AP Statistics Quiz: Concluding Tests Population Proportion

What this quiz covers

This quiz focuses on Concluding Tests Population Proportion, 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. Reject $H_0$; there is convincing evidence that the true proportion of all registered voters in the state who are unaffiliated is less than 0.15. (correct answer)
2. Reject $H_0$; there is a 4.1% chance that fewer than 15% of voters are unaffiliated.
3. Fail to reject $H_0$; since the sample had 103 unaffiliated voters, the population proportion must be 103/800.
4. Reject $H_0$; being unaffiliated causes the proportion to be less than 0.15.
5. Reject $H_0$; therefore fewer than 15% of the sampled voters are unaffiliated.

Explanation: The skill is drawing valid conclusions from a left-tailed one-sample z-test for a population proportion. Since the p-value of 0.041 < α=0.05, we reject H0: p=0.15, finding convincing evidence for Ha: p<0.15 that the true proportion of unaffiliated voters is less than 0.15. This conclusion correctly references the population and evidence. Distractors include misinterpreting p-value as probability of the alternative (choice B), equating sample proportion to population (choice C), or suggesting causation (choice D). Mini-lesson: Rejection provides statistical support for Ha, but it's about evidence, not absolute proof or causal links. Always specify the hypothesis context and avoid sample-only statements.

Question 2

1. Because 118 out of 200 in the sample support the plan, more than half of all residents definitely support building the library.
2. At the 5% level, there is sufficient evidence that the population proportion of all residents who support the plan is greater than 0.50. (correct answer)
3. Since the $p$-value is 0.028, there is a 2.8% chance that $H_0$ is true.
4. Rejecting $H_0$ proves that building a new library will cause more residents to support the city council.
5. There is sufficient evidence that more than 50% of the 200 sampled residents support the plan.

Explanation: This question tests understanding of concluding a one-proportion z-test for population proportions. Since the p-value (0.028) is less than α (0.05), we reject H₀ and conclude there is sufficient evidence that the population proportion exceeds 0.50. Option B correctly states this conclusion about the population parameter. Option A incorrectly makes a definitive claim about all residents based on sample data. Option C misinterprets the p-value as the probability that H₀ is true. Option D incorrectly claims causation and goes beyond the scope of the hypothesis test. Option E only addresses the sample, not the population. Remember: hypothesis test conclusions are always about population parameters, not sample statistics.

Question 3

1. Because the $p$-value is 0.41, there is a 41% chance the true click rate is exactly 60%.
2. At the 5% level, there is not sufficient evidence that the population click proportion differs from 0.60. (correct answer)
3. Failing to reject $H_0$ proves that exactly 60% of all visitors click the button.
4. Since 286 out of 500 clicked, the population proportion must be 0.572.
5. There is not sufficient evidence that the sample click proportion differs from 0.60.

Explanation: This question involves a two-tailed test where we fail to reject H₀. The p-value (0.41) is greater than α (0.05), so we fail to reject H₀ and conclude there is not sufficient evidence that the population proportion differs from 0.60. Option B correctly states this conclusion. Option A misinterprets the p-value as a probability about the parameter. Option C incorrectly claims that failing to reject proves H₀ true. Option D confuses the sample proportion with the population proportion. Option E incorrectly focuses on the sample rather than the population. Key concept: in two-tailed tests, we look for evidence of any difference from the hypothesized value.

Question 4

1. At the 5% level, there is not sufficient evidence that the population proportion of flights delayed more than 30 minutes differs from 0.12. (correct answer)
2. Since only 9 of 100 flights were delayed, the true delay rate is less than 12%.
3. Failing to reject $H_0$ proves the airline's claim is correct.
4. The $p$-value of 0.39 means there is a 39% chance that $H_0$ is true.
5. There is not sufficient evidence that the sample proportion differs from 0.12.

Explanation: This question involves a two-tailed test where we fail to reject H₀. The p-value (0.39) is greater than α (0.05), so we fail to reject H₀ and conclude there is not sufficient evidence that the population proportion differs from 0.12. Option A correctly states this conclusion. Option B makes an incorrect definitive claim based on the sample. Option C wrongly claims that failing to reject proves the airline's claim. Option D misinterprets the p-value as the probability H₀ is true. Option E incorrectly focuses on the sample rather than the population. Key principle: large p-values indicate the sample result is consistent with H₀.

Question 5

1. Rejecting $H_0$ shows that getting 8 hours of sleep causes better attendance at the school.
2. At the 10% level, there is sufficient evidence that the population proportion of students who get at least 8 hours of sleep is less than 0.30. (correct answer)
3. Because 28 out of 120 students reported 8 hours, fewer than 30% of all students definitely get at least 8 hours of sleep.
4. The $p$-value of 0.072 means there is a 7.2% chance that $H_0$ is true.
5. There is sufficient evidence that fewer than 30% of the sampled students get at least 8 hours of sleep.

Explanation: This problem tests understanding of left-tailed tests for population proportions. Since the p-value (0.072) is less than α (0.10), we reject H₀ and conclude there is sufficient evidence that the population proportion is less than 0.30. Option B correctly states this conclusion about the population parameter. Option A incorrectly implies causation between sleep and attendance. Option C makes a definitive claim rather than a statistical conclusion. Option D misinterprets the p-value. Option E only addresses the sample, not the population. Remember: statistical significance at the 10% level means we have evidence to support the alternative hypothesis about the population.

Question 6

1. There is sufficient evidence that the population proportion of students who own a bicycle is not 0.40.
2. At the 1% level, there is not sufficient evidence that the population proportion of students who own a bicycle differs from 0.40. (correct answer)
3. Since the $p$-value is 0.018, there is a 1.8% chance that $H_0$ is true.
4. Failing to reject $H_0$ shows that exactly 40% of all students own a bicycle.
5. There is not sufficient evidence that the sample proportion differs from 0.40.

Explanation: This question involves a two-tailed test where we fail to reject $H_0$ at the 1% level. The p-value (0.018) is greater than $\alpha$ (0.01), so we fail to reject $H_0$ and conclude there is not sufficient evidence that the population proportion differs from 0.40. Option B correctly states this conclusion. Option A would be correct at the 5% level but not at the 1% level used here. Option C misinterprets the p-value. Option D incorrectly claims that failing to reject proves $H_0$. Option E focuses on the sample instead of the population. Key insight: the significance level determines our decision threshold.

Question 7

1. There is sufficient evidence at the 0.05 level that the true proportion of all subscribers who watch at least one documentary each month is different from 0.60. (correct answer)
2. Rejecting $H_0$ proves that exactly 55.2% of all subscribers watch at least one documentary each month.
3. Because the $p$-value is 0.03, there is a 3% chance the claim $p=0.60$ is correct.
4. Since 138 out of 250 watched a documentary, the service's claim is false for the sample.
5. Watching documentaries causes the subscriber proportion to differ from 0.60.

Explanation: This question examines conclusions from a two-tailed test where H₀ is rejected. With p-value (0.03) less than α (0.05), we reject H₀: p = 0.60 in favor of Hₐ: p ≠ 0.60. Choice A correctly states there is sufficient evidence that the true proportion differs from 0.60. Choice B incorrectly claims we can determine an exact population proportion from rejecting H₀—we only know it's different from 0.60, not what it equals. Choice C misinterprets the p-value as the probability of H₀ being true. In two-tailed tests, rejecting H₀ means the parameter is significantly different from the hypothesized value in either direction, but doesn't specify the exact value or direction without examining the sample statistic.

Question 8

1. Reject $H_0$; there is convincing evidence that the true proportion of all county adults without a flu shot is greater than 0.25. (correct answer)
2. Reject $H_0$; there is a 0.9% chance that more than 25% of adults lack a flu shot.
3. Fail to reject $H_0$; a $p$-value this small means the null is likely correct.
4. Reject $H_0$; not getting a flu shot causes the clinic to believe the proportion is greater than 0.25.
5. Reject $H_0$; since 118 of the sample lacked a flu shot, more than 25% of the sample lacked a flu shot.

Explanation: The skill involves correctly concluding a right-tailed one-sample z-test for a population proportion. With a p-value of 0.009 < α=0.05, we reject H0: p=0.25, supporting Ha: p>0.25 with convincing evidence that the true proportion of all county adults without a flu shot is greater than 0.25. This is the appropriate interpretation, avoiding errors like misstating p-value meaning (choice B) or implying low p-value supports the null (choice C). Distractors often introduce causation (choice D) or limit statements to the sample (choice E). A mini-lesson: Low p-values indicate the observed data are improbable under H0, providing evidence for the alternative, but correlation does not imply causation. Always tie conclusions back to the population and the test's evidential strength.

Question 9

1. Fail to reject $H_0$; at the 0.01 level, there is not convincing evidence that the true proportion of all town residents who donated differs from 0.35. (correct answer)
2. Reject $H_0$; because the $p$-value is 0.022, the proportion must differ from 0.35.
3. Fail to reject $H_0$; therefore the true proportion of residents who donated is exactly 0.35.
4. Because the $p$-value is 0.022, there is a 2.2% chance that $H_0$ is true.
5. Fail to reject $H_0$; since 58 of the sampled residents donated, the conclusion is that 58 residents in the town donated.

Explanation: This question assesses interpreting a two-sided one-sample z-test for a population proportion. Since the p-value of 0.022 > α=0.01, we fail to reject H0: p=0.35, concluding there is not convincing evidence at the 0.01 level that the true proportion differs from 0.35. This avoids erroneous rejection (choice B) or claiming proof of the null (choice C). Distractors often misstate p-value as probability H0 is true (choice D) or confuse sample with population counts (choice E). A key lesson: Significance levels determine the threshold for evidence; even if p is small but above alpha, we fail to reject without calling H0 true. Conclusions should be population-focused and evidence-based.

Question 10

1. At the 5% level, there is sufficient evidence that the population proportion of bulbs lasting 1,000 hours is less than 0.95. (correct answer)
2. Because 72 of 80 bulbs lasted 1,000 hours, fewer than 95% of all bulbs definitely last 1,000 hours.
3. Rejecting $H_0$ proves the company intentionally makes bulbs that fail early.
4. The $p$-value of 0.002 means there is a 0.2% chance the inspector made a mistake in counting.
5. There is sufficient evidence that fewer than 95% of the 80 tested bulbs last 1,000 hours.

Explanation: This problem tests concluding a left-tailed test about quality control. Since the p-value (0.002) is less than α (0.05), we reject H₀ and conclude there is sufficient evidence that the population proportion is less than 0.95. Option A correctly states this conclusion about all bulbs produced. Option B makes an incorrect definitive claim. Option C introduces causation and intent not addressed by the test. Option D misinterprets what the p-value represents. Option E only addresses the tested bulbs, not the population. Important: rejecting H₀ provides evidence against the manufacturer's claim but doesn't prove intent or causation.

Question 11

1. At the 0.05 level, there is convincing evidence that the true proportion of all district voters who support the candidate is less than 0.60. (correct answer)
2. Rejecting $H_0$ means fewer than 60% of voters will support the candidate on election day.
3. Because 280 out of 500 voters in the sample support the candidate, the campaign's claim is false for the sample, but nothing can be said about the population.
4. The $p$-value of 0.018 means there is a 1.8% chance that $H_0$ is correct.
5. The campaign caused support to fall below 60%.

Explanation: This question tests understanding of a left-tailed hypothesis test conclusion. The null hypothesis is p = 0.60 and the alternative is p < 0.60. With a p-value of 0.018, which is less than α = 0.05, we reject H₀. This provides convincing evidence at the 0.05 level that the true proportion of all district voters who support the candidate is less than 0.60. Option A correctly states this conclusion. Option D misinterprets the p-value as the probability that H₀ is correct, which is incorrect. The p-value represents the probability of observing our sample result or more extreme, assuming H₀ is true, not the probability that H₀ itself is true.

Question 12

1. At the 5% level, there is sufficient evidence that the population proportion of eligible voters who are registered is less than 0.65. (correct answer)
2. Because only 248 of 400 were registered, fewer than 65% of all eligible voters are registered for sure.
3. Rejecting $H_0$ proves that not being registered causes people to be ineligible to vote.
4. The $p$-value of 0.021 means there is a 2.1% chance that the alternative hypothesis is false.
5. There is sufficient evidence that fewer than 65% of the sampled voters are registered.

Explanation: This problem tests concluding a left-tailed test about voter registration. Since the p-value (0.021) is less than α (0.05), we reject H₀ and conclude there is sufficient evidence that the population proportion is less than 0.65. Option A correctly states this conclusion about all eligible voters. Option B makes an incorrect definitive claim. Option C introduces irrelevant causation about registration and eligibility. Option D misinterprets the p-value as relating to the alternative hypothesis. Option E only addresses the sample, not the population. Important: rejecting H₀ in a left-tailed test provides evidence the true proportion is below the hypothesized value.

Question 13

1. At the 1% level, there is not sufficient evidence that the population return rate exceeds 8%. (correct answer)
2. Because 20 out of 150 were returned, the true return rate is greater than 8%.
3. Failing to reject $H_0$ proves that exactly 8% of all phones are returned within 30 days.
4. The $p$-value of 0.034 means there is a 3.4% chance the sample had 20 returns.
5. There is not sufficient evidence that more than 8% of the 150 sampled phones were returned.

Explanation: This question involves concluding a one-proportion z-test when we fail to reject H₀. The p-value (0.034) is greater than α (0.01), so we fail to reject H₀ and conclude there is not sufficient evidence that the population proportion exceeds 0.08. Option A correctly states this conclusion at the 1% significance level. Option B makes an incorrect definitive claim about the true rate. Option C wrongly claims that failing to reject H₀ proves it true. Option D misinterprets what the p-value represents. Option E incorrectly focuses on the sample rather than the population. Key principle: failing to reject H₀ means insufficient evidence against it, not proof that it's true.

Question 14

1. At the 5% level, there is sufficient evidence that the population proportion of adults in the county who received a flu shot is greater than 0.50. (correct answer)
2. Because 171 adults in the sample received a flu shot, more than half of the population definitely received a flu shot.
3. Rejecting $H_0$ proves that getting a flu shot causes adults to participate in surveys.
4. The $p$-value of 0.006 means there is a 0.6% chance that $H_a$ is true.
5. There is sufficient evidence that more than half of the sampled adults received a flu shot.

Explanation: This problem tests understanding of right-tailed tests for population proportions. Since the p-value (0.006) is less than α (0.05), we reject H₀ and conclude there is sufficient evidence that the population proportion exceeds 0.50. Option A correctly states this conclusion about all adults in the county. Option B makes an incorrect definitive claim. Option C introduces irrelevant causation. Option D misinterprets the p-value as relating to H_a rather than H₀. Option E only addresses the sample, not the population. Remember: hypothesis test conclusions always refer to population parameters, and rejecting H₀ supports the alternative hypothesis.

Question 15

1. Because the $p$-value is 0.018, there is a 1.8% chance that $H_0$ is true.
2. Reject $H_0$; there is convincing evidence that the true proportion of all district high school students who get at least 8 hours of sleep is less than 0.30. (correct answer)
3. Fail to reject $H_0$; there is not convincing evidence that the proportion is less than 0.30.
4. Since 48 out of 200 students got at least 8 hours, fewer than 30% of the sampled students get at least 8 hours.
5. Reject $H_0$; getting at least 8 hours of sleep causes students to be in a district where fewer than 30% sleep 8 hours.

Explanation: This question assesses the skill of drawing appropriate conclusions from a one-sample z-test for a population proportion. The decision logic involves comparing the p-value of 0.018 to the significance level α=0.05; since 0.018 < 0.05, we reject the null hypothesis H0: p=0.30 in favor of Ha: p<0.30. The correct conclusion states that there is convincing evidence that the true proportion of all district high school students who get at least 8 hours of sleep is less than 0.30. Common distractors include misinterpreting the p-value as the probability that H0 is true (choice A), confusing the sample proportion with the population (choice D), or implying causation (choice E). In hypothesis testing, rejecting H0 provides statistical evidence supporting the alternative hypothesis, but it does not prove causation or make definitive statements about the sample alone. Remember, conclusions should always refer to the population parameter and the strength of evidence based on the test decision.

Question 16

1. Reject $H_0$; there is convincing evidence that the population proportion of students who have taken an online course is greater than 0.25. (correct answer)
2. Reject $H_0$; there is a 0.4% chance that more than 25% of students have taken an online course.
3. Fail to reject $H_0$; since 40/120 is just a sample, no inference can be made.
4. Reject $H_0$; therefore taking online courses causes students to be counted in the sample.
5. Reject $H_0$; we can conclude that more than 25% of the 120 sampled students have taken an online course.

Explanation: This is a right-tailed test for population proportion. Since p-value (0.004) is less than α (0.05), we reject H₀. When rejecting H₀: p = 0.25 in favor of Hₐ: p > 0.25, we conclude there is convincing evidence that the population proportion of students who have taken an online course is greater than 0.25. Choice B misinterprets the p-value. Choice C incorrectly fails to reject. Choice D incorrectly implies causation. Choice E only refers to the sample. The correct conclusion must reference the population parameter and state we have evidence that p > 0.25.

Question 17

1. Failing to reject $H_0$ means the proportion of all patients who rate their care as excellent is exactly 70%.
2. There is not sufficient evidence at the 0.05 level to conclude that more than 70% of all patients rate their care as excellent. (correct answer)
3. Because the $p$-value is 0.18, there is an 18% chance that $H_0$ is true.
4. Since 92 of the 120 sampled patients rated care as excellent, the hospital has proven that more than 70% of all patients do so.
5. Improving care would cause the proportion of excellent ratings to be greater than 70%.

Explanation: This question tests understanding of failing to reject H₀ in a right-tailed test. Since p-value (0.18) exceeds α (0.05), we fail to reject H₀: p = 0.70. Choice B correctly states there is not sufficient evidence to conclude that more than 70% rate care as excellent. Choice A incorrectly claims that failing to reject H₀ means the parameter equals exactly 0.70—we simply lack evidence against this value. Choice C misinterprets the p-value as the probability H₀ is true. When we fail to reject H₀, we're saying the data doesn't provide strong enough evidence against the null hypothesis, not that we've proven it true. The distinction between "failing to reject" and "accepting" H₀ is crucial in statistical inference.

Question 18

1. Rejecting $H_0$ shows that eating breakfast at school causes students to be less than 40% of the population.
2. There is sufficient evidence at the 0.10 level to conclude that the proportion of all students who eat breakfast at school is less than 0.40. (correct answer)
3. Because the $p$-value is 0.08, there is an 8% chance that $H_0$ is true.
4. Since 52 of 150 students ate breakfast at school, fewer than 40% of students in the sample ate breakfast at school.
5. This proves that less than 40% of all students eat breakfast at school.

Explanation: This question tests interpretation of rejecting H₀ in a left-tailed test. Since the p-value (0.08) is less than α (0.10), we reject H₀: p = 0.40 in favor of Hₐ: p < 0.40. Choice B correctly states that there is sufficient evidence at the 0.10 level to conclude the proportion is less than 0.40. Choice C misinterprets the p-value as the probability that H₀ is true. Choice D merely restates the sample calculation (52/150 = 0.347) without making an inference about the population. Choice E incorrectly uses the word "proves"—hypothesis tests provide evidence but never prove conclusions with certainty. Statistical conclusions should always acknowledge the significance level and use appropriate language like "sufficient evidence" rather than absolute statements.

Question 19

1. There is sufficient evidence at the 0.05 level to conclude that the proportion of all registered voters in the county who are undecided is less than 0.25. (correct answer)
2. Because $H_0$ was rejected, the outreach group is 95% certain that exactly 21% of voters are undecided.
3. The $p$-value of 0.01 means there is a 1% chance that $H_0$ is true.
4. Since 105 of the sampled voters were undecided, we can only conclude that 21% of the sample is undecided (and nothing about the county).
5. This test shows that being contacted by the outreach group causes voters to be less likely to be undecided.

Explanation: This question assesses interpretation of rejecting H₀ in a left-tailed test. With p-value (0.01) less than α (0.05), we reject H₀: p = 0.25 in favor of Hₐ: p < 0.25. Choice A correctly states there is sufficient evidence that the proportion of undecided voters is less than 0.25. Choice B incorrectly claims we can determine an exact percentage with certainty—hypothesis tests provide evidence about inequalities, not exact values. Choice C misinterprets the p-value as the probability that H₀ is true. The p-value represents the probability of observing sample results as extreme as ours if H₀ were true, not the probability that H₀ itself is true. Statistical conclusions should focus on the evidence for or against hypotheses, not absolute certainties.

Question 20

1. Because the $p$-value is 0.12, there is a 12% chance that $H_0$ is true.
2. Failing to reject $H_0$ proves that exactly 50% of all residents support the park.
3. There is not sufficient evidence at the 0.05 level to conclude that more than half of all city residents support building the park. (correct answer)
4. Since 118 of the 200 sampled residents support the park, a majority of the city residents must support it.
5. Building a new park would cause more than half of residents to support the city council member.

Explanation: This question tests understanding of conclusions when failing to reject the null hypothesis in a one-sample proportion test. Since the p-value (0.12) is greater than α (0.05), we fail to reject H₀, meaning we don't have sufficient evidence to support the alternative hypothesis that p > 0.50. Choice C correctly states this conclusion using appropriate statistical language. Choice A misinterprets the p-value as the probability that H₀ is true, when it actually represents the probability of getting sample results as extreme as observed if H₀ were true. Choice B incorrectly claims that failing to reject H₀ proves the null hypothesis. When we fail to reject H₀, we simply lack evidence against it—we never prove a null hypothesis true through hypothesis testing.