Sampling Distributions: AP Statistics Math Test Answer Key – PDF Download

AP Statistics Sampling Test Answer Key: questions 1–5

Question 1: Sample proportion of students (2 points)

1. \(\mu_{\hat{p}}=0.35\); \(\sigma_{\hat{p}}=\sqrt{\frac{0.35(0.65)}{120}}\approx 0.0435\).
2. The sample size \(120\) is less than \(10\%\) of \(4{,}000\) (that is \(400\)); \(np=42\ge 10\) and \(n(1-p)=78\ge 10\).

Scoring: (a) 1 point for both values. (b) 1 point for the 10% condition and both normal checks.

Question 2: Probability of a low proportion (3 points)

1. \(\sigma_{\hat{p}}=\sqrt{\frac{0.45(0.55)}{250}}\approx 0.0315\).
2. \(z=\frac{0.40-0.45}{0.0315}=-1.59\); \(P(\hat{p}< 0.40)=\) 0.0559.

Scoring: (a) 1 point. (b) 1 point for the correct z-score, 1 point for the correct area.

Question 3: Sample size and variability (2 points)

Both distributions are centered at \(0.50\). For \(n=100\): \(\sigma=\sqrt{\frac{0.5(0.5)}{100}}=0.05\). For \(n=400\): \(\sigma=\sqrt{\frac{0.5(0.5)}{400}}=0.025\).

The larger sample has half the standard deviation, so its sample proportions cluster more tightly around \(0.50\).

Scoring: 1 point for equal centers and both standard deviations; 1 point for the comparison of spread.

Question 4: Bias and variability (2 points)

Method A is centered at \(0.40\) and has little spread: low bias, low variability.

Method B is also centered at \(0.40\), so it is unbiased, but its results are widely spread (high variability). Method C has little spread but is centered near \(0.53\), above \(p\): it is biased.

Scoring: 1 point for choosing A with reasoning on center and spread; 1 point for correctly describing both B and C.

Question 5: Sample mean of wait times (3 points)

1. \(\mu_{\,\overline{x}}=45\); \(\sigma_{\,\overline{x}}=\frac{20}{\sqrt{50}}\approx 2.8284\).
2. Approximately normal: although the population is skewed, \(n=50\ge 30\), so the Central Limit Theorem applies.
3. \(z=\frac{50-45}{2.8284}=1.77\); \(P(\,\overline{x}> 50)=\) 0.0384.

Scoring: (a) 1 point for both values. (b) 1 point for normal and a justification using the Central Limit Theorem and \(n\). (c) 1 point for z-score and area.

AP Statistics Sampling Test Answer Key: questions 6–8

Question 6: Difference of two proportions (3 points)

1. Mean: \(0.62-0.55=0.07\). Standard deviation: \(\sqrt{\frac{0.62(0.38)}{150}+\frac{0.55(0.45)}{200}}=\sqrt{0.001571+0.001238}\approx 0.0530\).
2. \(z=\frac{0-0.07}{0.0530}=-1.32\); \(P(\hat{p}_{1}-\hat{p}_{2}< 0)=\) 0.0934.

Scoring: (a) 1 point for the mean, 1 point for the standard deviation using the sum of variances. (b) 1 point for the z-score and area.

Question 7: Difference of two means (3 points)

1. Mean: \(240-225=15\). Standard deviation: \(\sqrt{\frac{30^{2}}{40}+\frac{25^{2}}{35}}=\sqrt{22.5+17.857}=\sqrt{40.357}\approx 6.3527\).
2. \(z=\frac{20-15}{6.3527}=0.79\); \(P=1-0.7852=\) 0.2148.

Scoring: (a) 1 point for the mean, 1 point for the standard deviation. (b) 1 point for z-score and area.

Question 8: Is the claim believable? (2 points)

1. \(\sigma_{\hat{p}}=\sqrt{\frac{0.5(0.5)}{200}}\approx 0.0354\); \(z=\frac{0.43-0.50}{0.0354}=-1.98\); \(P(\hat{p}\le 0.43)=\) 0.0239.
2. If the claim were true, about \(2.4\%\) of samples would be this low or lower. That is unlikely, so the sample gives reason to doubt the claim.

Scoring: (a) 1 point for the correct probability with setup. (b) 1 point for a conclusion in context linked to the small probability.

Scoring summary

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