Concluding Tests Population Proportion - AP Statistics

A public health researcher tests whether the proportion of adults in a county who have received a flu shot is greater than 50%. In a random sample of 300 adults, 171 reported receiving a flu shot. A one-proportion $z$ test was conducted for $H_0: p=0.50$ versus $H_a: p>0.50$ at $\alpha=0.05$ . The $p$ -value was 0.006, so the researcher rejected $H_0$ . Which conclusion is appropriate?
Answer: 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. 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.
A company's HR department claims that more than 35% of employees are satisfied with their health insurance plan. A random sample of 180 employees finds 73 satisfied. A one-sample $z$ test for a population proportion is conducted with $H_0: p=0.35$ and $H_a: p>0.35$ at $\alpha=0.05$ . The $p$ -value is 0.003, so $H_0$ is rejected. Which conclusion is appropriate?
Answer: There is sufficient evidence at the 0.05 level to conclude that more than 35% of all employees are satisfied with the health insurance plan. This problem tests concluding a right-tailed z-test with rejection. Given p=0.003 < α=0.05, we reject H0: p=0.35, indicating sufficient evidence for Ha: p>0.35, matching choice C. Choice D is a distractor confusing sample proportion with population. Option B overstates by claiming 'proof' and guarantees, which tests do not provide. In hypothesis testing, we discuss evidence supporting the alternative without implying certainty or causation, as in choice E. Mini-lesson: always tie conclusions to the hypotheses and alpha level, avoiding causal language.
A hospital reports that 8% of patients are readmitted within 30 days. A researcher randomly samples 300 discharged patients and finds 18 readmitted. A one-sample proportion test at $\alpha=0.05$ was performed with $H_0: p=0.08$ versus $H_a: p\ne 0.08$ . The test resulted in a $p$ -value of 0.20, so the researcher failed to reject $H_0$ . Which conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not sufficient evidence that the population readmission proportion differs from 0.08. This involves a two-tailed test for population proportion. With p-value (0.20) greater than α (0.05), we fail to reject H₀. When failing to reject H₀: p = 0.08 with a two-tailed alternative, we conclude there is not sufficient evidence that the population readmission proportion differs from 0.08. Choice B incorrectly claims the rate is exactly 8%. Choice C incorrectly rejects H₀. Choice D makes an unjustified claim about the population value. Choice E incorrectly implies causation. Failing to reject means we lack evidence against H₀, not that we've proven the null hypothesis true.
A streaming service claims that 40% of its subscribers watch documentaries at least once per month. A random sample of 120 subscribers finds that 54 report watching documentaries at least once per month. A one-sample proportion test is performed at $\alpha=0.05$ with $H_0:p=0.40$ and $H_a:p\ne 0.40$ . The test yields a $p$ -value of 0.28, so the decision is to fail to reject $H_0$ . Which conclusion is appropriate?
Answer: At the 0.05 level, there is not convincing evidence that the proportion of all subscribers who watch documentaries monthly is different from 0.40. This question involves a two-tailed test where we fail to reject the null hypothesis. The null hypothesis is p = 0.40 and the alternative is p ≠ 0.40. With a p-value of 0.28, which is greater than α = 0.05, we fail to reject H₀. This means we do not have convincing evidence that the proportion differs from 0.40. Option C correctly states this conclusion. Option A incorrectly claims that failing to reject H₀ proves it true, while Option B would be correct if we had rejected H₀. When we fail to reject H₀, we cannot conclude that the parameter equals the null value with certainty; we simply lack evidence to reject it.
A voter-registration group wants to know if fewer than 45% of eligible voters in a county are registered. In a random sample of 500 eligible voters, 214 were registered. A one-sample $z$ test for a population proportion was run at $\alpha=0.10$ with $H_0: p=0.45$ and $H_a: p<0.45$ . The $p$ -value was 0.078, so the group rejected $H_0$ . Which conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the county's population proportion of registered eligible voters is less than 0.45. This is a left-tailed test with α = 0.10. Since p-value (0.078) is less than α (0.10), we reject H₀. When rejecting H₀: p = 0.45 in favor of Hₐ: p < 0.45, we conclude there is convincing evidence that the county's population proportion of registered eligible voters is less than 0.45. Choice B incorrectly states an exact value. Choice C incorrectly fails to reject when p < α. Choice D misinterprets the p-value. Choice E only addresses the sample. The correct conclusion references the population parameter and states we have evidence that p < 0.45.
A city council wants to know whether a majority of residents support a proposed recycling ordinance. A random sample of 400 residents found 228 in favor. A one-sample $z$ test for a population proportion was conducted at significance level $\alpha=0.05$ to test $H_0: p=0.50$ versus $H_a: p>0.50$ . The test produced a $p$ -value of 0.018, so the council rejected $H_0$ . Which conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the population proportion of residents who support the ordinance is greater than 0.50. This question tests understanding of concluding a one-sample z-test for population proportion. Since the p-value (0.018) is less than α (0.05), we reject H₀. When we reject H₀: p = 0.50 in favor of Hₐ: p > 0.50, we conclude there is convincing evidence that the true population proportion is greater than 0.50. Choice A incorrectly interprets the p-value as 98.2% probability. Choice C incorrectly implies causation. Choice D incorrectly fails to reject when p < α. Choice E only refers to the sample, not the population. The correct conclusion must reference the population parameter and state that we have evidence p > 0.50.
A smartphone company claims that only 8% of its phones are returned within 30 days. A consumer group randomly sampled 150 purchases and found 20 returns. A one-proportion $z$ test was performed for $H_0: p=0.08$ versus $H_a: p>0.08$ at $\alpha=0.01$ . The test resulted in a $p$ -value of 0.034, so the group failed to reject $H_0$ . Which conclusion is appropriate?
Answer: At the 1% level, there is not sufficient evidence that the population return rate exceeds 8%. 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.
A candidate's campaign claims that 60% of voters in the district currently support the candidate. A random sample of 500 registered voters finds 280 who say they support the candidate. A one-sample $z$ test for a population proportion is conducted at $\alpha=0.05$ with $H_0:p=0.60$ and $H_a:p<0.60$ . The test produces a $p$ -value of 0.018, so the decision is to reject $H_0$ . Which conclusion is appropriate?
Answer: 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. 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.
A university newspaper reports that 30% of students favor switching to pass/fail grading. A random sample of 250 students found that 86 favor the switch. A one-sample $z$ test for a population proportion was conducted with $H_0: p=0.30$ versus $H_a: p>0.30$ at significance level $\alpha=0.05$ . The test produced a $p$ -value of 0.067, so the null hypothesis was not rejected. Which conclusion is appropriate?
Answer: At the 5% level, there is not enough evidence to conclude that the population proportion of all students who favor pass/fail grading is greater than 0.30. This question tests understanding of failing to reject H₀ in a one-tailed test. Since the p-value (0.067) is greater than α (0.05), we fail to reject H₀: p = 0.30. Choice B correctly states there is not enough evidence to conclude p > 0.30. Choice A incorrectly claims failing to reject proves H₀ is true. Choice C only describes the sample calculation without proper inference. Choice D misinterprets the p-value as P(H₀ is true). Choice E introduces effects on learning outcomes not tested. The p-value being close to α but still greater means the evidence was suggestive but not quite strong enough at the 5% level.
A nutrition label claims that 40% of customers prefer a new granola bar flavor. A company surveys a random sample of 120 customers and finds 54 who prefer the new flavor. They run a one-sample $z$ test for a population proportion with $H_0: p=0.40$ versus $H_a: p\ne 0.40$ at $\alpha=0.05$ . The $p$ -value is 0.27, so they do not reject $H_0$ . Which conclusion is appropriate?
Answer: There is insufficient evidence at the 0.05 level to conclude that the true preference proportion differs from 40%. This question probes conclusions from a two-sided z-test where the null is not rejected. With p=0.27 > α=0.05, we fail to reject H0: p=0.40, indicating insufficient evidence that the true preference differs from 40%, as in choice B. Choice C distracts by misinterpreting the p-value as the probability that H0 is true, which it is not. Option D incorrectly generalizes the sample count directly to the population percentage. In such tests, not rejecting H0 means we lack evidence to contradict it, but it does not confirm the null or suggest causation, unlike choice E. A mini-lesson: p-values measure evidence against H0, not the probability of hypotheses.
A university wants to check whether the proportion of students who prefer online sections is less than 40%. In a random sample of 250 students, 112 said they prefer online sections. A one-sample $z$ test for a population proportion was conducted with $H_0: p=0.40$ and $H_a: p<0.40$ at $\alpha=0.05$ . The $p$ -value was 0.214, so the university failed to reject $H_0$ . Which conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the true proportion of all students who prefer online sections is less than 0.40. This question evaluates understanding of conclusions in a one-sample z-test for a population proportion with a left-tailed alternative. The p-value of 0.214 is compared to α=0.05; since 0.214 > 0.05, we fail to reject H0: p=0.40, meaning there is not convincing evidence for Ha: p<0.40. The correct statement is that there is not convincing evidence that the true proportion of all students who prefer online sections is less than 0.40. Distractors may claim failing to reject proves the null (choice B), base conclusions solely on sample comparisons (choice C), or misinterpret the p-value as the probability H0 is false (choice D). In hypothesis testing, failing to reject H0 suggests the data are consistent with it, but this is not proof; other factors like sample size could influence power. Conclusions should emphasize lack of evidence rather than affirming the null hypothesis.
A streaming service claims that at least 70% of its subscribers are satisfied. A random sample of 250 subscribers finds 162 satisfied. A one-sample $z$ test for a population proportion is conducted with $H_0: p=0.70$ and $H_a: p<0.70$ at $\alpha=0.05$ . The $p$ -value is 0.018, so $H_0$ is rejected. Which conclusion is appropriate?
Answer: There is sufficient evidence at the 0.05 level to conclude that the proportion of all subscribers who are satisfied is less than 70%. This scenario examines proper interpretation of rejecting the null in a left-tailed z-test for a proportion. Given p=0.018 < α=0.05, we reject H0: p=0.70, providing sufficient evidence that the true satisfaction proportion is less than 70%, corresponding to choice B. Choice A is a distractor that mistakes the sample statistic for the exact population parameter. Option D overreaches by claiming 'proof' with 95% certainty, misusing the significance level. Hypothesis tests offer evidence based on the p-value comparison to alpha, but they do not prove claims or imply causation, as incorrectly suggested in choice E. Always frame conclusions in terms of evidence for the alternative hypothesis without absolute certainty.
A voter-advocacy group believes that more than 65% of eligible voters in a county are registered to vote. In a random sample of 160 eligible voters, 112 were registered. A one-sample $z$ test for a population proportion was conducted with $H_0: p=0.65$ versus $H_a: p>0.65$ at significance level $\alpha=0.05$ . The test produced a $p$ -value of 0.19, so the null hypothesis was not rejected. Which conclusion is appropriate?
Answer: At the 5% level, there is not enough evidence to conclude that the population proportion of eligible voters in the county who are registered is greater than 0.65. This question tests understanding of failing to reject H₀ with a large p-value. Since the p-value (0.19) is much greater than α (0.05), we fail to reject H₀: p = 0.65. Choice D correctly states there is not enough evidence to conclude p > 0.65. Choice A incorrectly claims failing to reject proves H₀. Choice B incorrectly claims evidence for Hₐ when we failed to reject. Choice C only describes the sample proportion without proper inference. Choice E misinterprets the p-value as P(Hₐ is true). A large p-value indicates the sample result is quite consistent with H₀, providing no evidence for the alternative.
A manufacturer advertises that only 3% of its light bulbs are defective. A quality-control inspector takes a random sample of 400 bulbs and finds 20 defective. A one-sample proportion test is performed at $\alpha=0.01$ with hypotheses $H_0:p=0.03$ and $H_a:p>0.03$ . The test results in a $p$ -value of 0.004, so the decision is to reject $H_0$ . Which conclusion is appropriate?
Answer: There is convincing evidence at the 0.01 level that the true defective rate is greater than 3%. This question assesses understanding of conclusions when rejecting the null hypothesis. With a p-value (0.004) less than α (0.01), we reject H₀: p = 0.03 in favor of Hₐ: p > 0.03. Choice A correctly states that there is convincing evidence at the 0.01 level that the true defective rate exceeds 3%. Choice D misinterprets the p-value as the probability of making an error in rejecting H₀, when it actually represents the probability of observing such extreme sample results if H₀ were true. Choice E incorrectly implies causation, which cannot be established from a hypothesis test. Remember that rejecting H₀ provides evidence for the alternative hypothesis but doesn't prove exact values or establish causal relationships.
A restaurant chain wants to check whether the proportion of customers who rate their experience as "excellent" is still 0.60 after a menu change. A random sample of 180 customers found that 96 rated their experience as excellent. A one-sample $z$ test for a population proportion was conducted with $H_0: p=0.60$ versus $H_a: p\ne 0.60$ at significance level $\alpha=0.05$ . The test produced a $p$ -value of 0.022, so the null hypothesis was rejected. Which conclusion is appropriate?
Answer: At the 5% level, there is convincing evidence that the population proportion of all customers who rate their experience as excellent is different from 0.60. This question tests understanding of a two-tailed test when H₀ is rejected. Since the p-value (0.022) is less than α (0.05), we reject H₀: p = 0.60. Choice A correctly states there is convincing evidence the population proportion differs from 0.60. Choice B incorrectly attributes causation to the menu change. Choice C misinterprets the p-value as P(H₀ is true). Choice D confuses sample proportion with population proportion. Choice E contradicts the rejection decision. In two-tailed tests, rejecting H₀ means evidence the parameter differs from the hypothesized value, without specifying direction in the conclusion.
A city council member claims that more than half of registered voters in the city support a proposed park. A random sample of 200 registered voters finds 118 who support it. A one-sample $z$ test for a population proportion is conducted with hypotheses $H_0: p=0.50$ and $H_a: p>0.50$ at significance level $\alpha=0.05$ . The test results in a $p$ -value of 0.041, so the null hypothesis is rejected. Which conclusion is appropriate?
Answer: There is sufficient evidence at the 0.05 level to conclude that more than half of all registered voters in the city support the proposed park. This question tests the skill of drawing appropriate conclusions from a one-sample z-test for a population proportion in AP Statistics. Since the p-value of 0.041 is less than the significance level α=0.05, we reject the null hypothesis H0: p=0.50, providing sufficient evidence to support the alternative hypothesis Ha: p>0.50. The correct conclusion states there is sufficient evidence at the 0.05 level that more than half of all registered voters support the proposed park, which matches choice B. A common distractor is choice A, which incorrectly equates the sample proportion directly to the population without inference. Another misleading option is choice C, which uses the word 'prove' and misinterprets the confidence level as certainty. In hypothesis testing, we never prove hypotheses but rather assess evidence against the null; failing to reject means insufficient evidence, while rejecting supports the alternative without implying causation, as seen in choice D.
A streaming service claims that 60% of its subscribers watch at least one documentary each month. A random sample of 250 subscribers is selected, and 138 report watching at least one documentary in the last month. A one-sample proportion test is conducted at $\alpha=0.05$ with hypotheses $H_0:p=0.60$ and $H_a:p\ne0.60$ . The test produces a $p$ -value of 0.03, so the decision is to reject $H_0$ . Which conclusion is appropriate?
Answer: 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. 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.
A city council member claims that a majority of city residents support building a new public park. To test this, a random sample of 200 residents is surveyed and 118 say they support the park. A one-sample $z$ test for a population proportion is conducted at significance level $\alpha=0.05$ with hypotheses $H_0:p=0.50$ and $H_a:p>0.50$ . The test yields a $p$ -value of 0.12, so the decision is to fail to reject $H_0$ . Which conclusion is appropriate?
Answer: There is not sufficient evidence at the 0.05 level to conclude that more than half of all city residents support building the park. 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.
A health clinic believes that more than 25% of adults in its county have not received a flu shot this season. A random sample of 400 adults found 118 had not received a flu shot. A one-sample $z$ test for a population proportion was performed with $H_0: p=0.25$ versus $H_a: p>0.25$ at $\alpha=0.05$ . The test produced a $p$ -value of 0.009, so $H_0$ was rejected. Which conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the true proportion of all county adults without a flu shot is greater than 0.25. 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.
A manufacturer advertises that 10% of its lightbulbs are defective. A quality-control team randomly samples 200 bulbs and finds 30 defective. They run a one-sample proportion test at $\alpha=0.01$ with hypotheses $H_0: p=0.10$ and $H_a: p\ne 0.10$ . The test yields a $p$ -value of 0.006, so they rejected $H_0$ . Which conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the true defective rate differs from 10% for all bulbs produced. This question involves a two-tailed test for population proportion. With p-value (0.006) less than α (0.01), we reject H₀. When rejecting H₀: p = 0.10 with a two-tailed alternative (p ≠ 0.10), we conclude there is convincing evidence that the true defective rate differs from 10%. Choice B incorrectly fails to reject. Choice C wrongly attributes causation to sampling. Choice D misinterprets the p-value as certainty about H₀. Choice E only addresses the sample, not the population. The proper conclusion references the population parameter and states we have evidence that p ≠ 0.10.

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