Interpreting p-Values - AP Statistics

A city planner believes the mean commute time for residents is $\mu = 28$ minutes. A random sample of 80 residents is used to test $H_0: \mu = 28$ versus $H_a: \mu \ne 28$ at $\alpha = 0.01$ . The p-value from the test is $p = 0.012$ . Which interpretation of the p-value is correct?
Answer: If $\mu=28$ minutes, the probability of getting a sample mean at least as extreme as the observed one (in either direction) is 0.012. This question assesses interpreting p-values in a two-sided test for a population mean in AP Statistics. The p-value of 0.012 is the conditional probability of a sample mean at least as extreme as observed (in either direction) given H0: μ = 28 is true. A common distractor is choice C, which misinterprets the p-value as the probability that the alternative hypothesis is true, a misunderstanding of hypothesis testing logic. In a mini-lesson, p-values help decide if data is surprising under H0: here, 0.012 exceeds α = 0.01 slightly, so we fail to reject H0, but it would be significant at higher α. Choice B correctly notes the two-sided nature with 'at least as extreme... (in either direction)'. Avoid pitfalls in A, D, and E, such as equating p-values to probabilities of specific values or sample equalities.
A city tests whether the proportion of residents who support a new public transit tax is different from 50%. The hypotheses are $H_0: p=0.50$ versus $H_a: p\ne 0.50$ , using $\alpha=0.05$ . A random sample yields a p-value of 0.049. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of getting a sample proportion at least as far from 0.50 as the one observed (in either direction) is 0.049. This question involves interpreting a p-value from a two-tailed test about population proportions. The p-value represents the conditional probability of obtaining a sample proportion at least as extreme as observed in either direction, given that the null hypothesis is true. With $H_0$ : p = 0.50 and a two-tailed alternative, the p-value of 0.049 means that if exactly 50% of residents support the tax, there's a 4.9% chance of getting a sample proportion at least as far from 0.50 as observed. Choice B correctly captures this interpretation. Students often mistakenly think p-values represent the probability of hypotheses being true (Choices A and E) or relate to sampling error rates (Choice C). The key insight is that p-values are conditional probabilities calculated assuming $H_0$ is true, measuring how unusual our observed data would be under that assumption.
A wildlife biologist tests whether the mean weight of a certain fish species in a lake differs from $\mu = 2.5$ kg. Using a random sample, the biologist conducts a two-sided one-sample $t$ test: $H_0: \mu = 2.5$ versus $H_a: \mu \ne 2.5$ at $\alpha = 0.05$ . The p-value is $p = 0.08$ . Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of getting a sample mean at least as far from 2.5 kg as the observed sample mean (in either direction) is 0.08. This question assesses the skill of interpreting p-values in a two-sided t-test for a mean in AP Statistics. The p-value of 0.08 is the conditional probability of a sample mean at least as far from 2.5 kg as observed (in either direction) given H0: μ = 2.5 is true. A common distractor is choice E, which wrongly interprets the p-value as the probability that H0 is false, a frequent misunderstanding. In a mini-lesson, p-values evaluate surprise under H0: 0.08 is above α = 0.05, so we fail to reject H0, indicating insufficient evidence of a difference in mean weight. Choice B correctly includes the two-sided aspect with 'at least as far... (in either direction)'. Avoid errors in A, C, and D, such as equating p-values to chances of exact values or population proportions.
A hospital investigates whether a new hand-washing protocol reduces the mean number of hospital-acquired infections per month compared with the previous mean of 12. The hypotheses are $H_0: \mu=12$ versus $H_a: \mu<12$ , with $\alpha=0.10$ . The p-value from the test is 0.08. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a sample mean number of infections as low as (or lower than) the one observed is 0.08. This question tests understanding of p-value interpretation in a left-tailed test about hospital infections. The p-value represents the conditional probability of observing data at least as extreme as what was observed, assuming the null hypothesis is true. For this left-tailed test ( $H_a$ : μ < 12), the p-value of 0.08 means that if the true mean number of infections is 12, there's an 8% chance of observing a sample mean as low as or lower than what was observed. Choice C correctly states this interpretation. Common errors include thinking the p-value represents the probability of $H_0$ being false (Choice A) or the probability under the alternative hypothesis (Choice B). Remember that p-values are always calculated under the assumption that $H_0$ is true - they measure how surprising our data would be if there were no effect.
A manufacturer claims its light bulbs last an average of 1000 hours. A consumer group tests this claim using $H_0: \mu=1000$ versus $H_a: \mu\ne 1000$ at $\alpha=0.01$ . The test yields a p-value of 0.18. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of getting a test statistic at least as extreme as the one observed (in either direction) is 0.18. This question involves interpreting a p-value from a two-tailed test about light bulb lifetimes. The p-value represents the conditional probability of obtaining a test statistic at least as extreme as observed in either direction, assuming the null hypothesis is true. With a two-tailed test ( $H_a$ : μ ≠ 1000) and p-value of 0.18, this means if the true mean is 1000 hours, there's an 18% chance of getting a sample mean at least as far from 1000 as observed. Choice A correctly captures this interpretation. Students often confuse p-values with the probability of hypotheses being true or false (Choices B and C), or misinterpret what constitutes "strong evidence" - a p-value of 0.18 actually suggests weak evidence against $H_0$ . The p-value is a conditional probability: P(data as extreme or more | $H_0$ is true), not P( $H_0$ is true | data).
A nutrition label claims a cereal box contains a mean of $\mu = 14$ oz of cereal. A quality-control analyst tests $H_0: \mu = 14$ versus $H_a: \mu \ne 14$ at $\alpha = 0.05$ using a random sample of boxes and obtains p-value $p = 0.74$ . Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a sample mean at least as far from 14 oz as the observed sample mean (in either direction) is 0.74. This question evaluates the skill of interpreting p-values in a two-sided hypothesis test for a mean in AP Statistics. The p-value of 0.74 is the conditional probability of observing a sample mean at least as far from 14 oz as the observed one (in either direction) if H0: μ = 14 is true. A frequent distractor is choice A, which flips the interpretation to the probability that H0 is true, a common confusion with Bayesian concepts. As a mini-lesson, p-values measure data compatibility with H0: a large p-value like 0.74 means the data is very plausible under H0, so we fail to reject it at α = 0.05. Choice B accurately describes the two-tailed calculation. Errors in C, D, and E include treating p-values as chances of exact parameters or proportions in the population.
A coffee shop owner believes the mean amount of coffee dispensed by a machine is $\mu = 12$ oz. After maintenance, a technician tests $H_0: \mu = 12$ versus $H_a: \mu > 12$ at $\alpha = 0.01$ using a random sample of 40 pours and obtains a p-value of $p = 0.18$ . Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, there is an 18% chance of getting a sample mean at least as large as the observed sample mean (in the direction of $H_a$ ) due to random sampling variability. This question evaluates the skill of interpreting p-values in a one-sided hypothesis test for a population mean in AP Statistics. The p-value of 0.18 is the conditional probability of observing a sample mean at least as large as the one obtained, given that H0: μ = 12 is true, accounting for random sampling variability in the direction of Ha: μ > 12. A frequent distractor is choice A, which omits the directionality of the alternative hypothesis, making it seem like a two-sided interpretation instead of one-sided. As a mini-lesson, p-values quantify how compatible the data is with the null hypothesis: a larger p-value like 0.18 suggests the data is not surprising under H0, so we fail to reject it at α = 0.01. Choice C accurately reflects the one-tailed nature by specifying 'at least as large as the observed sample mean (in the direction of Ha)'. Misinterpretations like those in B, D, and E confuse p-values with probabilities of hypotheses or population parameters.
A tech company tests whether a new website layout increases the proportion of visitors who make a purchase above the current rate of 0.12. The hypotheses are $H_0: p=0.12$ versus $H_a: p>0.12$ , with $\alpha=0.05$ . The p-value from the test is 0.001. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a sample proportion at least as large as the one observed is 0.001. This question involves interpreting a very small p-value in a right-tailed test about website conversion rates. The p-value represents the conditional probability of obtaining a sample proportion at least as extreme as observed, assuming the null hypothesis is true. For this right-tailed test ( $H_a$ : p > 0.12), the p-value of 0.001 means that if the true purchase rate is 0.12, there's only a 0.1% chance of observing a sample proportion as large as or larger than what was observed. Choice C correctly captures this interpretation. The very small p-value provides strong evidence against $H_0$ . Common errors include confusing the p-value with the probability of $H_0$ being true (Choice D) or false (Choice A), or misinterpreting it as a descriptive statistic about the sample (Choice E). P-values are always conditional probabilities calculated under the assumption that $H_0$ is true.
A technology company claims that the proportion of users who enable two-factor authentication is $p = 0.35$ . A random sample of users is selected and a one-proportion test is performed for $H_0: p = 0.35$ versus $H_a: p > 0.35$ at $\alpha = 0.05$ . The p-value is $p = 0.006$ . Which interpretation of the p-value is correct?
Answer: If $p=0.35$ , the probability of getting a sample proportion at least as large as the observed sample proportion (in the direction of $H_a$ ) is 0.006. This question evaluates interpreting p-values in a one-sided right-tailed test for a proportion in AP Statistics. The p-value of 0.006 is the conditional probability of getting a sample proportion at least as large as observed, given H0: p = 0.35 is true, in the direction of Ha: p > 0.35. A frequent distractor is choice C, which mistakes the p-value for the probability that H0 is true. As a mini-lesson, p-values assess evidence by computing data extremity under H0: a small 0.006 provides strong grounds to reject H0 at α = 0.05, suggesting more than 35% enable authentication. Choice B accurately specifies 'at least as large... (in the direction of Ha)'. Misinterpretations in A, D, and E include reversing conditioning or applying to population subsets.
A quality-control engineer tests whether the mean diameter of ball bearings differs from the target of 5.00 mm. The hypotheses are $H_0: \mu=5.00$ versus $H_a: \mu\ne 5.00$ , at $\alpha=0.10$ . The p-value is 0.07. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a sample mean diameter at least as far from 5.00 mm as the one observed (in either direction) is 0.07. This question involves interpreting a p-value from a two-tailed test in quality control. The p-value represents the conditional probability of obtaining a sample mean at least as extreme as observed in either direction, assuming the null hypothesis is true. With $H_0$ : μ = 5.00 and a two-tailed alternative, the p-value of 0.07 means that if the true mean diameter is exactly 5.00 mm, there's a 7% chance of observing a sample mean at least as far from 5.00 mm as what was observed. Choice B correctly captures this interpretation. Since 0.07 < 0.10 (the significance level), we would reject $H_0$ at the 10% level. Common errors include thinking p-values represent probabilities about individual measurements (Choice E) or about the truth of conclusions (Choice C). P-values measure how surprising our data would be if $H_0$ were true, not the probability that $H_0$ is true.
A researcher tests whether a new tutoring program increases the mean math score above 75. For a random sample of students in the program, a one-sample $t$ test is run for $H_0: \mu = 75$ versus $H_a: \mu > 75$ at $\alpha = 0.05$ , resulting in p-value $p = 0.049$ . Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of getting a sample mean at least as large as the observed sample mean is 0.049. This question tests interpreting p-values in a one-sided right-tailed t-test in AP Statistics. The p-value of 0.049 is the conditional probability of getting a sample mean at least as large as observed, assuming H0: μ = 75 is true. A common distractor is choice E, which incorrectly conditions on the alternative hypothesis instead of the null. In a mini-lesson, p-values provide evidence against H0 by quantifying extremeness under it: 0.049 is just below α = 0.05, suggesting borderline evidence to reject H0 and conclude the program increases scores. Choice B correctly specifies the one-tailed direction 'at least as large'. Avoid mistakes like those in A, C, and D, which confuse p-values with probabilities of hypotheses or direct population percentages.
A school district tests whether a new tutoring program increases the mean math score above the historical mean of 70. The hypotheses are $H_0: \mu=70$ versus $H_a: \mu>70$ , and the test is conducted at significance level $\alpha=0.05$ . The resulting p-value is 0.03. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of obtaining a sample mean at least as large as the one observed (in the direction of $H_a$ ) is 0.03. This question tests understanding of p-value interpretation in a one-tailed hypothesis test. The p-value represents the conditional probability of obtaining a test statistic at least as extreme as observed, given that the null hypothesis is true. Since this is a right-tailed test ( $H_a$ : μ > 70), the p-value of 0.03 means there's a 3% chance of getting a sample mean at least as large as the one observed if the true mean is actually 70. Choice B correctly states this conditional probability interpretation. Common misconceptions include thinking the p-value is the probability that $H_0$ is true (Choice A) or that it represents the probability of the alternative hypothesis being true (Choice C). Remember: p-values are always calculated assuming $H_0$ is true, and they tell us about the likelihood of our data, not the likelihood of hypotheses.
A nutritionist tests whether the mean sodium content of a brand of soup is less than the label claim of 800 mg. The hypotheses are $H_0: \mu=800$ versus $H_a: \mu<800$ , at $\alpha=0.05$ . The p-value is 0.60. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of obtaining a sample mean sodium content as low as (or lower than) the observed sample mean is 0.60. This question tests understanding of p-value interpretation in a left-tailed test about sodium content. The p-value represents the conditional probability of observing a test statistic at least as extreme as what was observed, assuming the null hypothesis is true. For this left-tailed test ( $H_a$ : μ < 800), a p-value of 0.60 means that if the true mean sodium content is 800 mg, there's a 60% chance of observing a sample mean as low as or lower than what was observed. Choice A correctly states this interpretation. A large p-value like 0.60 indicates the observed data is quite likely under $H_0$ , providing no evidence against the null hypothesis. Common mistakes include thinking the p-value represents probabilities about the truth of hypotheses (Choices B and C) or about individual measurements (Choice E). Remember: p-values tell us about the probability of data given $H_0$ , not the probability of $H_0$ given data.
An environmental agency tests whether the mean concentration of a pollutant in a river exceeds the legal limit of 10 ppm. The hypotheses are $H_0: \mu=10$ versus $H_a: \mu>10$ , and the test uses $\alpha=0.01$ . The p-value is 0.009. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a sample mean concentration at least as large as the one observed is 0.009. This question involves interpreting a p-value in a right-tailed test about pollutant concentration. The p-value represents the conditional probability of observing a sample mean at least as extreme as what was observed, assuming the null hypothesis is true. For this right-tailed test ( $H_a$ : μ > 10), the p-value of 0.009 means that if the true mean concentration is exactly 10 ppm, there's a 0.9% chance of observing a sample mean as large as or larger than what was observed. Choice B correctly captures this interpretation. Since 0.009 < 0.01 (the significance level), we would reject $H_0$ , providing strong evidence that the mean exceeds the legal limit. Common mistakes include thinking the p-value represents probabilities about hypotheses being true (Choice A) or about Type I error rates for specific tests (Choice C). Remember: p-values are calculated assuming $H_0$ is true.
A political scientist tests whether the proportion of voters who approve of a policy is less than 0.40. The hypotheses are $H_0: p=0.40$ versus $H_a: p<0.40$ , with $\alpha=0.05$ . The p-value is 0.20. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a sample proportion as low as (or lower than) the one observed is 0.20. This question tests understanding of p-value interpretation in a left-tailed test about voter approval. The p-value represents the conditional probability of obtaining a sample proportion at least as extreme as observed, given that the null hypothesis is true. For this left-tailed test ( $H_a$ : p < 0.40), the p-value of 0.20 means that if the true approval proportion is 0.40, there's a 20% chance of observing a sample proportion as low as or lower than what was observed. Choice C correctly states this interpretation. A p-value of 0.20 is quite large, indicating the observed data is reasonably likely under $H_0$ and providing no evidence against it. Students often confuse p-values with probabilities about hypotheses (Choices A and D) or misinterpret them as descriptive statistics (Choice E). The key is remembering that p-values are conditional probabilities: P(data | $H_0$ is true).
A hospital reports that 8% of patients return within 30 days. An auditor takes a random sample of 150 discharged patients and performs a one-proportion $z$ test for $H_0: p = 0.08$ versus $H_a: p < 0.08$ at $\alpha = 0.05$ . The p-value is $p = 0.21$ . Which interpretation of the p-value is correct?
Answer: If $p=0.08$ , the probability of getting a sample proportion as small as (or smaller than) the observed sample proportion is 0.21. This question examines interpreting p-values in a one-sided left-tailed z-test for a proportion in AP Statistics. The p-value of 0.21 is the conditional probability of a sample proportion as small as or smaller than observed if H0: p = 0.08 is true. A typical distractor is choice C, which misstates the p-value as the chance that H0 is false, inverting the conditioning. For a mini-lesson, p-values indicate how likely the data is under H0: a larger value like 0.21 means no strong evidence against H0 at α = 0.05, so we fail to reject the reported return rate. Choice A properly uses 'as small as (or smaller than)' for the left tail. Choices B, D, and E err by confusing p-values with posterior probabilities or alternative conditioning.
A manufacturer claims its light bulbs last an average of $\mu = 1000$ hours. A consumer group tests a random sample of 25 bulbs and performs a one-sample $t$ test for $H_0: \mu = 1000$ versus $H_a: \mu < 1000$ at $\alpha = 0.05$ . The test produces p-value $p = 0.002$ . Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of obtaining a sample mean as low as (or lower than) the observed sample mean is 0.002. This question examines the skill of interpreting p-values in a one-sided left-tailed t-test for a mean in AP Statistics. The p-value of 0.002 is the conditional probability of obtaining a sample mean as low as or lower than observed, assuming H0: μ = 1000 is true. A typical distractor is choice A, which incorrectly states the p-value as the probability that H0 is correct, reversing the conditioning. For a mini-lesson, p-values assess evidence against H0 by calculating the probability of data extremes under it: a very small p-value like 0.002 provides strong evidence to reject H0 at α = 0.05, indicating the bulbs likely last less than claimed. Choice B properly specifies the one-tailed direction 'as low as (or lower than)'. Choices C, D, and E exemplify errors like confusing p-values with unconditional probabilities or switching to the alternative hypothesis.
A school district claims that the mean time students spend on homework per night is $\mu = 90$ minutes. A random sample of 60 students reports an average of 84 minutes. A one-sample $t$ test is performed for $H_0: \mu = 90$ versus $H_a: \mu \ne 90$ at significance level $\alpha = 0.05$ , and the p-value is $p = 0.03$ . Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, there is a 3% chance of obtaining a sample mean at least as far from 90 minutes as the one observed (in either direction) just by random sampling variability. This question assesses the skill of interpreting p-values in the context of a two-sided hypothesis test for a population mean in AP Statistics. The p-value of 0.03 represents the conditional probability of obtaining a sample mean at least as extreme as 84 minutes (in either direction from 90) given that the null hypothesis H0: μ = 90 is true, due to random sampling variability. A common distractor is choice A, which incorrectly interprets the p-value as the probability that H0 is true, confusing it with posterior probability rather than the conditional probability under H0. In a mini-lesson on p-values, remember that they measure the strength of evidence against the null hypothesis: a small p-value like 0.03 indicates the observed data would be rare if H0 were true, potentially leading to rejection at α = 0.05. Choice B correctly captures this by emphasizing the assumption of H0 and the extremeness in both tails for a two-sided test. Avoid mistaking p-values for probabilities of hypotheses or specific sample outcomes, as seen in choices C, D, and E.
A company advertises that 60% of its customers renew their subscription. A random sample of 200 customers finds 112 renewals. A one-proportion $z$ test is conducted for $H_0: p = 0.60$ versus $H_a: p \ne 0.60$ at $\alpha = 0.10$ , yielding p-value $p = 0.041$ . Which interpretation of the p-value is correct?
Answer: If the true renewal rate is 60%, the probability of getting a sample proportion at least as far from 0.60 as 0.56 (in either direction) is 0.041. This question tests the skill of interpreting p-values in a two-sided one-proportion z-test in AP Statistics. The p-value of 0.041 is the conditional probability of getting a sample proportion at least as far from 0.60 as 0.56 (in either direction) if H0: p = 0.60 is true. A common distractor is choice B, which wrongly treats the p-value as the probability that the null hypothesis is true, a classic error mixing conditional and posterior probabilities. In a mini-lesson, p-values indicate the likelihood of the observed data or more extreme under H0: here, 0.041 is small enough to reject H0 at α = 0.10, suggesting evidence against the claimed 60% renewal rate. Choice A correctly includes the two-sided extremeness and the assumption of H0. Avoid errors like those in C, D, and E, which misapply the p-value to exact outcomes or alternative hypotheses.
A researcher tests whether the mean reaction time for a task is different from 250 ms. The hypotheses are $H_0: \mu=250$ versus $H_a: \mu\ne 250$ , using $\alpha=0.05$ . The p-value is 0.051. Which interpretation of the p-value is correct?
Answer: If $H_0$ is true, the probability of observing a result at least as extreme as the sample result (in either direction) is 0.051. This question tests understanding of p-value interpretation in a two-tailed test about reaction times. The p-value represents the conditional probability of obtaining a test statistic at least as extreme as observed in either direction, given that the null hypothesis is true. With $H_0$ : μ = 250 and a two-tailed alternative, the p-value of 0.051 means that if the true mean reaction time is 250 ms, there's a 5.1% chance of observing a sample mean at least as far from 250 ms as what was observed. Choice B correctly states this interpretation. With α = 0.05, we would fail to reject $H_0$ since 0.051 > 0.05, but this doesn't mean we "accept" $H_0$ as true (Choice C is incorrect). The p-value doesn't tell us about individual measurements (Choice E) or probabilities under the alternative hypothesis (Choice D).

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