AP Statistics Quiz: Interpreting P Values

20 questions · exam conditions

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Interpreting P ValuesQuestion 1 of 20

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?

There is a 0.6% chance that

p

is greater than 0.35.

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.

There is a 0.6% chance that the null hypothesis is true.

A p-value of 0.006 means 0.6% of users enable two-factor authentication.

If the alternative hypothesis is true, there is a 0.6% chance of observing the sample result.

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AP Statistics Quiz

AP Statistics Quiz: Interpreting P Values

Practice Interpreting P Values in AP Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting P Values, 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.

All questions

Question 1

There is a 0.6% chance that $p$ is greater than 0.35.
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. (correct answer)
There is a 0.6% chance that the null hypothesis is true.
A p-value of 0.006 means 0.6% of users enable two-factor authentication.
If the alternative hypothesis is true, there is a 0.6% chance of observing the sample result.

Explanation: 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.

Question 2

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?

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. (correct answer)
There is a 4.1% chance that the true renewal rate is 60%.
There is a 4.1% chance that exactly 112 out of 200 customers renew.
Because $p=0.041$ , 4.1% of customers will not renew.
If the alternative hypothesis is true, the probability of observing a sample proportion of 0.56 is 0.041.

Explanation: 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.

Question 3

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?

There is a 0.2% chance that the manufacturer's claim $\mu=1000$ is correct.
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. (correct answer)
There is a 0.2% chance that the sample mean is less than 1000 hours.
A p-value of 0.002 means the probability the bulbs last less than 1000 hours is 0.002.
If $H_0$ is false, there is a 0.2% chance of getting the observed sample mean.

Explanation: 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.

Question 4

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?

There is a 0.9% chance that the true mean concentration is at most 10 ppm.
If $H_0$ is true, the probability of observing a sample mean concentration at least as large as the one observed is 0.009. (correct answer)
There is a 0.9% chance that the agency made a Type I error.
If the true mean concentration exceeds 10 ppm, the probability of obtaining the observed sample mean is 0.009.
A p-value of 0.009 means 0.9% of river samples exceed 10 ppm.

Explanation: 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.

Question 5

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?

There is a 5.1% chance that the mean reaction time is exactly 250 ms.
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. (correct answer)
Because the p-value is 0.051, the null hypothesis must be accepted as true.
If the true mean is not 250 ms, the probability of getting the observed sample mean is 0.051.
A p-value of 0.051 means 5.1% of individual reaction times are different from 250 ms.

Explanation: 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).

Question 6

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?

If $H_0$ is true, there is an 18% chance of getting a sample mean of 12 oz or more.
There is an 18% chance that $\mu$ is greater than 12 oz.
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. (correct answer)
There is an 18% chance that the null hypothesis is false.
An 18% p-value means 18% of all pours exceed 12 oz.

Explanation: 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.

Question 7

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?

There is an 8% chance the fish in the lake have mean weight exactly 2.5 kg.
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. (correct answer)
There is an 8% chance that the sample mean equals the observed value.
A p-value of 0.08 means 8% of fish weigh 2.5 kg.
There is an 8% chance that the null hypothesis is false.

Explanation: 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.

Question 8

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?

Because $p=0.012$ , the probability the mean commute time is 28 minutes is 0.012.
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. (correct answer)
There is a 1.2% chance that the alternative hypothesis is true.
There is a 1.2% chance of selecting a resident with a 28-minute commute.
If $H_0$ is true, 1.2% of all samples will have a mean exactly equal to the observed sample mean.

Explanation: 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.

Question 9

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?

There is a 4.9% chance that the tutoring program does not increase the mean score above 75.
If $H_0$ is true, the probability of getting a sample mean at least as large as the observed sample mean is 0.049. (correct answer)
There is a 4.9% chance that the sample mean is greater than 75.
A p-value of 0.049 means 4.9% of students scored above 75.
If the alternative hypothesis is true, the probability of observing a result at least as extreme as the sample is 0.049.

Explanation: 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.

Question 10

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?

There is a 3% chance that the null hypothesis $H_0$ is true.
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. (correct answer)
There is a 3% chance that the sample mean equals 84 minutes when $\mu=90$ .
Because $p=0.03$ , 3% of students spend exactly 84 minutes on homework per night.
If the alternative hypothesis is true, there is a 3% chance of obtaining a result like the sample mean of 84 minutes.

Explanation: 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.

Question 11

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?

There is a 74% chance that the null hypothesis is true.
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. (correct answer)
There is a 74% chance that $\mu$ is exactly 14 oz.
A p-value of 0.74 means 74% of boxes have more than 14 oz.
If the alternative hypothesis is true, there is a 74% chance of getting the observed sample mean.

Explanation: 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.

Question 12

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?

If $p=0.08$ , the probability of getting a sample proportion as small as (or smaller than) the observed sample proportion is 0.21. (correct answer)
There is a 21% chance that the true return rate is less than 8%.
There is a 21% chance that the null hypothesis is false.
A p-value of 0.21 means 21% of patients return within 30 days.
If the alternative hypothesis is true, the probability of getting the observed sample proportion is 0.21.

Explanation: 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.

Question 13

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?

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. (correct answer)
There is a 60% chance the soup's true mean sodium content is below 800 mg.
There is a 60% chance that the label claim is correct.
If the true mean sodium content is less than 800 mg, the probability of observing the sample mean is 0.60.
A p-value of 0.60 means 60% of soup cans have sodium content less than 800 mg.

Explanation: 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.

Question 14

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?

There is a 4.9% chance that exactly 50% of residents support the tax.
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. (correct answer)
There is a 4.9% chance that the sample proportion is incorrect due to random sampling.
If the true proportion differs from 0.50, the probability that the null hypothesis will be rejected is 0.049.
Because the p-value is 0.049, the probability that $H_0$ is true is 0.049.

Explanation: 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.

Question 15

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?

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. (correct answer)
There is an 18% chance that $H_0$ is false.
There is an 18% chance that the mean lifetime of all bulbs is exactly 1000 hours.
If the true mean lifetime is not 1000 hours, the probability of observing the sample mean is 0.18.
Because the p-value is 0.18, there is strong evidence that the mean differs from 1000 hours.

Explanation: 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).

Question 16

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?

There is a 3% chance that the null hypothesis $H_0$ is true.
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. (correct answer)
There is a 3% chance that the tutoring program increases the mean score above 70.
If the tutoring program truly increases the mean score above 70, the probability of getting the observed sample mean is 0.03.
Because the p-value is 0.03, 3% of all students in the district scored above 70 due to the tutoring program.

Explanation: 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.

Question 17

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?

There is a 7% chance that the mean diameter of all ball bearings is exactly 5.00 mm.
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. (correct answer)
There is a 7% chance that the engineer's conclusion is correct.
If the true mean diameter is not 5.00 mm, the probability of getting the observed sample mean is 0.07.
A p-value of 0.07 means 7% of individual ball bearings have diameter different from 5.00 mm.

Explanation: 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.

Question 18

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?

There is an 8% probability that the new protocol does not reduce infections.
If the new protocol truly reduces infections, there is an 8% probability of observing the sample result.
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. (correct answer)
There is an 8% chance that $H_a$ is true.
The p-value 0.08 means that 8% of months will have fewer than 12 infections under the new protocol.

Explanation: 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.

Question 19

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?

There is a 0.1% chance that the new layout does not increase the purchase rate.
If the true purchase rate is greater than 0.12, the probability of obtaining the observed sample proportion is 0.001.
If $H_0$ is true, the probability of observing a sample proportion at least as large as the one observed is 0.001. (correct answer)
The probability that $H_0$ is true is 0.001.
A p-value of 0.001 means that 0.1% of visitors made a purchase in the sample.

Explanation: 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.

Question 20

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?

There is a 20% chance that the true approval proportion is exactly 0.40.
If the true approval proportion is less than 0.40, the probability of observing the sample proportion is 0.20.
If $H_0$ is true, the probability of observing a sample proportion as low as (or lower than) the one observed is 0.20. (correct answer)
There is a 20% probability that $H_0$ is false.
A p-value of 0.20 means that 20% of voters disapprove of the policy.

Explanation: 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).