AP Statistics Quiz: Sampling Distributions For Sample Proportions

What this quiz covers

This quiz focuses on Sampling Distributions For Sample Proportions, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The sampling distribution of $\hat p$ may not be approximately Normal because the success-failure condition may fail. (correct answer)
2. The sampling distribution of $\hat p$ is approximately Normal because $p$ is close to 0.50.
3. The sampling distribution of $\hat p$ is exactly Normal because sampling is random.
4. The sampling distribution of $\hat p$ has no spread because $p$ is fixed.
5. The sampling distribution of $\hat p$ is centered at 15.

Explanation: This question examines conditions for normality in sampling distributions for sample proportions in AP Statistics. With p=0.53 and n=15, np=7.95 <10 and n(1-p)=7.05 <10, so the distribution may not be approximately normal. The center is at 0.53, and spread is (\sqrt{0.53 \times 0.47 / 15} $\approx$ 0.129), but normality isn't guaranteed. Choice E is a distractor, wrongly centering at n=15 instead of p. Mini-lesson: For the sampling distribution of (\hat{p}) to be useful, it should be nearly normal, requiring at least 10 expected successes and failures; without this, shape can be skewed, affecting probability calculations. Always check these conditions before applying normal models.

Question 2

1. The 10% condition is satisfied because $150$ is less than 10% of $2{,}000$. (correct answer)
2. The 10% condition is not satisfied because $p=0.06$ is small.
3. The 10% condition is not satisfied because $n$ must be less than 10% of $p$.
4. The sampling distribution of $\hat p$ has no variability when sampling from a finite population.
5. If sampling is without replacement, $\hat p$ is biased and not centered at 0.06.

Explanation: Focus on 10% condition for finite population sampling in AP Statistics. 150 is 7.5% of 2000 (<10%), satisfied, so A correct. Distractor B says not because p small, but unrelated. Mini-lesson: Condition justifies ignoring finite correction; mean p, hat p unbiased unlike E suggests.

Question 3

1. The sampling distribution of $\hat p$ is centered at 0.19. (correct answer)
2. The sampling distribution of $\hat p$ is centered at 500.
3. The sampling distribution of $\hat p$ is centered at 0.81.
4. The sampling distribution of $\hat p$ has no variability because 19% is fixed.
5. The sampling distribution of $\hat p$ is centered at $\sqrt{0.19(0.81)/500}$.

Explanation: This question probes the center of sampling distributions for sample proportions in AP Statistics. The sampling distribution of (\hat{p}) is centered at p = 0.19, not at n=500 or the complement 0.81. Its spread is (\sqrt{0.19 \times 0.81 / 500} $\approx$ 0.018), showing some variability. Distractor E mistakenly identifies the center as the standard deviation, a mix-up of concepts. Mini-lesson: Sampling distributions for proportions are centered at the true p, with spread decreasing as n increases; they're approximately normal if np and n(1-p) ≥ 10, here both are satisfied (95 and 405). This framework underlies confidence intervals and hypothesis tests.

Question 4

1. The sampling distribution of $\hat p$ has mean 0.40 and standard deviation $\sqrt{0.40(0.60)/100}$. (correct answer)
2. The sampling distribution of $\hat p$ has mean 0.40 and standard deviation $\sqrt{0.40(0.60)\cdot 100}$.
3. The sampling distribution of $\hat p$ has mean $\sqrt{0.40(0.60)/100}$ and standard deviation 0.40.
4. The sampling distribution of $\hat p$ has mean 100 and standard deviation 0.40.
5. The sampling distribution of $\hat p$ has mean 0 and standard deviation 0 because $p$ is fixed.

Explanation: This question focuses on the mean and standard deviation formulas for sampling distributions of sample proportions in AP Statistics. The distribution is centered at $p=0.40$ with standard deviation $\sqrt{0.40 \times 0.60 / 100}$, capturing the center and spread accurately. Choice B incorrectly multiplies by n instead of dividing, inflating the spread. A common distractor is choice E, suggesting mean and sd of zero because p is fixed, but statistics vary while parameters do not. In a mini-lesson, the sampling distribution of $\hat p$ has mean $p$ and sd $\sqrt{p(1-p)/n}$, and is approximately normal if $n p \geq 10$ and $n(1-p) \geq 10$. Here, $100 \times 0.40 = 40$ and $100 \times 0.60 = 60$, both sufficient. This allows probabilistic statements about $\hat p$'s range.

Question 5

1. The distribution of $\hat p$ is more concentrated around 0.50 than it would be if $n=25$. (correct answer)
2. The distribution of $\hat p$ is more concentrated around 0.50 than it would be if $n=400$.
3. The distribution of $\hat p$ has no spread because $p=0.50$.
4. The distribution of $\hat p$ is centered at 100 because $n=100$.
5. The distribution of $\hat p$ is centered at $\sqrt{0.50(0.50)/100}$.

Explanation: This question examines how sample size affects the spread of sampling distributions for sample proportions in AP Statistics. The sampling distribution is centered at p=0.50, and with larger n=100 compared to n=25, the spread sqrt(p(1-p)/n) is smaller, making it more concentrated. For n=400, it would be even more concentrated, so choice B is incorrect. A distractor like choice C claims no spread because p=0.50, ignoring that variability exists regardless of p's value. In a mini-lesson, sampling distributions show that as n increases, the standard deviation decreases, leading to more precise estimates of p. Here, conditions for normality are met since n p=50 ≥ 10 and n(1-p)=50 ≥ 10. This principle underscores why larger samples are preferred in statistical inference.

Question 6

1. The sampling distribution of $\hat p$ may not be approximately Normal because $n(1-p)$ may be less than 10. (correct answer)
2. The sampling distribution of $\hat p$ is approximately Normal because $p$ is close to 1.
3. The sampling distribution of $\hat p$ is exactly Normal whenever $n=20$.
4. The sampling distribution of $\hat p$ has no spread because $p=0.75$.
5. The sampling distribution of $\hat p$ cannot be studied because $\hat p$ changes each sample.

Explanation: This question evaluates conditions for normality in sampling distributions for sample proportions in AP Statistics. The distribution may not be approximately normal because n(1-p)=20*0.25=5 <10, failing the success-failure condition. The center is p=0.75, with spread sqrt(p(1-p)/n), but shape is skewed when conditions fail. Choice B is a distractor, incorrectly linking normality to p near 1 rather than the conditions. In a mini-lesson, for î p's distribution to be normal, need random sample, n p ≥ 10, n(1-p) ≥ 10, and for finite pops, n<10% N. Here, n p=15 ≥ 10 but n(1-p)=5<10, so questionable. This affects reliability of normal-based inferences.

Question 7

1. The sampling distribution of $\hat p$ is centered at 0.05 and typically varies from sample to sample. (correct answer)
2. The sampling distribution of $\hat p$ is centered at 0.05 and has no variation because $p$ is fixed.
3. The sampling distribution of $\hat p$ is centered at 500 because $n=500$.
4. The sampling distribution of $\hat p$ is centered at 0.95 because most do not respond.
5. The sampling distribution of $\hat p$ is centered at $\sqrt{0.05(0.95)/500}$.

Explanation: This question assesses the center and variability of sampling distributions for sample proportions in AP Statistics. The distribution is centered at $p=0.05$ and varies from sample to sample, with spread $\sqrt{p(1-p)/n}$. Choice B wrongly claims no variation, confusing the fixed $p$ with the variable $\hat p$. A distractor like choice C centers it at $n=500$, mistaking count for proportion. In a mini-lesson, sampling distributions describe the long-run behavior of $\hat p$: mean $p$, sd decreasing with larger $n$, and normality if success-failure conditions hold. Here, $n p=500 \times 0.05=25 \geq 10$, $n(1-p)=475 \geq 10$, so normal approximation is good. This is crucial for confidence intervals and hypothesis tests.

Question 8

1. The independence assumption is reasonable because $80$ is less than 10% of $1{,}200$. (correct answer)
2. The independence assumption fails because sampling is without replacement.
3. The sampling distribution of $\hat p$ will be centered at 80.
4. The sampling distribution of $\hat p$ has mean $\sqrt{0.45(0.55)/80}$.
5. The sampling distribution of $\hat p$ has mean 0.55, the proportion not in clubs.

Explanation: This question tests the independence assumption via 10% condition in AP Statistics. 80 is 6.67% of 1200 (<10%), reasonable, confirming A. Distractor B says fails without replacement, but condition allows it. Mini-lesson: For without replacement, independence approx if n<10% N; mean p, spread adjusted but approx sqrt{p(1-p)/n}.

Question 9

1. The sampling distribution of $\hat p$ is likely approximately Normal and has relatively small variability. (correct answer)
2. The sampling distribution of $\hat p$ is likely approximately Normal but has no variability.
3. The sampling distribution of $\hat p$ is centered at 1000.
4. The sampling distribution of $\hat p$ is centered at $\sqrt{0.57(0.43)/1000}$.
5. The sampling distribution of $\hat p$ is not approximately Normal because $n$ is too large.

Explanation: This question evaluates understanding of sampling distributions for sample proportions in AP Statistics, emphasizing normality and variability with large n. The sampling distribution of (\hat{p}) is centered at p = 0.57, with a small standard deviation (\sqrt{0.57 \times 0.43 / 1000} $\approx$ 0.016), indicating low variability. Given np = 570 ≥ 10 and n(1-p) = 430 ≥ 10, the distribution is approximately normal. A frequent distractor is choice E, which incorrectly suggests that large n prevents normality, when actually larger n improves the normal approximation. Mini-lesson: Sampling distributions describe how statistics like (\hat{p}) vary across many random samples; for proportions, the mean is p, spread decreases with larger n, and normality holds when success and failure counts are sufficient. This setup allows us to model probabilities about sample results.

Question 10

1. The mean of $\hat p$ is 0.33, and the standard deviation is $\sqrt{0.33(0.67)/150}$. (correct answer)
2. The mean of $\hat p$ is $\sqrt{0.33(0.67)/150}$, and the standard deviation is 0.33.
3. The mean of $\hat p$ is 150, and the standard deviation is 0.33.
4. The mean of $\hat p$ is 0.67, and the standard deviation is $\sqrt{0.33(0.67)/150}$.
5. The standard deviation of $\hat p$ is 0 because the population proportion is fixed.

Explanation: This question examines the mean and sd of sampling distributions for sample proportions in AP Statistics. The mean is p=0.33, sd sqrt(0.330.67/150), describing center and spread. Choice B swaps mean and sd, a common error. A distractor is choice E, claiming sd=0 due to fixed p, ignoring sampling variation. In a mini-lesson, sampling distributions have mean p, sd sqrt(p(1-p)/n), normal if conditions met: here, 1500.33=49.5 ≥ 10, 150*0.67=100.5 ≥ 10. This enables probability calculations for î p.

Question 11

1. The sampling distribution of $\hat p$ is approximately Normal and centered at $0.52$. (correct answer)
2. The sampling distribution of $\hat p$ is exactly Normal because $\hat p$ is a proportion.
3. The sampling distribution of $\hat p$ is centered at $0.52$ and has standard deviation $\sqrt{\frac{\hat p(1-\hat p)}{n}}$ because $\hat p$ is the true parameter.
4. The sampling distribution of $\hat p$ is centered at $0.50$ because random samples should balance out to half.
5. The sampling distribution of $\hat p$ has no variability since the population proportion is fixed at $0.52$.

Explanation: This question evaluates understanding of sampling distributions for sample proportions in AP Statistics. The center of the sampling distribution of (\hat{p}) is the population proportion p = 0.52, and with n = 50, np = 26 and n(1-p) = 24 both exceeding 10, it's approximately normal, supporting choice A. The spread is quantified by the standard deviation (\sqrt{\frac{p(1-p)}{n}}), which reflects variability around the center. Distractor choice D suggests centering at 0.50 due to 'balancing,' but the distribution is unbiased and centered at p, not 0.50. Choice E wrongly asserts no variability, overlooking that samples fluctuate around p. Mini-lesson: sampling distributions describe how statistics like (\hat{p}) vary across many samples; they're centered at the parameter, have spread inversely related to sqrt(n), and approximate normality when expected successes and failures are at least 10.

Question 12

1. The sampling distribution of $\hat p$ will have more variability than it would if $n=100$. (correct answer)
2. The sampling distribution of $\hat p$ will have no variability because $p=0.35$.
3. The sampling distribution of $\hat p$ is centered at 25.
4. The sampling distribution of $\hat p$ is centered at $\sqrt{0.35(0.65)/25}$.
5. The sampling distribution of $\hat p$ becomes wider as $n$ increases.

Explanation: This evaluates spread in sampling distributions of proportions in AP Statistics. For $n=25$, variability is more than for $n=100$ since SD increases as $n$ decreases, making A correct. Distractor E says wider with larger $n$, but opposite is true. Mini-lesson: Distribution of $\hat p$ centered at $p$, spread $\sqrt{p(1-p)/n}$; smaller $n$ means wider distribution, more variable $\hat p$.

Question 13

1. The sampling distribution of $\hat p$ is centered at 0.12. (correct answer)
2. The sampling distribution of $\hat p$ is centered at 150.
3. The sampling distribution of $\hat p$ is centered at $1-0.12=0.88$.
4. The sampling distribution of $\hat p$ has no variability because 12% is fixed.
5. The sampling distribution of $\hat p$ is centered at $\sqrt{0.12(0.88)/150}$.

Explanation: This question checks the mean of $\hat p$'s distribution in AP Statistics. Center at $p=0.12$ for $n=150$, confirming A. Choice E distracts by using SD $\sqrt{0.12(0.88)/150}$ as center. Mini-lesson: Sampling distributions describe $\hat p$ variability; mean $p$, spread $\sqrt{p(1-p)/n}$, variability always present unlike choice D.

Question 14

1. The 10% condition is violated because $75$ is more than 10% of $600$. (correct answer)
2. The 10% condition is satisfied because $75$ is less than 10% of $600$.
3. The 10% condition depends on the sample proportion, so it cannot be checked in advance.
4. The 10% condition is always satisfied when $n\ge 30$.
5. The 10% condition is unnecessary for proportions, only for means.

Explanation: This question assesses understanding of the 10% condition in sampling distributions for sample proportions in AP Statistics. The sampling distribution of the sample proportion î p is approximately normal when certain conditions are met, including the 10% condition for finite populations when sampling without replacement. Here, the population size is 600, and the sample size n=75 exceeds 10% of 600 (which is 60), so the condition is violated, potentially affecting the independence assumption and the spread of the distribution. The center of the sampling distribution remains at the population proportion p=0.40, but the spread might not be accurately approximated by the usual formula due to the violation. A common distractor is choice B, which incorrectly states the condition is satisfied, misunderstanding that 75 is actually greater than 60. In a mini-lesson on sampling distributions, remember that for proportions, we need random sampling, n p ≥ 10 and n(1-p) ≥ 10 for normality, and n < 10% N for finite populations to ensure the standard deviation formula sqrt(p(1-p)/n) applies well. Always check these conditions before proceeding with inferences.

Question 15

1. The standard deviation of $\hat p$ is $\sqrt{0.28(0.72)/75}$. (correct answer)
2. The standard deviation of $\hat p$ is $\sqrt{0.28(0.72)}$.
3. The standard deviation of $\hat p$ is $\sqrt{75/(0.28\cdot0.72)}$.
4. The standard deviation of $\hat p$ is 75.
5. The standard deviation of $\hat p$ is 0 because 28% is fixed.

Explanation: This question tests the standard deviation formula for sampling distributions of sample proportions in AP Statistics. The sd is sqrt(p(1-p)/n) = sqrt(0.280.72/75), measuring spread around the center p=0.28. Choice B omits the division by n, incorrectly calculating variance instead. A distractor is choice E, claiming sd=0 because p is fixed, but sampling variability always exists. In a mini-lesson, the sampling distribution of î p is centered at p with sd sqrt(p(1-p)/n), and approximates normal when n p and n(1-p) ≥ 10. Here, 750.28≈21 and 75*0.72≈54, both fine. Larger n reduces spread, improving precision.

Question 16

1. The sampling distribution of $\hat p$ is centered at 0.62 and has some spread from sample to sample. (correct answer)
2. The sampling distribution of $\hat p$ is centered at 0.62 and has zero spread because $p$ is fixed.
3. The sampling distribution of $\hat p$ is centered at 100 and has spread 0.62.
4. The sampling distribution of $\hat p$ is centered at the first observed $\hat p$ and has no spread.
5. The sampling distribution of $\hat p$ is centered at $\sqrt{0.62(0.38)/100}$.

Explanation: This question tests knowledge of the center and spread in sampling distributions for sample proportions in AP Statistics. The sampling distribution of î p is centered at the true population proportion p=0.62, and it has variability from sample to sample due to sampling error. The spread is measured by the standard deviation sqrt(p(1-p)/n), which is not zero, reflecting that different samples yield different î p values. Choice B is a distractor, wrongly claiming zero spread because p is fixed, but the population parameter is fixed while sample statistics vary. In a mini-lesson, sampling distributions describe how statistics like î p behave over many samples: unbiased with mean p, and spread decreasing as n increases. Here, with n=100, the distribution is likely approximately normal if success-failure conditions hold. Understanding this helps in making inferences about populations from samples.

Question 17

1. The sampling distribution of $\hat p$ is centered at 0.48. (correct answer)
2. The sampling distribution of $\hat p$ is centered at 500.
3. The sampling distribution of $\hat p$ is centered at 0.52.
4. Because the population proportion is 0.48, $\hat p$ cannot be above 0.48.
5. The sampling distribution of $\hat p$ is centered at $\sqrt{0.48(0.52)/500}$.

Explanation: This assesses the mean of hat p's sampling distribution in AP Statistics. With p=0.48 and n=500, the center is 0.48, making A correct. Choice E is a distractor confusing center with standard deviation sqrt{0.48(0.52)/500}. Mini-lesson: The sampling distribution of hat p shows how proportions vary; mean equals p, spread is sqrt{p(1-p)/n}, smaller for larger n. hat p can exceed or fall below p due to variability, unlike choice D suggests.

Question 18

1. The mean of the sampling distribution of $\hat p$ is 0.65. (correct answer)
2. The mean of the sampling distribution of $\hat p$ is 80.
3. The sampling distribution of $\hat p$ has no spread because the population proportion is 0.65.
4. The sampling distribution of $\hat p$ is centered at the observed sample proportion from one sample.
5. The mean of the sampling distribution of $\hat p$ equals $\sqrt{0.65(0.35)/80}$.

Explanation: The focus is on the center of the sampling distribution of hat p in AP Statistics. For p=0.65 and n=80, the mean is 0.65, supporting choice A. Distractor E mixes up the mean with sqrt{0.65(0.35)/80}, the standard deviation for spread. Mini-lesson: Sampling distributions illustrate variation in hat p over repeated samples; centered at p, with spread sqrt{p(1-p)/n} that narrows as n increases. The distribution has spread, contrary to choice C, because samples vary randomly.

Question 19

1. The sampling distribution of $\hat p$ may be approximately Normal because $np$ and $n(1-p)$ are likely at least 10. (correct answer)
2. The sampling distribution of $\hat p$ is approximately Normal because $n$ is a perfect square.
3. The sampling distribution of $\hat p$ is exactly Normal because $\hat p$ is a proportion.
4. The sampling distribution of $\hat p$ is approximately Normal only if $p=0.44$ is unknown.
5. The sampling distribution of $\hat p$ has no variability because $p=0.44$ is fixed.

Explanation: This question assesses normality conditions for sampling distributions of sample proportions in AP Statistics. May be normal as n=36, for p=0.44, n p=15.84 ≥ 10, n(1-p)=20.16 ≥ 10. Center 0.44, spread sqrt(p(1-p)/n). Choice B is distractor, irrelevant perfect square. In a mini-lesson, success-failure conditions key, not other factors. Enables normal use despite modest n.

Question 20

1. The sampling distribution of $\hat p$ may not be approximately Normal because $np$ and/or $n(1-p)$ may be less than 10. (correct answer)
2. The sampling distribution of $\hat p$ is approximately Normal because $p$ is not close to 0 or 1.
3. The sampling distribution of $\hat p$ is exactly Normal because sampling is random.
4. The sampling distribution of $\hat p$ has no variability because $p$ is fixed.
5. The sampling distribution of $\hat p$ is centered at 20.

Explanation: This question tests knowledge of sampling distributions for sample proportions in AP Statistics, focusing on the conditions for normality. The sampling distribution of the sample proportion (\hat{p}) is centered at the true population proportion p = 0.41, with standard deviation (\sqrt{p(1-p)/n} = \sqrt{0.41 \times 0.59 / 20}). However, for the distribution to be approximately normal, both np and n(1-p) should be at least 10; here, np = 8.2 < 10, so the normality condition fails. A common distractor is choice E, which mistakenly centers the distribution at the sample size n=20 instead of p. In a mini-lesson on sampling distributions: when we repeatedly sample from a population, the sample proportions vary around p, and the distribution is approximately normal if the sample size is large enough relative to p, ensuring np ≥ 10 and n(1-p) ≥ 10. This variability reflects sampling error, not a fixed value.