Concluding Tests Population Proportion
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AP Statistics › Concluding Tests Population Proportion
A state agency wants to determine whether fewer than 15% of registered voters in the state are unaffiliated with any political party. In a random sample of 800 registered voters, 103 were unaffiliated. A one-sample $z$ test for a population proportion was performed with $H_0: p=0.15$ and $H_a: p<0.15$ at $\alpha=0.05$. The $p$-value was 0.041, so the agency rejected $H_0$. Which conclusion is appropriate?
Reject $H_0$; there is a 4.1% chance that fewer than 15% of voters are unaffiliated.
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.
Fail to reject $H_0$; since the sample had 103 unaffiliated voters, the population proportion must be 103/800.
Reject $H_0$; being unaffiliated causes the proportion to be less than 0.15.
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.
A city council wants to know whether a majority of residents support building a new public library. A random sample of 200 residents found that 118 support the plan. A one-proportion $z$ test was conducted for $H_0: p=0.50$ versus $H_a: p>0.50$ at significance level $\alpha=0.05$. The test produced a $p$-value of 0.028, so the council rejected $H_0$. Which conclusion is appropriate?
Since the $p$-value is 0.028, there is a 2.8% chance that $H_0$ is true.
There is sufficient evidence that more than 50% of the 200 sampled residents support the plan.
Because 118 out of 200 in the sample support the plan, more than half of all residents definitely support building the library.
Rejecting $H_0$ proves that building a new library will cause more residents to support the city council.
At the 5% level, there is sufficient evidence that the population proportion of all residents who support the plan is greater than 0.50.
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.
A website designer claims that 60% of visitors click a certain button. After a random sample of 500 visitors, 286 clicked the button. A one-proportion $z$ test was carried out for $H_0: p=0.60$ versus $H_a: p\neq 0.60$ at $\alpha=0.05$. The test produced a $p$-value of 0.41, so the designer failed to reject $H_0$. Which conclusion is appropriate?
There is not sufficient evidence that the sample click proportion differs from 0.60.
Because the $p$-value is 0.41, there is a 41% chance the true click rate is exactly 60%.
At the 5% level, there is not sufficient evidence that the population click proportion differs from 0.60.
Failing to reject $H_0$ proves that exactly 60% of all visitors click the button.
Since 286 out of 500 clicked, the population proportion must be 0.572.
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.
An airline states that 12% of its flights are delayed by more than 30 minutes. A random sample of 100 flights found 9 such delays. A one-proportion $z$ test was run for $H_0: p=0.12$ versus $H_a: p\neq 0.12$ at $\alpha=0.05$. The $p$-value was 0.39, so the analyst failed to reject $H_0$. Which conclusion is appropriate?
The $p$-value of 0.39 means there is a 39% chance that $H_0$ is true.
At the 5% level, there is not sufficient evidence that the population proportion of flights delayed more than 30 minutes differs from 0.12.
There is not sufficient evidence that the sample proportion differs from 0.12.
Failing to reject $H_0$ proves the airline’s claim is correct.
Since only 9 of 100 flights were delayed, the true delay rate is less than 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.
A school nurse believes that fewer than 30% of students get at least 8 hours of sleep on school nights. In a random sample of 120 students, 28 reported getting at least 8 hours. A one-proportion $z$ test was conducted for $H_0: p=0.30$ versus $H_a: p<0.30$ at $\alpha=0.10$. The $p$-value was 0.072, so the nurse rejected $H_0$. Which conclusion is appropriate?
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.
There is sufficient evidence that fewer than 30% of the sampled students get at least 8 hours of sleep.
Rejecting $H_0$ shows that getting 8 hours of sleep causes better attendance at the school.
The $p$-value of 0.072 means there is a 7.2% chance that $H_0$ is true.
Because 28 out of 120 students reported 8 hours, fewer than 30% of all students definitely 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.
A university wants to check whether the proportion of students who own a bicycle is 40%. A random sample of 250 students found that 112 own a bicycle. A one-proportion $z$ test was performed for $H_0: p=0.40$ versus $H_a: p \neq 0.40$ at $\alpha=0.01$. The $p$-value was 0.018, so the university failed to reject $H_0$. Which conclusion is appropriate?
There is not sufficient evidence that the sample proportion differs from 0.40.
At the 1% level, there is not sufficient evidence that the population proportion of students who own a bicycle differs from 0.40.
Failing to reject $H_0$ shows that exactly 40% of all students own a bicycle.
There is sufficient evidence that the population proportion of students who own a bicycle is not 0.40.
Since the $p$-value is 0.018, there is a 1.8% chance that $H_0$ is true.
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.
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?
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.
Since 138 out of 250 watched a documentary, the service’s claim is false for the sample.
Rejecting $H_0$ proves that exactly 55.2% of all subscribers watch at least one documentary each month.
Because the $p$-value is 0.03, there is a 3% chance the claim $p=0.60$ is correct.
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.
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?
Reject $H_0$; not getting a flu shot causes the clinic to believe the proportion is greater than 0.25.
Reject $H_0$; there is convincing evidence that the true proportion of all county adults without a flu shot is greater than 0.25.
Reject $H_0$; there is a 0.9% chance that more than 25% of adults lack a flu shot.
Reject $H_0$; since 118 of the sample lacked a flu shot, more than 25% of the sample lacked a flu shot.
Fail to reject $H_0$; a $p$-value this small means the null is likely correct.
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.
A nonprofit organization believes that the proportion of residents in a town who have donated to any charity in the past year is not 35%. A random sample of 180 residents found that 58 had donated. A one-sample $z$ test for a population proportion was performed with $H_0: p=0.35$ and $H_a: p\ne 0.35$ at $\alpha=0.01$. The $p$-value was 0.022, so the nonprofit failed to reject $H_0$. Which conclusion is appropriate?
Because the $p$-value is 0.022, there is a 2.2% chance that $H_0$ is true.
Reject $H_0$; because the $p$-value is 0.022, the proportion must differ from 0.35.
Fail to reject $H_0$; since 58 of the sampled residents donated, the conclusion is that 58 residents in the town donated.
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.
Fail to reject $H_0$; therefore the true proportion of residents who donated is exactly 0.35.
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.
A manufacturer advertises that at least 95% of its light bulbs last 1,000 hours. A quality inspector randomly tested 80 bulbs and found 72 lasted 1,000 hours. A one-proportion $z$ test was conducted for $H_0: p=0.95$ versus $H_a: p<0.95$ at $\alpha=0.05$. The $p$-value was 0.002, so the inspector rejected $H_0$. Which conclusion is appropriate?
Because 72 of 80 bulbs lasted 1,000 hours, fewer than 95% of all bulbs definitely last 1,000 hours.
Rejecting $H_0$ proves the company intentionally makes bulbs that fail early.
There is sufficient evidence that fewer than 95% of the 80 tested bulbs last 1,000 hours.
At the 5% level, there is sufficient evidence that the population proportion of bulbs lasting 1,000 hours is less than 0.95.
The $p$-value of 0.002 means there is a 0.2% chance the inspector made a mistake in counting.
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.