Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 9.2.13a

Chapter 9, Problem 9.2.13a

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)

Bicycle Commuting A researcher used two different bicycles to commute to work. One bicycle was steel and weighed 30.0 lb; the other was carbon and weighed 20.9 lb. The commuting times (minutes) were recorded with the results shown below (based on data from “Bicycle Weights and Commuting Time,” by Jeremy Groves, British Medical Journal).

a. Use a 0.05 significance level to test the claim that the mean commuting time with the heavier bicycle is the same as the mean commuting time with the lighter bicycle.

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: The mean commuting time with the heavier bicycle is equal to the mean commuting time with the lighter bicycle (μ₁ = μ₂). H₁: The mean commuting time with the heavier bicycle is not equal to the mean commuting time with the lighter bicycle (μ₁ ≠ μ₂). This is a two-tailed test.

Step 2: Identify the sample statistics provided in the problem. For the heavier bicycle: n₁ = 30, x̄₁ = 107.8, s₁ = 4.9. For the lighter bicycle: n₂ = 26, x̄₂ = 108.4, s₂ = 6.3.

Step 3: Calculate the test statistic using the formula for a two-sample t-test for independent samples: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)). Substitute the values for x̄₁, x̄₂, s₁, s₂, n₁, and n₂ into the formula.

Step 4: Determine the degrees of freedom (df) using the formula: df = min(n₁ - 1, n₂ - 1). In this case, df = min(30 - 1, 26 - 1) = 25. Use this df to find the critical t-value from Table A-3 or technology for a two-tailed test at a significance level of 0.05.

Step 5: Compare the calculated t-value to the critical t-value. If the absolute value of the calculated t-value exceeds the critical t-value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this context, the null hypothesis (H0) posits that there is no difference in mean commuting times between the two bicycles, while the alternative hypothesis (H1) suggests that a difference exists. The significance level (alpha) determines the threshold for rejecting the null hypothesis, with a common choice being 0.05.

Independent Samples

Independent samples refer to two or more groups that are not related or paired in any way. In this scenario, the commuting times for the heavier and lighter bicycles are collected from different samples of cyclists, meaning the performance of one group does not influence the other. This independence is crucial for applying certain statistical tests, such as the t-test for comparing means.

T-test for Two Independent Samples

The t-test for two independent samples is a statistical test used to compare the means of two groups when the population standard deviations are unknown and assumed to be unequal. This test calculates a t-statistic based on the sample means, sizes, and standard deviations, which is then compared to a critical value from the t-distribution to determine if the observed difference is statistically significant.

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Textbook Question

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9-1 along with “Table” answers based on Table A-3 with df equal to the smaller of n1-1 and n2-1)

Queues Listed on the next page are waiting times (seconds) of observed cars at a Delaware inspection station. The data from two waiting lines are real observations, and the data from the single waiting line are modeled from those real observations. These data are from Data Set 30 “Queues” in Appendix B. The data were collected by the author.

a. Use a 0.01 significance level to test the claim that cars in two queues have a mean waiting time equal to that of cars in a single queue.

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Textbook Question

Second-Hand Smoke Samples from Data Set 15 “Passive and Active Smoke” include cotinine levels measured in a group of smokers ( n = 40, x_bar = 172.48 ng/mL, 119.50 ng/mL ) and a group of nonsmokers not exposed to tobacco smoke ( n = 40, x_bar = 16.35 ng/mL, 62.53 ng/mL ). Cotinine is a metabolite of nicotine, meaning that when nicotine is absorbed by the body, cotinine is produced.

a. Use a 0.05 significance level to test the claim that the variation of cotinine in smokers is greater than the variation of cotinine in nonsmokers not exposed to tobacco smoke.

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Textbook Question

Cigarette Pack Warnings A study was conducted to find the effects of cigarette pack warnings that consisted of text or pictures. Among 1078 smokers given cigarette packs with text warnings, 366 tried to quit smoking. Among 1071 smokers given cigarette packs with warning pictures, 428 tried to quit smoking. (Results are based on data from “Effect of Pictorial Cigarette Pack Warnings on Changes in Smoking Behavior,” by Brewer et al., Journal of the American Medical Association.) Use a 0.01 significance level to test the claim that the proportion of smokers who tried to quit in the text warning group is less than the proportion in the picture warning group.

a. Test the claim using a hypothesis test.

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Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.

Do Men Talk Less than Women? Listed below are word counts of males and females in couple relationships (from Data Set 14 “Word Counts” in Appendix B).

a. Use a 0.05 significance level to test the claim that men talk less than women.

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Textbook Question

Can Dogs Detect Malaria? A study was conducted to determine whether dogs could detect malaria from socks worn by malaria patients and socks worn by patients without malaria. Among 175 socks worn by malaria patients, the dogs made correct identifications 123 times. Among 145 socks worn by patients without malaria, the dogs made correct identifications 131 times (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Use a 0.05 significance level to test the claim of no difference between the two rates of correct responses.

a. Test the claim using a hypothesis test.

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Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.

Measured and Reported Weights Listed below are measured and reported weights (lb) of random female subjects (from Data Set 4 “Measured and Reported” in Appendix B).

a. Use a 0.05 significance level to test the claim that for females, the measured weights tend to be higher than the reported weights.

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