Carrying Out Test for Population Mean - AP Statistics

An online retailer wants to check whether the mean delivery time for a certain shipping option is 2 days. A random sample of 100 deliveries is selected, and a one-sample $t$ test is performed with $H_0:\mu=2$ versus $H_a:\mu\ne 2$ at $\alpha=0.05$ . The p-value is 0.58. What conclusion is appropriate?
Answer: Fail to reject $H_0$ because $p=0.58>0.05$ ; there is not convincing evidence that the population mean delivery time differs from 2 days. This question evaluates interpreting a one-sample t-test for a population mean. With p=0.58 > α=0.05, we fail to reject H₀: μ=2, indicating no convincing evidence that the population mean delivery time differs from 2 days. Choice D is a distractor that wrongly takes p as the chance H₀ is true, whereas p assumes H₀ for its calculation. Choice C overclaims that the mean is exactly 2, but failing to reject does not prove this. In a mini-lesson: for mean tests, p > α means insufficient evidence against H₀, not confirmation of it; always conclude about the population, maintain probabilistic phrasing, and differentiate from sample-specific statements.
A quality-control manager tests whether the mean fill amount for a brand of cereal differs from the labeled 16 oz. A random sample of 40 boxes is taken, and conditions for a one-sample $t$ test are met. The hypotheses are $H_0: \mu=16$ and $H_a: \mu\neq 16$ at significance level $\alpha=0.05$ . The test is completed and the p-value is 0.032. What conclusion is appropriate?
Answer: Because the p-value is less than $\alpha$ , reject $H_0$ ; there is convincing evidence that the population mean fill amount is different from 16 oz. This question assesses the skill of carrying out a one-sample t-test for a population mean, focusing on interpreting the p-value and drawing an appropriate conclusion. The p-value of 0.032 is less than the significance level α=0.05, so we reject the null hypothesis H0: μ=16, providing convincing evidence that the true population mean fill amount differs from 16 oz. A common distractor, like choice B, incorrectly states that the p-value is greater than α, which would lead to failing to reject H0, but here the p-value is indeed smaller. In hypothesis testing for a population mean, rejecting H0 supports the alternative hypothesis but does not prove it definitively; it simply indicates the sample data are unlikely under H0. Remember, the conclusion should reference the population parameter and avoid claiming causation or absolutes, as seen in distractors like C and E that overstate the results. Always compare the p-value directly to α to decide on rejection, and phrase conclusions in terms of evidence for the alternative.
A hospital states that the mean wait time in its emergency department is $\mu=30$ minutes. A random sample of 35 patients is selected, and a one-sample $t$ test is performed with $H_0:\mu=30$ and $H_a:\mu>30$ at $\alpha=0.01$ . The p-value is 0.012. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the population mean wait time is greater than 30 minutes. This problem tests interpretation when p-value slightly exceeds α in a one-tailed test. With p-value = 0.012 and α = 0.01, we fail to reject H₀ because 0.012 > 0.01, meaning there is not convincing evidence at the 0.01 level that the population mean wait time is greater than 30 minutes. Choice A would be correct if α were 0.05, but the stricter 0.01 level requires stronger evidence. Choice C misinterprets what the p-value represents. Choice E makes an unsupported causal claim about staffing changes. When using α = 0.01, we require very strong evidence (p < 0.01) to reject H₀.
A nutrition label claims a certain cereal has a mean of 210 calories per serving. A random sample of 30 servings is measured, and a one-sample $t$ test is performed with $H_0: \mu = 210$ calories and $H_a: \mu < 210$ calories at $\alpha=0.05$ . The p-value from the test is 0.51. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not enough evidence that the population mean calories per serving is less than 210. This question tests understanding of hypothesis test conclusions when the p-value is large. With a p-value of 0.51, which is much greater than α = 0.05, we fail to reject H₀. This means there is not enough evidence to conclude that the mean calories per serving is less than 210. The large p-value suggests the sample data is quite consistent with the null hypothesis. Choice A would only be correct if the p-value were less than 0.05. Choice C misinterprets the p-value as a probability about the parameter. Choice D confuses sample statistics with population parameters. Choice E introduces causation that isn't part of the hypothesis test.
A gym claims that the mean weight loss for members after 8 weeks on a program is $\mu=6$ pounds. A random sample of 16 members is selected, and a one-sample $t$ test is conducted: $H_0:\mu=6$ vs. $H_a:\mu>6$ , using $\alpha=0.05$ . The p-value is 0.27. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the population mean weight loss is greater than 6 pounds. This question tests understanding of hypothesis test conclusions in a right-tailed test with a large p-value. Since p-value = 0.27 >> α = 0.05, we fail to reject H₀. When we fail to reject in a right-tailed test (Hₐ: μ > 6), we conclude there is not convincing evidence that the population mean weight loss is greater than 6 pounds. Choice B correctly states this conclusion. Common errors include: rejecting when p is large (A), claiming failure to reject proves H₀ (C), misinterpreting p-values as data proportions (D), or confusing evidence with proof (E). Remember: a large p-value indicates the data are consistent with H₀, not that H₀ is proven true.
A city transit agency claims the mean wait time for a bus on a certain route is $\mu=8$ minutes. Riders suspect the mean wait time is longer. A random sample of 50 wait times is recorded, and a one-sample $t$ test is conducted with $H_0:\mu=8$ versus $H_a:\mu>8$ at $\alpha=0.05$ . The p-value is 0.051. What conclusion is appropriate?
Answer: Because $p=0.051>0.05$ , fail to reject $H_0$ ; there is not quite sufficient evidence that the population mean wait time is greater than 8 minutes. In this one-tailed test for mean wait time, the p-value (0.051) is just slightly greater than α (0.05), so we fail to reject the null hypothesis. The conclusion is that there is not quite sufficient evidence that the population mean wait time is greater than 8 minutes. Choice B incorrectly rejects H₀ when p > α. Choice C misinterprets the p-value as P(μ > 8) rather than P(data | μ = 8). Choice D confuses inference about the sample with inference about the population. This case illustrates the importance of the significance level as a decision threshold - even a p-value very close to α still leads to failing to reject H₀ when p > α.
A bottling plant targets an average fill volume of $\mu=2.00$ liters. A random sample of 50 bottles is measured, and a one-sample $t$ test is carried out with $H_0:\mu=2.00$ and $H_a:\mu\neq 2.00$ at $\alpha=0.05$ . The p-value is 0.62. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the population mean fill volume differs from 2.00 liters. This problem tests understanding of a two-tailed test with a large p-value. With p-value = 0.62 and α = 0.05, we fail to reject H₀ because 0.62 > 0.05, indicating no convincing evidence that the population mean fill volume differs from 2.00 liters. Choice C incorrectly concludes that failing to reject H₀ proves the mean equals exactly 2.00 liters. Choice D misinterprets the p-value as the probability H₀ is true. Choice E discusses only the sample mean rather than making an inference about the population. A large p-value suggests our sample data is consistent with H₀, but we cannot conclude H₀ is definitely true.
A fitness app claims that the mean number of steps per day for its users is more than $\mu=8000$ . A random sample of 50 users is selected and a one-sample $t$ test is performed with $H_0:\mu=8000$ versus $H_a:\mu>8000$ at $\alpha=0.05$ . The p-value is 0.006. What conclusion is appropriate?
Answer: Reject $H_0$ because $p=0.006<0.05$ ; there is convincing evidence that the population mean steps per day is greater than 8000. This problem tests the skill of conducting a one-sample t-test for a population mean and drawing appropriate conclusions. Since the p-value of 0.006 is less than α=0.05, we reject H₀: μ=8000 in favor of $H_a$ : μ>8000, providing convincing evidence that the population mean steps per day exceeds 8000. A typical distractor is choice D, which misstates the p-value as the probability that the mean is greater than 8000, but it actually assumes H₀ and assesses data extremity. Choice E incorrectly shifts the conclusion to the sample mean, missing that inference is about the population. Mini-lesson on mean test conclusions: rejecting H₀ supports $H_a$ with evidence at the given α level, but does not imply causation; always frame conclusions in terms of the population and evidence strength, avoiding misinterpretation of p as the probability of $H_a$ .
A sleep researcher wants to know whether the mean number of hours of sleep for students at a large high school is less than 8 hours. A random sample of 25 students reports their sleep, and a one-sample $t$ test is conducted: $H_0:\mu=8$ vs. $H_a:\mu<8$ , using $\alpha=0.01$ . The p-value is 0.047. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the population mean sleep is less than 8 hours. This question tests understanding of hypothesis test conclusions when p-value > α. Since p-value = 0.047 > α = 0.01, we fail to reject H₀. When we fail to reject the null hypothesis, we conclude there is not convincing evidence for the alternative hypothesis (that μ < 8). Choice B correctly states this conclusion. Common errors include: rejecting H₀ by comparing to the wrong significance level (A), claiming failure to reject proves H₀ is true (C), using language of proof rather than evidence (D), or misinterpreting the p-value as probability of Hₐ (E). Key principle: failing to reject H₀ means insufficient evidence for Hₐ, not proof that H₀ is true.
A teacher believes students in her class score higher than the district average of $\mu=72$ on a standardized quiz. She randomly selects 20 students from her class and conducts a one-sample $t$ test with $H_0:\mu=72$ and $H_a:\mu>72$ at $\alpha=0.05$ . The p-value is 0.004. What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the population mean quiz score for her class is greater than 72. This question involves a one-tailed test with a very small p-value. With p-value = 0.004 and α = 0.05, we reject H₀ because 0.004 < 0.05, providing convincing evidence that the population mean quiz score for her class is greater than 72. Choice A incorrectly states we should fail to reject when p < 0.05. Choice C makes an unsupported causal claim about the teaching method. Choice D misinterprets what the p-value represents. Choice E overgeneralizes to individual students rather than the population mean. A small p-value provides strong evidence against H₀, leading us to conclude there is convincing evidence for H₁.
A bottled water company states its bottles have a mean volume of $\mu=500$ mL. A regulator tests whether the mean differs from the label. A random sample of 20 bottles is measured, and a one-sample $t$ test is performed with $H_0:\mu=500$ versus $H_a:\mu\neq 500$ at $\alpha=0.05$ . The p-value is 0.0004. What conclusion is appropriate?
Answer: Because $p=0.0004<0.05$ , reject $H_0$ ; there is very strong evidence that the population mean bottle volume differs from 500 mL. This two-tailed test examines whether the population mean bottle volume differs from 500 mL. With an extremely small p-value (0.0004) compared to α (0.05), we reject the null hypothesis with very strong evidence. The conclusion is that there is very strong evidence the population mean bottle volume differs from 500 mL. Choice B incorrectly fails to reject when p < α. Choice C misinterprets the p-value as P(H₀ is true). Choice D overgeneralizes to every bottle rather than the population mean. Very small p-values indicate the observed data would be extremely unlikely if H₀ were true, providing strong evidence against the null hypothesis.
A manufacturer advertises that its batteries last an average of $\mu=10$ hours. A consumer group suspects the mean lifetime is less. A random sample of 18 batteries is tested, and a one-sample $t$ test is conducted with $H_0:\mu=10$ versus $H_a:\mu<10$ at $\alpha=0.10$ . The p-value is 0.12. What conclusion is appropriate?
Answer: Fail to reject $H_0$ because $p=0.12>0.10$ ; there is not convincing evidence that the population mean lifetime is less than 10 hours. This question focuses on interpreting a one-sample t-test for a population mean. The p-value of 0.12 exceeds α=0.10, so we fail to reject H₀: μ=10, meaning there is not convincing evidence that the population mean lifetime is less than 10 hours. Choice D is a common distractor, incorrectly presenting the p-value as the chance H₀ is false, when it is conditional on H₀ being true. Choice C overreaches by claiming the mean is >=10 for all batteries, but failing to reject only indicates insufficient evidence for $H_a$ . In a mini-lesson for t-test conclusions: compare p to α carefully; if p > α, do not support $H_a$ , but refrain from affirming H₀ as definitively true—conclusions are probabilistic and apply to the population, not guaranteeing outcomes for every individual case.
A scientist tests whether the mean pH of water in a lake is different from neutral ( $\mu=7.0$ ). A random sample of 18 water measurements is taken, and a one-sample $t$ test is performed: $H_0:\mu=7.0$ vs. $H_a:\mu\ne 7.0$ , at $\alpha=0.05$ . The p-value is 0.049. What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the population mean pH differs from 7.0. This question tests understanding of hypothesis test conclusions when p-value is just below α in a two-tailed test. Since p-value = 0.049 < α = 0.05, we reject H₀. In a two-tailed test (Hₐ: μ ≠ 7.0), rejecting H₀ means we have convincing evidence that the population mean pH differs from 7.0. Choice B correctly states this conclusion. Common mistakes include: failing to reject when p < α (A), making specific directional claims from a two-tailed test (C), misinterpreting p-values (D), or inferring causation (E). Note: a two-tailed test only tells us the mean differs from 7.0, not whether it's higher or lower.
A city engineer believes the mean time for a traffic light cycle at an intersection is greater than 90 seconds. A random sample of 50 cycles is recorded, and a one-sample $t$ test is conducted with $H_0: \mu = 90$ seconds and $H_a: \mu > 90$ seconds at $\alpha=0.05$ . The p-value is 0.049. What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the population mean cycle time is greater than 90 seconds. In this one-tailed test about a population mean, the p-value (0.049) is less than the significance level α = 0.05. Therefore, we reject H₀ and conclude there is convincing evidence that the population mean cycle time is greater than 90 seconds. The engineer's belief is supported by the data. Choice A incorrectly fails to reject H₀ when p < α. Choice C misinterprets the p-value as the probability that the alternative hypothesis is true. Choice D makes an inappropriate leap from sample observations to population conclusions without proper testing. Choice E introduces causation that isn't tested by the hypothesis.
A package delivery company advertises that the mean delivery time for local shipments is $\mu=2.0$ days. A random sample of 60 shipments is selected, and a one-sample $t$ test is run: $H_0:\mu=2.0$ vs. $H_a:\mu>2.0$ , at $\alpha=0.10$ . The p-value is 0.082. What conclusion is appropriate?
Answer: Reject $H_0$ ; there is convincing evidence that the population mean delivery time is greater than 2.0 days. This question tests understanding of hypothesis test conclusions with a one-tailed test when p-value < α. Since p-value = 0.082 < α = 0.10, we reject H₀. In a right-tailed test (Hₐ: μ > 2.0), rejecting H₀ means we have convincing evidence that the population mean delivery time is greater than 2.0 days. Choice A correctly states this conclusion. Common mistakes include: failing to reject when p < α (B), thinking the sample mean must equal the hypothesized value (C), inferring causation from a hypothesis test (D), or misinterpreting p-values as probabilities about hypotheses (E). Remember: the significance level α = 0.10 is the threshold for decision-making, not the common 0.05.
A hospital states that the mean length of stay for patients undergoing a certain procedure is $\mu=3.5$ days. A researcher suspects the mean length of stay has decreased with a new protocol. A random sample of 22 patients is collected, and a one-sample $t$ test is conducted with $H_0:\mu=3.5$ versus $H_a:\mu<3.5$ at $\alpha=0.05$ . The p-value is 0.049. What conclusion is appropriate?
Answer: Reject $H_0$ because $p=0.049<0.05$ ; there is convincing evidence that the population mean length of stay is less than 3.5 days. This problem tests skills in performing a one-sample t-test for a population mean and concluding appropriately. Since p=0.049 < α=0.05, we reject H₀: μ=3.5, concluding convincing evidence that the population mean length of stay is less than 3.5 days. Distractor choice C misinterprets p as the chance H₀ is true, but p gauges data probability assuming H₀. Choice B attributes causation to the protocol, which is not supported by the test alone. Mini-lesson on conclusions: rejecting H₀ supports $H_a$ with evidence, but avoid causal claims unless the study design allows; conclusions target the population mean, use precise language about evidence, and never confuse p with the probability of hypotheses.
A fitness app claims that users who follow its plan will have a mean resting heart rate of $\mu=70$ beats per minute after 8 weeks. A researcher believes the mean is lower. A random sample of 35 users completes the plan, and a one-sample $t$ test is run with $H_0:\mu=70$ versus $H_a:\mu<70$ at $\alpha=0.01$ . The p-value is 0.009. What conclusion is appropriate?
Answer: Because $p=0.009<0.01$ , reject $H_0$ ; there is convincing evidence that the population mean resting heart rate after 8 weeks is less than 70 bpm. This problem involves testing whether the population mean resting heart rate is less than 70 bpm. Since p-value (0.009) < α (0.01), we reject the null hypothesis and conclude there is convincing evidence that the population mean resting heart rate after 8 weeks is less than 70 bpm. Choice B incorrectly fails to reject when p < α. Choice C misinterprets the p-value as the probability that μ < 70. Choice D overgeneralizes to every individual user rather than making an inference about the population mean. When conducting hypothesis tests, we make conclusions about population parameters based on sample evidence, not about individual values or certainties.
A sleep researcher tests whether the mean number of hours of sleep per night for students at a certain college is less than 7 hours. A random sample of 25 students is selected, and conditions for a one-sample $t$ test are met. The hypotheses are $H_0: \mu=7$ and $H_a: \mu<7$ with $\alpha=0.01$ . The test is completed and the p-value is 0.084. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the population mean sleep time is less than 7 hours. This question evaluates the ability to perform a one-sample t-test for a population mean, emphasizing the decision rule based on the p-value. With a p-value of 0.084 greater than α=0.01, we fail to reject H0: μ=7, meaning there is not convincing evidence that the population mean sleep time is less than 7 hours. Choice A, a distractor, wrongly suggests rejecting H0 despite the high p-value, which would be incorrect for this left-tailed test. In conducting tests for means, failing to reject H0 does not prove it true; it only indicates insufficient evidence against it, as opposed to choice C's erroneous claim of proof. A mini-lesson: always state conclusions in context, avoiding absolutes like 'proves' or assumptions about the sample directly implying population truths, as in distractors D and E. The key is comparing p to α without confusing the direction of the alternative hypothesis.
A university states that the mean time to graduate for its students is $\mu=4.0$ years. A random sample of 50 recent graduates is selected, and a one-sample $t$ test is conducted: $H_0:\mu=4.0$ vs. $H_a:\mu\ne 4.0$ , at $\alpha=0.01$ . The p-value is 0.018. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; there is not convincing evidence that the population mean time to graduate differs from 4.0 years. This question tests understanding of hypothesis test conclusions when p-value falls between α and a larger value. Since p-value = 0.018 > α = 0.01, we fail to reject H₀ at the 0.01 significance level. Choice B correctly states we fail to reject H₀ and lack convincing evidence that μ differs from 4.0 years. Common mistakes include: using the wrong significance level (A), misinterpreting p-values as probabilities of H₀ (C), thinking the sample mean must equal the hypothesized value (D), or inferring causation (E). This illustrates that the same p-value can lead to different conclusions depending on the chosen significance level.
A cereal manufacturer advertises that boxes contain an average of $\mu=500$ grams of cereal. A quality-control analyst takes a random sample of 25 boxes and performs a one-sample $t$ test with $H_0:\mu=500$ and $H_a:\mu\neq 500$ using $\alpha=0.01$ . The p-value from the test is 0.043. What conclusion is appropriate?
Answer: Fail to reject $H_0$ ; at the 0.01 level, there is not convincing evidence that the population mean differs from 500 grams. This question requires comparing a p-value to the significance level in a two-tailed test. With p-value = 0.043 and α = 0.01, we fail to reject H₀ because 0.043 > 0.01, meaning there is not convincing evidence at the 0.01 level that the population mean differs from 500 grams. Choice C misinterprets the p-value as the probability of H₁ being true. Choice D incorrectly concludes that failing to reject H₀ means the population mean equals exactly 500 grams. Choice E makes a causal claim that cannot be supported by this observational study. When the p-value exceeds α, we fail to reject H₀ but cannot conclude H₀ is true—we simply lack sufficient evidence against it.

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