AP Statistics Quiz: Setting Up Tests For Population Mean

20 questions · exam conditions

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Setting Up Tests For Population MeanQuestion 1 of 20

A city's water department states that the mean lead concentration in tap water is 5 parts per billion (ppb). A public health researcher wants to check whether the true mean lead concentration is different from 5 ppb. A random sample of 50 homes is tested, and the sample mean lead concentration is 5.8 ppb. Which hypotheses are appropriate for a one-sample test of a population mean lead concentration?

H_0: \mu=5

vs.

H_a: \mu>5

H_0: \mu=5

vs.

H_a: \mu\ne5

H_0: \bar{x}=5.8

vs.

H_a: \bar{x}\ne5.8

H_0: p=0.05

vs.

H_a: p\ne0.05

H_0: \mu=5.8

vs.

H_a: \mu\ne5.8

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AP Statistics Quiz

AP Statistics Quiz: Setting Up Tests For Population Mean

Practice Setting Up Tests For Population Mean 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 Setting Up Tests For Population Mean, 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

$H_0: \mu=5$ vs. $H_a: \mu>5$
$H_0: \mu=5$ vs. $H_a: \mu\ne5$ (correct answer)
$H_0: \bar{x}=5.8$ vs. $H_a: \bar{x}\ne5.8$
$H_0: p=0.05$ vs. $H_a: p\ne0.05$
$H_0: \mu=5.8$ vs. $H_a: \mu\ne5.8$

Explanation: This question tests hypothesis setup for a one-sample population mean test regarding lead concentration in water. The water department claims a mean of 5 ppb, and the researcher wants to check if it's different, implying a two-tailed test with H₀: μ = 5 vs. Hₐ: μ ≠ 5 to capture any deviation. This aligns perfectly with the 'different from' language, using the population mean μ and the claimed value in the null. Distractors include one-tailed tests that assume a specific direction (like >5), using the sample mean (5.8 ppb) in hypotheses, employing \bar{x} instead of μ, or switching to a proportion test with p, which doesn't fit since lead concentration is a quantitative mean, not a proportion. Mini-lesson: For non-directional suspicions ('different' or 'changed'), use a two-sided alternative Hₐ: μ ≠ μ₀, where μ₀ is the stated value. Always focus on the population parameter μ, not sample statistics. This setup allows evidence to support rejection of the null if the mean is either higher or lower than claimed.

Question 2

A city transit agency reports that the mean wait time for a bus on a certain route is 12 minutes. A commuter advocacy group randomly samples $n=35$ bus arrivals and records wait times, obtaining a sample mean of $\bar{x}=13.1$ minutes. The group wants to test whether the true mean wait time is different from 12 minutes. Which hypotheses are appropriate?

$H_0: \mu=12$ vs. $H_a: \mu\ne 12$ (correct answer)
$H_0: \mu\ne 12$ vs. $H_a: \mu=12$
$H_0: \bar{x}=12$ vs. $H_a: \bar{x}\ne 12$
$H_0: p=12$ vs. $H_a: p\ne 12$
$H_0: \mu=13.1$ vs. $H_a: \mu\ne 13.1$

Explanation: This question tests setting up a two-tailed hypothesis test for a population mean. The transit agency reports a mean of 12 minutes, giving us H₀: μ = 12. Since the group wants to test if the mean is 'different from' 12 minutes (not specifically higher or lower), we need a two-tailed test: Hₐ: μ ≠ 12. Option B incorrectly reverses the null and alternative hypotheses. Option C uses x̄ (sample mean) instead of μ (population mean) in the hypotheses. Option D uses p, which is for proportions, not means. Option E incorrectly uses the sample mean value (13.1) in the hypotheses—we test the claimed value, not the observed value. For 'different from' questions, always use ≠ in the alternative hypothesis.

Question 3

A cereal manufacturer states that the mean weight of cereal in its boxes is 18.0 ounces. A quality-control inspector randomly selects $n=12$ boxes and finds a sample mean weight of $\bar{x}=17.8$ ounces. The inspector wants to test whether the true mean weight is less than 18.0 ounces. Which hypotheses are appropriate?

$H_0: \mu=18.0$ vs. $H_a: \mu<18.0$ (correct answer)
$H_0: \mu=17.8$ vs. $H_a: \mu<17.8$
$H_0: \mu\le 18.0$ vs. $H_a: \mu>18.0$
$H_0: \bar{x}=18.0$ vs. $H_a: \bar{x}<18.0$
$H_0: p=18.0$ vs. $H_a: p<18.0$

Explanation: This question involves setting up a left-tailed test for a population mean. The manufacturer claims μ = 18.0 ounces, which becomes our null hypothesis: H₀: μ = 18.0. The inspector wants to test if the true mean is less than 18.0 ounces, so we use a left-tailed alternative: Hₐ: μ < 18.0. Option B incorrectly uses the sample mean (17.8) in the hypotheses—we test claims about population parameters, not sample statistics. Option C has the inequality in H₀ pointing the wrong way for a 'less than' test. Option D uses x̄ instead of μ for the population parameter. Option E uses p, which is for proportions, not means. When the research question asks about 'less than,' use < in the alternative hypothesis.

Question 4

A smartphone manufacturer states that the mean time to fully charge its new phone model is 80 minutes. A reviewer randomly tests $n=15$ phones and finds a sample mean charge time of $\bar{x}=84$ minutes. The reviewer wants to test whether the true mean charge time is greater than 80 minutes. Which hypotheses are appropriate?

$H_0: \mu=80$ vs. $H_a: \mu>80$ (correct answer)
$H_0: \mu=84$ vs. $H_a: \mu>84$
$H_0: \bar{x}=80$ vs. $H_a: \bar{x}>80$
$H_0: p=80$ vs. $H_a: p>80$
$H_0: \mu\ge 80$ vs. $H_a: \mu<80$

Explanation: This question requires setting up a right-tailed hypothesis test for a population mean. The manufacturer states the mean is 80 minutes, so H₀: μ = 80. The reviewer wants to test if the true mean is greater than 80 minutes, making this a right-tailed test with Hₐ: μ > 80. Option B incorrectly uses the sample mean value (84) in the hypotheses—we test the claimed population value. Option C uses x̄ (sample mean) instead of μ (population mean). Option D uses p, which is for proportions, not means. Option E has an inequality in H₀ and reverses the direction of the test. For 'greater than' questions, the alternative hypothesis uses > to indicate the direction of interest.

Question 5

A university dining hall claims the mean sodium content of a particular lunch entrée is 900 mg per serving. A nutrition student randomly samples $n=30$ servings and measures sodium content, finding a sample mean of $\bar{x}=872$ mg. The student wants to test whether the true mean sodium content is less than 900 mg. Which hypotheses are appropriate?

$H_0: \mu=872$ vs. $H_a: \mu<872$
$H_0: \mu=900$ vs. $H_a: \mu<900$ (correct answer)
$H_0: \mu=900$ vs. $H_a: \mu>900$
$H_0: \bar{x}=900$ vs. $H_a: \bar{x}<900$
$H_0: p=900$ vs. $H_a: p<900$

Explanation: This question involves setting up a left-tailed test for a population mean. The dining hall claims the mean sodium content is 900 mg, so H₀: μ = 900. The student wants to test if the true mean is less than 900 mg, making this a left-tailed test with Hₐ: μ < 900. Option A incorrectly uses the sample mean value (872) in the null hypothesis—we test the claimed value, not the observed value. Option C has the wrong direction in the alternative hypothesis (> instead of <). Option D uses x̄ instead of μ for the population parameter. Option E uses p, which is for proportions, not means. Remember that hypothesis tests always involve population parameters (μ) and test claimed values, not sample statistics.

Question 6

A cereal company claims that the mean net weight of its cereal boxes is 18.0 ounces. A quality-control manager wants to test whether the true mean net weight is not equal to 18.0 ounces. A random sample of 30 boxes is selected, and the sample mean net weight is 17.8 ounces. Which hypotheses are appropriate for a one-sample test of a population mean net weight?

$H_0: \mu=18.0$ vs. $H_a: \mu\ne18.0$ (correct answer)
$H_0: \mu=17.8$ vs. $H_a: \mu\ne17.8$
$H_0: \mu=18.0$ vs. $H_a: \mu<18.0$
$H_0: \bar{x}=18.0$ vs. $H_a: \bar{x}\ne18.0$
$H_0: p=18.0$ vs. $H_a: p\ne18.0$

Explanation: This question focuses on hypothesis setup for a one-sample mean test of cereal box weights. The company claims 18.0 ounces mean, manager tests if not equal, implying two-tailed: H₀: μ = 18.0 vs. Hₐ: μ ≠ 18.0, using μ and claimed value. This fits 'not equal to' phrasing. Distractors use sample mean (17.8), one-tailed, \bar{x}, or p, wrong for weight mean. Mini-lesson: Two-sided Hₐ: μ ≠ μ₀ for 'not equal' tests, null equality. Population parameter, sample for assessment. Tests for any deviation from claim.

Question 7

A coffee shop claims that the mean amount of caffeine in its "medium" brewed coffee is 210 mg per cup. A student suspects the true mean caffeine content is higher than claimed. The student randomly selects 35 medium coffees from this shop over several days and measures the caffeine content; the sample mean is 223 mg. Which hypotheses are appropriate for a one-sample test of a population mean?

$H_0: \bar{x}=210$ vs. $H_a: \bar{x}>210$
$H_0: \mu=223$ vs. $H_a: \mu>223$
$H_0: \mu=210$ vs. $H_a: \mu>210$ (correct answer)
$H_0: p=0.210$ vs. $H_a: p>0.210$
$H_0: \mu=210$ vs. $H_a: \mu<210$

Explanation: This question assesses the skill of setting up hypotheses for a one-sample test of a population mean in AP Statistics. The coffee shop claims a mean caffeine content of 210 mg, but the student suspects it is higher, indicating a right-tailed test with the null hypothesis stating equality to the claimed value and the alternative reflecting the suspicion of a greater mean. The appropriate hypotheses are H₀: μ = 210 vs. Hₐ: μ > 210, where μ represents the true population mean caffeine content. Common distractors include using the sample mean (223 mg) in the hypotheses, employing the sample mean symbol \bar{x} instead of μ, or testing a proportion p, which are incorrect because hypotheses must focus on the population parameter μ and use the claimed value in the null. A mini-lesson on mean test setup: always state the null as H₀: μ = μ₀ (claimed value), and choose Hₐ based on the direction of the claim—greater than for suspected increases, less than for decreases, or not equal for differences. Here, the sample mean of 223 mg is evidence but not part of the hypotheses themselves. Remember, the goal is to test if there's sufficient evidence to reject the null in favor of the alternative.

Question 8

An online retailer claims that the mean delivery time for a certain product is 3.5 days. A customer advocacy group suspects the mean delivery time is longer than 3.5 days. They randomly sample 40 deliveries and compute a sample mean delivery time of 3.9 days. Which hypotheses are appropriate for a one-sample test of a population mean delivery time?

$H_0: \mu=3.5$ vs. $H_a: \mu<3.5$
$H_0: \bar{x}=3.5$ vs. $H_a: \bar{x}>3.5$
$H_0: \mu=3.9$ vs. $H_a: \mu>3.9$
$H_0: \mu=3.5$ vs. $H_a: \mu>3.5$ (correct answer)
$H_0: p=3.5$ vs. $H_a: p>3.5$

Explanation: This question tests hypothesis setup for a one-sample population mean test on delivery times. The retailer claims 3.5 days mean, but the group suspects longer (greater), suggesting a right-tailed test: H₀: μ = 3.5 vs. Hₐ: μ > 3.5, with μ as the population mean. This aligns with the 'longer than' wording. Distractors use sample mean (3.9 days) in null, wrong direction like <3.5, \bar{x} instead of μ, or proportion p, inappropriate for days as a mean. Mini-lesson: For 'longer' or 'greater' suspicions, set Hₐ: μ > μ₀, null as equality to claim. Hypotheses are about population, sample informs testing. This tests if data supports rejecting the claim for a higher mean.

Question 9

A gym advertises that the mean number of calories burned in its 45-minute cycling class is 400 calories. A member suspects the true mean calories burned is less than 400. The member records calories burned for a random sample of 12 classes and finds a sample mean of 385 calories. Which hypotheses are appropriate for a one-sample test of a population mean calories burned?

$H_0: \mu=400$ vs. $H_a: \mu>400$
$H_0: \mu=400$ vs. $H_a: \mu<400$ (correct answer)
$H_0: \mu=385$ vs. $H_a: \mu<385$
$H_0: \bar{x}=400$ vs. $H_a: \bar{x}<400$
$H_0: p=0.400$ vs. $H_a: p<0.400$

Explanation: This question tests setting up hypotheses for a one-sample population mean test on calories burned. The gym advertises 400 calories mean, member suspects less, warranting left-tailed: H₀: μ = 400 vs. Hₐ: μ < 400, with population mean μ. This aligns with 'less than' suspicion. Distractors feature sample mean (385) in null, right-tailed, \bar{x}, or proportion p, incorrect for calorie mean. Mini-lesson: Left-tailed Hₐ: μ < μ₀ for 'less' claims, null equality to advertised. Hypotheses on population, sample for statistic. Evaluates if evidence shows lower mean than claimed.

Question 10

A hospital reports that the mean waiting time in its emergency room is 42 minutes. A local news station believes the mean waiting time has changed and may not be 42 minutes anymore. They record waiting times for a random sample of 60 patients and find a sample mean of 45 minutes. Which hypotheses are appropriate for a one-sample test of a population mean waiting time?

$H_0: \mu=42$ vs. $H_a: \mu>42$
$H_0: \mu=45$ vs. $H_a: \mu\ne45$
$H_0: \mu=42$ vs. $H_a: \mu\ne42$ (correct answer)
$H_0: \bar{x}=42$ vs. $H_a: \bar{x}\ne42$
$H_0: p=0.42$ vs. $H_a: p\ne0.42$

Explanation: This question assesses setting up hypotheses for a one-sample mean test on emergency room waiting times. The hospital reports 42 minutes mean, but the news station believes it has changed (not 42 anymore), indicating a two-tailed test: H₀: μ = 42 vs. Hₐ: μ ≠ 42, using μ and the reported value. This captures the non-directional 'changed' suspicion. Distractors involve one-tailed alternatives, sample mean (45 minutes) in null, \bar{x} symbols, or proportion p, which doesn't apply to time as a mean. Mini-lesson: Use Hₐ: μ ≠ μ₀ for suspicions of change without direction, keeping null as equality. Focus on population mean μ, not sample. The sample mean helps evaluate if evidence rejects the null for any difference.

Question 11

A nutrition label on a cereal box states that the mean sugar content per serving is no more than 9 grams. A consumer group randomly samples 60 boxes and measures sugar per serving; the sample mean is $\bar{x}=9.4$ grams. Which hypotheses are appropriate for a test of the label's claim about the population mean sugar content $\mu$ ?

$H_0: \mu=9$ g \quad vs. \quad $H_a: \mu\ne 9$ g
$H_0: \mu\le 9$ g \quad vs. \quad $H_a: \mu>9$ g (correct answer)
$H_0: \mu\ge 9$ g \quad vs. \quad $H_a: \mu<9$ g
$H_0: \bar{x}\le 9$ g \quad vs. \quad $H_a: \bar{x}>9$ g
$H_0: p=9$ \quad vs. \quad $H_a: p>9$

Explanation: This question tests understanding of how to translate "no more than" into proper hypotheses. The claim "no more than 9 grams" means μ ≤ 9 grams. When testing against such a claim, the null hypothesis includes this inequality (H₀: μ ≤ 9), and the alternative hypothesis represents the opposite direction (Hₐ: μ > 9). This tests whether the cereal exceeds the stated sugar limit. Option A incorrectly uses equality in the null with a two-sided alternative. Option C tests in the wrong direction. Option D incorrectly uses x̄ instead of μ and includes the sample statistic in the hypothesis. Option E incorrectly uses p (proportion) instead of μ. The correct answer B properly sets up H₀: μ ≤ 9 g vs. Hₐ: μ > 9 g to test if the mean sugar content exceeds the label's claim.

Question 12

A nutrition label on a granola bar says the mean sugar content is 12 grams per bar. A dietitian suspects the true mean sugar content is more than 12 grams. The dietitian randomly selects 25 bars and measures sugar content; the sample mean is 12.7 grams. Which hypotheses are appropriate for a one-sample test of a population mean sugar content?

$H_0: \mu=12$ vs. $H_a: \mu<12$
$H_0: \mu=12.7$ vs. $H_a: \mu>12.7$
$H_0: \mu=12$ vs. $H_a: \mu>12$ (correct answer)
$H_0: \bar{x}=12$ vs. $H_a: \bar{x}>12$
$H_0: p=12$ vs. $H_a: p>12$

Explanation: This question focuses on hypothesizing for a one-sample mean test about sugar content in granola bars. The label claims 12 grams mean, but the dietitian suspects more, warranting a right-tailed test: H₀: μ = 12 vs. Hₐ: μ > 12, using μ for the population mean and the claimed value in the null. This matches the 'more than' suspicion directly. Distractors feature the sample mean (12.7 grams) in the null, wrong direction (like <12), using \bar{x} symbols inappropriately, or testing a proportion p, which is incorrect for a mean measurement like grams of sugar. Mini-lesson: Set up the null as H₀: μ = claimed value, and for suspicions of excess ('more' or 'higher'), use Hₐ: μ > μ₀. Sample results like the mean of 12.7 inform the p-value calculation, not the hypotheses. This structure tests if evidence supports the alternative over the null claim.

Question 13

A hospital claims that the mean length of stay for patients undergoing a certain procedure is 3.5 days. A researcher collects a random sample of $n=20$ patients and finds a sample mean length of stay of $\bar{x}=3.2$ days. The researcher wants to test whether the true mean length of stay is less than 3.5 days. Which hypotheses are appropriate?

$H_0: \mu=3.5$ vs. $H_a: \mu<3.5$ (correct answer)
$H_0: \mu=3.2$ vs. $H_a: \mu<3.2$
$H_0: \mu\le 3.5$ vs. $H_a: \mu>3.5$
$H_0: \bar{x}=3.5$ vs. $H_a: \bar{x}<3.5$
$H_0: p=3.5$ vs. $H_a: p<3.5$

Explanation: This problem involves setting up a left-tailed test for a population mean. The hospital claims μ = 3.5 days, which forms our null hypothesis: H₀: μ = 3.5. The researcher wants to test if the true mean is less than 3.5 days, so we use a left-tailed alternative: Hₐ: μ < 3.5. Option B incorrectly uses the sample mean (3.2) in the hypotheses—we test claims about the population parameter, not sample statistics. Option C has an inequality in H₀ and points in the wrong direction for a 'less than' test. Option D uses x̄ instead of μ for the population parameter. Option E uses p (for proportions) instead of μ (for means). When testing if something is 'less than' a claimed value, use < in the alternative hypothesis.

Question 14

A company that makes AA batteries claims its batteries have a mean lifetime of 10.0 hours under a standard test. An engineer tests a random sample of $n=25$ batteries and finds a sample mean lifetime of $\bar{x}=9.6$ hours. The engineer wants to determine whether the true mean lifetime is different from 10.0 hours. Which hypotheses are appropriate?

$H_0: \mu=10.0$ vs. $H_a: \mu\ne 10.0$ (correct answer)
$H_0: \mu=9.6$ vs. $H_a: \mu\ne 9.6$
$H_0: \bar{x}=10.0$ vs. $H_a: \bar{x}\ne 10.0$
$H_0: p=10.0$ vs. $H_a: p\ne 10.0$
$H_0: \mu\ne 10.0$ vs. $H_a: \mu=10.0$

Explanation: This question requires setting up a two-tailed test for a population mean. The null hypothesis states the claimed value: H₀: μ = 10.0 hours. Since the engineer wants to test if the mean is 'different from' 10.0 hours (not specifically greater or less), we need a two-tailed alternative: Hₐ: μ ≠ 10.0. Option B incorrectly uses the sample mean value (9.6) in the hypotheses—we always test claims about the population parameter, not sample statistics. Option C incorrectly uses x̄ instead of μ. Option D uses p (for proportions) instead of μ. Option E reverses the null and alternative hypotheses. Remember that for 'different from' questions, use a two-tailed test with ≠ in the alternative hypothesis.

Question 15

A farmer's cooperative claims that the mean weight of its apples is 150 grams. A random sample of $n=50$ apples is weighed, and the sample mean is $\bar{x}=151.6$ grams. The cooperative wants to test whether the true mean apple weight is different from 150 grams. Which hypotheses are appropriate?

$H_0: \mu=150$ vs. $H_a: \mu\ne 150$ (correct answer)
$H_0: \mu=151.6$ vs. $H_a: \mu\ne 151.6$
$H_0: p=150$ vs. $H_a: p\ne 150$
$H_0: \mu\ne 150$ vs. $H_a: \mu=150$
$H_0: \bar{x}=150$ vs. $H_a: \bar{x}\ne 150$

Explanation: This problem tests setting up a two-tailed hypothesis test for a population mean. The cooperative claims μ = 150 grams, which becomes our null hypothesis: H₀: μ = 150. Since they want to test if the mean is 'different from' 150 grams (not specifically heavier or lighter), we need a two-tailed alternative: Hₐ: μ ≠ 150. Option B incorrectly uses the sample mean (151.6) in the hypotheses—we test claims about population parameters. Option C uses p (for proportions) instead of μ (for means). Option D reverses the null and alternative hypotheses. Option E uses x̄ instead of μ for the population parameter. When the question asks about 'different from,' always use a two-tailed test with ≠ in the alternative hypothesis.

Question 16

A textbook publisher claims that the mean weight of its hardcover calculus book is 4.5 pounds. A university bookstore manager wants to check if the mean weight is actually different, because shipping costs depend on weight. A random sample of 20 books has a sample mean weight of $\bar{x}=4.62$ pounds. Which hypotheses are appropriate for a test about the population mean book weight $\mu$ ?

$H_0: \mu=4.62$ lb \quad vs. \quad $H_a: \mu\ne 4.62$ lb
$H_0: \mu=4.5$ lb \quad vs. \quad $H_a: \mu\ne 4.5$ lb (correct answer)
$H_0: \mu\ne 4.5$ lb \quad vs. \quad $H_a: \mu=4.5$ lb
$H_0: \bar{x}=4.5$ lb \quad vs. \quad $H_a: \bar{x}\ne 4.5$ lb
$H_0: p=4.5$ \quad vs. \quad $H_a: p\ne 4.5$

Explanation: This question requires setting up a two-sided hypothesis test because the bookstore manager wants to check if the mean weight is "actually different" from the claimed value. The publisher claims μ = 4.5 pounds, and we test for any difference using a two-sided alternative. The proper setup is H₀: μ = 4.5 vs. Hₐ: μ ≠ 4.5. Option A incorrectly uses the sample value 4.62 in the null hypothesis. Option C incorrectly reverses the null and alternative hypotheses. Option D incorrectly uses the sample mean x̄ instead of the population parameter μ. Option E incorrectly uses p (proportion) instead of μ for weight data. The correct answer B properly sets up a two-sided test to determine if the population mean book weight differs from the claimed 4.5 pounds.

Question 17

A city's transportation department claims that the mean time to clear a minor accident from a roadway is 30 minutes. A random sample of 16 minor accidents is observed, and the sample mean clearance time is $\bar{x}=33.5$ minutes. Which hypotheses are appropriate for testing whether the true mean clearance time $\mu$ is longer than the claimed value?

$H_0: \mu=33.5$ min \quad vs. \quad $H_a: \mu>33.5$ min
$H_0: \mu=30$ min \quad vs. \quad $H_a: \mu>30$ min (correct answer)
$H_0: \mu=30$ min \quad vs. \quad $H_a: \mu<30$ min
$H_0: \bar{x}=30$ min \quad vs. \quad $H_a: \bar{x}>30$ min
$H_0: p=30$ \quad vs. \quad $H_a: p>30$

Explanation: This question requires setting up a one-sided test to determine if the true mean clearance time is "longer than" (greater than) the claimed 30 minutes. The city claims μ = 30 minutes, and we want to test if the actual mean exceeds this value. For a "greater than" test, we use H₀: μ = 30 vs. Hₐ: μ > 30. Option A incorrectly uses the sample value 33.5 instead of the claimed value 30 in the hypotheses. Option C tests in the wrong direction (less than instead of greater than). Option D incorrectly uses the sample mean x̄ instead of the population parameter μ. Option E incorrectly uses p (proportion) instead of μ for this quantitative variable. The correct answer B properly sets up the test to determine if the mean clearance time exceeds the claimed 30 minutes.

Question 18

A city transit agency reports that the mean wait time for a bus on a certain route during rush hour is 12 minutes. Riders believe the mean wait time is longer. A random sample of $n=45$ rush-hour waits has mean $overline{x}=13.4$ minutes. Which hypotheses are appropriate for testing the riders' claim about the population mean wait time, $\mu$ ?

$H_0: \mu = 12$ vs. $H_a: \mu < 12$
$H_0: \mu = 13.4$ vs. $H_a: \mu > 13.4$
$H_0: \mu = 12$ vs. $H_a: \mu > 12$ (correct answer)
$H_0: \overline{x} = 12$ vs. $H_a: \overline{x} > 12$
$H_0: p = 12$ vs. $H_a: p > 12$

Explanation: This question involves setting up a one-sided hypothesis test for a population mean. The transit agency claims the mean wait time is 12 minutes, so H₀: μ = 12. Riders believe the wait time is longer than claimed, leading to Ha: μ > 12. Choice A has the wrong direction for the alternative hypothesis (less than instead of greater than). Choice B incorrectly uses the sample mean 13.4 in the hypotheses. Choice D uses the sample mean x̄ instead of the population parameter μ. Choice E uses p (proportion) instead of μ (mean). When the claim involves "longer," "more," or "greater," the alternative hypothesis uses the > symbol.

Question 19

A bottled water company states that the mean volume in its 500 mL bottles is 500 mL. Inspectors take a random sample of $n=50$ bottles and measure a sample mean of $\bar{x}=498.6$ mL. They want to test whether the true mean volume is different from 500 mL. Which hypotheses are appropriate for a test of a population mean?

$H_0: \mu=498.6$ vs. $H_a: \mu\ne 498.6$
$H_0: \bar{x}=500$ vs. $H_a: \bar{x}\ne 500$
$H_0: \mu=500$ vs. $H_a: \mu\ne 500$ (correct answer)
$H_0: p=500$ vs. $H_a: p\ne 500$
$H_0: \mu=500$ vs. $H_a: \mu<500$

Explanation: Focusing on AP Statistics hypothesis setup for population means, the correct hypotheses are H₀: μ = 500 vs. Hₐ: μ ≠ 500, fitting the inspectors' test of whether the true mean volume differs from 500 mL. This two-sided test matches the non-directional query about difference. Distractors include choice B using x-bar, choice D with p, and choice A centering on the sample mean 498.6 incorrectly. Mini-lesson: Always employ μ, placing the stated claim in H₀ as equality and using ≠ in Hₐ for two-tailed tests of difference. The sample mean x-bar = 498.6 helps calculate the test statistic but isn't in the hypotheses.

Question 20

A hospital reports that the mean length of stay for patients undergoing a certain procedure is 3.5 days. A random sample of $n=28$ recent patients shows a mean stay of $\bar{x}=3.9$ days. An administrator wants to test whether the true mean length of stay has increased above 3.5 days. Which hypotheses are appropriate for a test of a population mean?

$H_0: \mu=3.5$ vs. $H_a: \mu>3.5$ (correct answer)
$H_0: \mu=3.9$ vs. $H_a: \mu>3.9$
$H_0: \bar{x}=3.5$ vs. $H_a: \bar{x}>3.5$
$H_0: p=3.5$ vs. $H_a: p>3.5$
$H_0: \mu=3.5$ vs. $H_a: \mu<3.5$

Explanation: This question in AP Statistics tests hypothesis formulation for a population mean with a directional claim. Correct are H₀: μ = 3.5 vs. Hₐ: μ > 3.5, as the administrator checks if the true mean stay has increased above 3.5 days, aligning with a right-tailed test. This setup reflects the suspicion of lengthening stays. Distractors like choice C use x-bar, choice D employs p, and choice E reverses to less than, mismatching the intent. Mini-lesson: Frame with μ, H₀ as equal to the reported value, and Hₐ > for testing increases. Sample data x-bar = 3.9 provides evidence but not hypothesis content.