd and z-score, 1 point for the probability.

Question 7 – Conditions [2 pts]

\(np=30(0.05)=1.5\), less than 10.

Not approximately normal.

Scoring: 1 point for np = 1.5, 1 point for the conclusion.

Question 8 – Within two standard errors [2 pts]

Sampling distribution of the mean for n = 16.

\(\sigma_{\overline{x}}=\frac{12}{4}=3\). \(50\pm 2(3)=(44,56)\).

About 44 to 56.

Scoring: 1 point for the sd, 1 point for the interval.

Total score

Total

20 points

Key facts behind this sampling distributions test answers

Key facts

A sampling distribution describes how a statistic varies from sample to sample.
p-hat: mean p, sd sqrt(p(1-p)/n); approximately normal if np >= 10 and n(1-p) >= 10.
x-bar: mean mu, sd sigma/sqrt(n); normal if the population is normal or n is large (CLT).
Differences of independent statistics: means subtract, variances add.
Larger n reduces variability but never corrects bias; the 10% condition is needed for independence.

How to use this sampling distributions 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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