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AP Statistics Help: Setting Up Tests For Population Mean

Review real example questions for Setting Up Tests For Population Mean in AP Statistics.

Question 1 / 10

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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?

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.