AP Statistics Quiz: Setting Up Tests For Population Mean

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.

Question 1

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

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

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

1. $H_0: \mu=80$ vs. $H_a: \mu>80$ (correct answer)
2. $H_0: \mu=84$ vs. $H_a: \mu>84$
3. $H_0: \bar{x}=80$ vs. $H_a: \bar{x}>80$
4. $H_0: p=80$ vs. $H_a: p>80$
5. $H_0: \mu\ge 80$ 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

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

1. $H_0: \mu=9$ g \quad vs. \quad $H_a: \mu\ne 9$ g
2. $H_0: \mu\le 9$ g \quad vs. \quad $H_a: \mu>9$ g (correct answer)
3. $H_0: \mu\ge 9$ g \quad vs. \quad $H_a: \mu<9$ g
4. $H_0: \bar{x}\le 9$ g \quad vs. \quad $H_a: \bar{x}>9$ g
5. $H_0: p=9$ \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 7

1. $H_0: \mu=12$ vs. $H_a: \mu<12$
2. $H_0: \mu=12.7$ vs. $H_a: \mu>12.7$
3. $H_0: \mu=12$ vs. $H_a: \mu>12$ (correct answer)
4. $H_0: \bar{x}=12$ vs. $H_a: \bar{x}>12$
5. $H_0: p=12$ 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 ar{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 8

1. $H_0: \mu=3.5$ vs. $H_a: \mu<3.5$ (correct answer)
2. $H_0: \mu=3.2$ vs. $H_a: \mu<3.2$
3. $H_0: \mu\le 3.5$ vs. $H_a: \mu>3.5$
4. $H_0: \bar{x}=3.5$ vs. $H_a: \bar{x}<3.5$
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 9

1. $H_0: \mu=10.0$ vs. $H_a: \mu\ne 10.0$ (correct answer)
2. $H_0: \mu=9.6$ vs. $H_a: \mu\ne 9.6$
3. $H_0: \bar{x}=10.0$ vs. $H_a: \bar{x}\ne 10.0$
4. $H_0: p=10.0$ vs. $H_a: p\ne 10.0$
5. $H_0: \mu\ne 10.0$ 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 10

1. $H_0: \mu=150$ vs. $H_a: \mu\ne 150$ (correct answer)
2. $H_0: \mu=151.6$ vs. $H_a: \mu\ne 151.6$
3. $H_0: p=150$ vs. $H_a: p\ne 150$
4. $H_0: \mu\ne 150$ vs. $H_a: \mu=150$
5. $H_0: \bar{x}=150$ 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 11

1. $H_0: \mu=4.62$ lb \quad vs. \quad $H_a: \mu\ne 4.62$ lb
2. $H_0: \mu=4.5$ lb \quad vs. \quad $H_a: \mu\ne 4.5$ lb (correct answer)
3. $H_0: \mu\ne 4.5$ lb \quad vs. \quad $H_a: \mu=4.5$ lb
4. $H_0: \bar{x}=4.5$ lb \quad vs. \quad $H_a: \bar{x}\ne 4.5$ lb
5. $H_0: p=4.5$ \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 12

1. $H_0: \mu=33.5$ min \quad vs. \quad $H_a: \mu>33.5$ min
2. $H_0: \mu=30$ min \quad vs. \quad $H_a: \mu>30$ min (correct answer)
3. $H_0: \mu=30$ min \quad vs. \quad $H_a: \mu<30$ min
4. $H_0: \bar{x}=30$ min \quad vs. \quad $H_a: \bar{x}>30$ min
5. $H_0: p=30$ \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 13

1. $H_0: \mu = 12$ vs. $H_a: \mu < 12$
2. $H_0: \mu = 13.4$ vs. $H_a: \mu > 13.4$
3. $H_0: \mu = 12$ vs. $H_a: \mu > 12$ (correct answer)
4. $H_0: \overline{x} = 12$ vs. $H_a: \overline{x} > 12$
5. $H_0: p = 12$ 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 14

1. $H_0: \mu=498.6$ vs. $H_a: \mu\ne 498.6$
2. $H_0: \bar{x}=500$ vs. $H_a: \bar{x}\ne 500$
3. $H_0: \mu=500$ vs. $H_a: \mu\ne 500$ (correct answer)
4. $H_0: p=500$ vs. $H_a: p\ne 500$
5. $H_0: \mu=500$ 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 15

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

Question 16

1. $H_0: \mu=9.1$ vs. $H_a: \mu>9.1$
2. $H_0: \mu=8$ vs. $H_a: \mu<8$
3. $H_0: \bar{x}=8$ vs. $H_a: \bar{x}>8$
4. $H_0: \mu=8$ vs. $H_a: \mu>8$ (correct answer)
5. $H_0: p=8$ vs. $H_a: p>8$

Explanation: In AP Statistics, this question tests the ability to set up hypotheses for a population mean test. The proper hypotheses are H₀: μ = 8 vs. Hₐ: μ > 8, as the advocate group suspects the true mean wait time exceeds the stated 8 minutes, aligning with a right-tailed test. This setup correctly places the claimed value in H₀ and the greater-than suspicion in Hₐ. Common distractors are choice C, using sample mean x-bar instead of μ, choice E with proportion p, or choice B, which reverses the direction incorrectly. A mini-lesson reminder: Always frame hypotheses around the population parameter μ, with H₀ as equality to the null value and Hₐ directional based on the test's goal—here, greater than for investigating an increase. The sample mean x-bar = 9.1 supports evidence collection but doesn't appear in the hypotheses.

Question 17

1. $H_0: \mu=1.1$ vs. $H_a: \mu<1.1$
2. $H_0: \mu=1.5$ vs. $H_a: \mu<1.5$ (correct answer)
3. $H_0: \mu=1.5$ vs. $H_a: \mu>1.5$
4. $H_0: \bar{x}=1.5$ vs. $H_a: \bar{x}<1.5$
5. $H_0: p=1.5$ vs. $H_a: p<1.5$

Explanation: In AP Statistics, this assesses setting up hypotheses for a population mean test against a claim. The appropriate setup is H₀: μ = 1.5 vs. Hₐ: μ < 1.5, since the critic tests if the true mean weight loss is less than the claimed 1.5 pounds, using a left-tailed alternative. This matches the directional critique of overstatement. Distractors include choice D with x-bar, choice E using p, and choice C flipping to greater than, which contradicts the suspicion. Mini-lesson: Use μ in both, with H₀ equaling the claim and Hₐ < for testing below; sample x-bar = 1.1 is for later analysis, not hypotheses.

Question 18

1. $H_0: \mu=12$ vs. $H_a: \mu<12$ (correct answer)
2. $H_0: \mu=11.3$ vs. $H_a: \mu<11.3$
3. $H_0: \bar{x}=12$ vs. $H_a: \bar{x}<12$
4. $H_0: \mu=12$ vs. $H_a: \mu>12$
5. $H_0: p=12$ vs. $H_a: p<12$

Explanation: This AP Statistics question involves setting up hypotheses for testing a population mean claim. The right choice is H₀: μ = 12 vs. Hₐ: μ < 12, as management tests if the true mean check-in time is less than the claimed 12 minutes, using a left-tailed alternative. This directly aligns with the goal of checking for improvement below the claim. Distractors are choice C with x-bar, choice E using p for proportions, and choice D flipping to greater than, which doesn't fit. Mini-lesson: Hypotheses center on μ, with H₀ as the claimed equality and Hₐ directional per the test—less than here for under the value. Sample stats like x-bar = 11.3 are computational tools, not part of the hypotheses.

Question 19

1. $H_0: \bar{x}=18$ vs. $H_a: \bar{x}<18$
2. $H_0: \mu=18$ vs. $H_a: \mu<18$ (correct answer)
3. $H_0: \mu=17.8$ vs. $H_a: \mu<17.8$
4. $H_0: p=18$ vs. $H_a: p<18$
5. $H_0: \mu=18$ vs. $H_a: \mu>18$

Explanation: This question assesses the skill of setting up hypothesis tests for a population mean in AP Statistics. The appropriate hypotheses are H₀: μ = 18 vs. Hₐ: μ < 18, as the quality-control manager wants to test if the true mean fill weight is less than the advertised 18 ounces, using the population parameter μ rather than the sample mean x-bar. This aligns with the scenario because the null hypothesis assumes the advertised mean is correct, while the alternative reflects the suspicion of underfilling. A common distractor is choice A, which incorrectly uses the sample mean x-bar instead of μ, or choice D, which uses proportion p inappropriate for means. In a mini-lesson on mean test setup, remember that hypotheses always involve the population mean μ, with H₀ stating equality to the claimed value and Hₐ indicating the direction of the test based on the research question. The sample data like x-bar = 17.8 informs the test statistic later, not the hypotheses themselves.

Question 20

1. $H_0: \bar{x}=210$ vs. $H_a: \bar{x}>210$
2. $H_0: \mu=223$ vs. $H_a: \mu>223$
3. $H_0: \mu=210$ vs. $H_a: \mu>210$ (correct answer)
4. $H_0: p=0.210$ vs. $H_a: p>0.210$
5. $H_0: \mu=210$ 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 ar{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.