Setting Up Tests for Population Mean - AP Statistics

A manager of a call center claims that the mean time customers spend on hold is no more than 3.0 minutes. To check the claim, a random sample of $n=55$ calls is selected, and the sample mean hold time is $\bar{x}=3.2$ minutes. Which hypotheses are appropriate for a test of the population mean hold time, $\mu$ ?
Answer: $H_0: \mu=3.0$ min\quad vs.\quad $H_a: \mu>3.0$ min This question requires translating "no more than 3.0 minutes" into proper hypothesis notation for a population mean test. The phrase "no more than" means "less than or equal to" (≤ 3.0), which becomes the null hypothesis claim. To test against this claim, we need to show the mean is greater than 3.0 minutes. Option A correctly states H₀: μ = 3.0 min vs. Hₐ: μ > 3.0 min, using the boundary value with equality in the null hypothesis. Students often confuse sample statistics (x̄) with population parameters (μ) or use the wrong inequality direction. Remember that when testing a claim of "at most" or "no more than," the alternative hypothesis tests for values greater than the boundary, making this a right-tailed test.
A shipping company claims that the mean delivery time for a certain route is at least 2.5 days. A competitor believes the mean delivery time is actually less than 2.5 days. A random sample of $n=45$ deliveries on that route is selected, and the sample mean delivery time is $\bar{x}=2.3$ days. Which hypotheses are appropriate for a test of the population mean delivery time, $\mu$ ?
Answer: $H_0: \mu=2.5$ days\quad vs.\quad $H_a: \mu<2.5$ days This question requires careful interpretation of "at least 2.5 days" to set up proper hypotheses for a population mean test. The shipping company claims the mean is "at least" (≥) 2.5 days, but in hypothesis testing, we always use equality in the null hypothesis at the boundary value. Since the competitor believes the mean is less than 2.5 days, we test H₀: μ = 2.5 days vs. Hₐ: μ < 2.5 days. Option C correctly states these hypotheses. Common mistakes include writing H₀: μ ≥ 2.5 or using sample statistics instead of population parameters. When a claim involves "at least" or "at most," convert it to an equality at the boundary for H₀, then set up Hₐ to test in the opposite direction of the original claim.
A coffee shop claims the mean amount of time customers wait in line during the morning rush is 4 minutes. A manager believes a new ordering system has reduced the mean wait time. After implementation, a random sample of $n=35$ waits has mean $ overline{x}=3.6$ minutes. Which hypotheses are appropriate for testing the manager's claim about the population mean wait time, $\mu$ ?
Answer: $H_0: \mu = 4$ vs. $H_a: \mu < 4$ This question involves setting up hypotheses to test if a new system has reduced the mean wait time. The coffee shop claims μ = 4 minutes, so H₀: μ = 4. The manager believes the new system has reduced wait time, making Ha: μ < 4. Choice A has the wrong direction (greater than instead of less than). Choice B incorrectly uses the sample mean 3.6 in the hypotheses. Choice D uses x̄ instead of μ. Choice E tests the standard deviation σ instead of the mean μ. When testing if something has been "reduced" or "decreased," the alternative hypothesis uses the < symbol.
A teacher believes that the mean score on a certain standardized math test for students in the district is 500. A principal suspects the district's true mean score is higher than 500. A random sample of 80 students takes the test, and the sample mean score is 512. Which hypotheses are appropriate for a one-sample test of a population mean score?
Answer: $H_0: \mu=500$ vs. $H_a: \mu>500$ This question evaluates hypothesizing for a one-sample mean test on standardized test scores. The teacher believes mean is 500, but principal suspects higher, leading to right-tailed: H₀: μ = 500 vs. Hₐ: μ > 500, using population mean μ. This matches 'higher than' suspicion. Distractors include sample mean (512) in null, left-tailed alternative, \bar{x} symbols, or proportion p, not fitting scores as means. Mini-lesson: Right-tailed Hₐ: μ > μ₀ for 'higher' claims, null equality to believed value. Population-focused, sample for evidence. Setup checks if scores exceed the believed mean significantly.
A city water department claims that the mean lead concentration in household tap water is 5 parts per billion (ppb). A public health scientist suspects the mean lead concentration is greater than 5 ppb. A random sample of $n=20$ homes is tested, and the sample mean lead concentration is $\bar{x}=6.1$ ppb. Which hypotheses are appropriate for a test of the population mean lead concentration, $\mu$ ?
Answer: $H_0: \mu=5$ ppb\quad vs.\quad $H_a: \mu>5$ ppb This question requires setting up a one-sided test when a scientist suspects the mean lead concentration exceeds the city's claimed value of 5 ppb. The null hypothesis states the city's claim (μ = 5 ppb) with equality, while the alternative hypothesis reflects the scientist's suspicion that the true mean is greater (μ > 5 ppb). Option B correctly shows H₀: μ = 5 ppb vs. Hₐ: μ > 5 ppb. Students often err by using sample statistics (x̄) in hypotheses or by incorporating the sample mean (6.1 ppb) into the hypothesis statements. For right-tailed tests about population means, always use the population parameter μ with the claimed value in H₀, and the > symbol in Hₐ when testing for values greater than the claim.
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?
Answer: $H_0: \mu=500$ vs. $H_a: \mu\ne 500$ 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.
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?
Answer: $H_0: \mu=80$ vs. $H_a: \mu>80$ 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.
A coffee shop advertises that the mean amount of caffeine in its medium brewed coffee is at least 190 mg. A student randomly selects 36 medium coffees from this shop and measures the caffeine content; the sample mean is $\bar{x}=184.2$ mg. Which hypotheses are appropriate for a significance test of the shop's advertising claim about the population mean caffeine content $\mu$ ?
Answer: $H_0: \mu\ge 190$ mg \quad vs. \quad $H_a: \mu<190$ mg This question tests understanding of setting up hypotheses for a one-sided test about a population mean. The coffee shop claims the mean caffeine content is "at least" 190 mg, which translates to μ ≥ 190 mg. When testing a claim of "at least," the null hypothesis includes the equality and the inequality in the same direction (H₀: μ ≥ 190), while the alternative hypothesis represents the opposite (Hₐ: μ < 190). Option A incorrectly uses the sample mean x̄ instead of the population parameter μ in the hypotheses. Option C incorrectly uses p (a proportion parameter) instead of μ. Option E incorrectly uses the sample value 184.2 in the null hypothesis. The correct answer D properly states H₀: μ ≥ 190 mg vs. Hₐ: μ < 190 mg, which tests whether the mean is less than the advertised "at least 190 mg" claim.
A fitness app claims that the mean number of steps its users take per day is 8,000. A researcher suspects the true mean for all users is lower. A random sample of $n=60$ users shows a mean of $ overline{x}=7,650$ steps. Which hypotheses are appropriate for testing the researcher's claim about the population mean steps, $\mu$ ?
Answer: $H_0: \mu = 8000$ vs. $H_a: \mu < 8000$ This question tests setting up hypotheses for a population mean with a one-sided alternative (less than). The fitness app claims μ = 8,000 steps, which forms our null hypothesis: H₀: μ = 8000. The researcher suspects the true mean is lower, so Ha: μ < 8000. Choice B incorrectly uses ≤ in the null hypothesis; the null must contain equality (=). Choice C uses x̄ instead of the population parameter μ. Choice D incorrectly uses the sample mean 7650 instead of the claimed value 8000. Choice E uses p (proportion) instead of μ (mean). The null hypothesis always states equality with the claimed value, never an inequality.
A farm cooperative claims the mean weight of its apples is 180 grams. A grocery chain suspects the cooperative's apples are heavier on average. A random sample of $n=48$ apples has mean weight $ overline{x}=186$ grams. Which hypotheses are appropriate for testing the grocery chain's claim about the population mean apple weight, $\mu$ ?
Answer: $H_0: \mu = 180$ vs. $H_a: \mu > 180$ This question tests setting up hypotheses for a population mean when the alternative is one-sided (greater than). The farm cooperative claims μ = 180 grams, forming H₀: μ = 180. The grocery chain suspects apples are heavier on average, so Ha: μ > 180. Choice B has the wrong direction (less than instead of greater than). Choice C uses x̄ instead of μ. Choice D incorrectly reverses the hypotheses and puts inequality in the null. Choice E uses p (proportion) instead of μ (mean). When testing if something is "heavier," "larger," or "more than" claimed, use > in the alternative hypothesis.
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?
Answer: $H_0: \mu=900$ vs. $H_a: \mu<900$ 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.
A school district claims the mean time high school students spend on homework per night is 2.5 hours. A parent group believes students spend more time than that. A random sample of $n=55$ students reports a mean of $ overline{x}=2.8$ hours. Which hypotheses are appropriate for testing the parent group's claim about the population mean homework time, $\mu$ ?
Answer: $H_0: \mu = 2.5$ vs. $H_a: \mu > 2.5$ This question requires setting up hypotheses for a population mean with a one-sided alternative (greater than). The school district claims μ = 2.5 hours, so this becomes our null hypothesis: H₀: μ = 2.5. The parent group believes students spend more time than claimed, making the alternative hypothesis Ha: μ > 2.5. Choice B incorrectly uses the sample mean 2.8 instead of the claimed value 2.5. Choice C reverses the hypotheses incorrectly. Choice D uses the sample mean x̄ instead of the population parameter μ. Choice E uses p (proportion) instead of μ (mean). Remember that hypothesis tests always compare the population parameter to a claimed value, not to the sample statistic.
A city transit authority states that the mean wait time for a bus on a certain route during rush hour is 8 minutes. A random sample of $n=60$ riders records an average wait time of $\bar{x}=9.1$ minutes. An advocate group wants to test whether the true mean wait time is greater than 8 minutes. Which hypotheses are appropriate for a test of a population mean?
Answer: $H_0: \mu=8$ vs. $H_a: \mu>8$ 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.
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?
Answer: $H_0: \mu=12$ vs. $H_a: \mu>12$ 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.
A cereal company advertises that the mean amount of cereal in its 18-ounce boxes is 18 ounces. A random sample of $n=40$ boxes from the production line has a sample mean fill weight of $\bar{x}=17.8$ ounces. The quality-control manager wants to test whether the true mean fill weight for all boxes is less than advertised. Which hypotheses are appropriate for a test of a population mean?
Answer: $H_0: \mu=18$ vs. $H_a: \mu<18$ 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.
A streaming service reports that the mean number of hours its subscribers watch per week is 10 hours. A marketing analyst believes the true mean weekly viewing time is not 10 hours. A random sample of $n=80$ subscribers has mean viewing time $ overline{x}=9.4$ hours. Which hypotheses are appropriate for testing the analyst's claim about the population mean weekly viewing time, $\mu$ ?
Answer: $H_0: \mu = 10$ vs. $H_a: \mu \ne 10$ This question requires setting up a two-sided hypothesis test for a population mean. The streaming service reports μ = 10 hours, which becomes H₀: μ = 10. The analyst believes the true mean is "not 10 hours" (could be either higher or lower), requiring Ha: μ ≠ 10. Choice B incorrectly uses a one-sided alternative. Choice C incorrectly uses the sample mean 9.4 in the hypotheses. Choice D uses x̄ instead of μ. Choice E uses p (proportion) instead of μ (mean). The phrase "is not" or "differs from" without specifying direction indicates a two-sided test with ≠ in the alternative hypothesis.
A manufacturer advertises that its AA batteries have a mean lifetime of 8.0 hours under a standard test. A consumer group believes the batteries last less than advertised. They test a random sample of 20 batteries and find a sample mean lifetime of 7.6 hours. Which hypotheses are appropriate for a one-sample test of a population mean battery lifetime?
Answer: $H_0: \mu=8.0$ vs. $H_a: \mu<8.0$ This question evaluates the ability to formulate hypotheses for a one-sample test of a population mean, specifically for battery lifetime. The manufacturer claims a mean of 8.0 hours, but the consumer group believes it is less, suggesting a left-tailed test where the null hypothesis equates the population mean μ to 8.0 and the alternative posits μ < 8.0. The correct setup is H₀: μ = 8.0 vs. Hₐ: μ < 8.0, aligning with the suspicion of a shorter lifetime. Distractors often involve using the sample mean (7.6 hours) in the null, substituting \bar{x} for μ, opting for a right-tailed test which contradicts the 'less than' belief, or mistakenly using a proportion p, as this is a mean, not a proportion problem. Mini-lesson: In setting up a t-test for means, the null always includes equality to the advertised or claimed value, while the alternative reflects the directional suspicion—use < for 'less than' claims. The sample data informs the test statistic later, not the hypotheses. This ensures we're testing the population parameter, not the sample.
A nutrition label on a snack bar says the mean sodium content is 140 mg per bar. A random sample of $n=30$ bars is tested, and the sample mean sodium content is $\bar{x}=145$ mg. A researcher wants to test the company's claim that the true mean sodium content is 140 mg (not higher or lower). Which hypotheses are appropriate for a test of a population mean?
Answer: $H_0: \mu=140$ vs. $H_a: \mu\ne 140$ This AP Statistics question evaluates setting up a two-sided hypothesis test for a population mean. The correct hypotheses are H₀: μ = 140 vs. Hₐ: μ ≠ 140, matching the researcher's aim to verify if the true mean sodium content differs from the labeled 140 mg in either direction. This two-tailed test aligns with testing the exact claim without directional bias. Distractors like choice C use x-bar erroneously, choice D confuses with proportion p, and choice B incorrectly centers on the sample mean 145. Mini-lesson on setup: Hypotheses for mean tests use μ, with H₀ equaling the claimed value and Hₐ as ≠ for checking deviations; direction matters only if specified. Sample data like x-bar = 145 informs the test but not the hypothesis statements.
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 $\mu$ ?
Answer: $H_0: \mu=4.5$ lb \quad vs. \quad $H_a: \mu\ne 4.5$ lb 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.
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?
Answer: $H_0: \mu=12$ vs. $H_a: \mu\ne 12$ 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.

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