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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 9 - Inferences from Two SamplesProblem 9.3.7a
Chapter 9, Problem 9.3.7a

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.


The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April.


a. Use a 0.01 significance level to test the claim that for the population of freshman male college students, the weights in September are less than the weights in the following April.

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ₁ = μ₂, which states that the mean weight in September is equal to the mean weight in April. The alternative hypothesis is H₁: μ₁ < μ₂, which states that the mean weight in September is less than the mean weight in April.
Step 2: Calculate the differences between the paired weights (April - September) for each student. For example, for the first student, the difference is 67 - 67 = 0. Repeat this for all pairs to create a list of differences.
Step 3: Compute the mean (d̄) and standard deviation (s_d) of the differences. Use the formulas: d̄ = (Σd) / n and s_d = sqrt((Σ(d - d̄)²) / (n - 1)), where d represents the differences and n is the number of pairs.
Step 4: Perform a t-test for paired samples. Calculate the test statistic t using the formula: t = (d̄ - 0) / (s_d / sqrt(n)), where 0 is the hypothesized mean difference under H₀. Determine the degrees of freedom (df = n - 1).
Step 5: Compare the calculated t-value to the critical t-value at a significance level of 0.01 and df = n - 1. If the calculated t-value is less than the critical t-value, reject H₀ and conclude that the mean weight in September is less than the mean weight in April. Otherwise, fail to reject H₀.

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Key Concepts

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

Paired Sample T-Test

A paired sample t-test is a statistical method used to compare the means of two related groups. In this context, it assesses whether the average weight of male college freshmen in September is significantly less than in April. This test accounts for the fact that the samples are related, as they consist of the same individuals measured at two different times.
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Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether a result is statistically significant. In this case, a significance level of 0.01 indicates that there is a 1% risk of concluding that a difference exists when there is none. This stringent level is often used in studies where the consequences of a Type I error (false positive) are particularly serious.
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Step 4: State Conclusion Example 4

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. The assumption of normality is crucial for the validity of the paired sample t-test, as it ensures that the test statistics follow a predictable distribution under the null hypothesis.
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Related Practice
Textbook Question

Forecast and Actual Temperatures Listed below are actual temperatures (°F) along with the temperatures that were forecast five days earlier (data collected by the author). Use a 0.05 significance level to test the claim that differences between actual temperatures and temperatures forecast five days earlier are from a population with a mean of 0°F.

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Textbook Question

F Test Statistic


a. If s2,1 represents the larger of two sample variances, can the F test statistic ever be less than 1?


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Textbook Question

Variation of Hospital Times Use the sample data given in Exercise 7 “Seat Belts” and test the claim that for children hospitalized after motor vehicle crashes, the numbers of days in intensive care units for those wearing seat belts and for those not wearing seat belts have the same variation. Use a 0.05 significance level.

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Textbook Question

Pulse Rates of Women and Men Using the samples of women and men included in Data Set 1 “Body Data,” we get this 95% confidence interval estimate of the difference between the population mean of pulse rates (bpm) of women and the population mean of pulse rates (bpm) of men: 1.7 bpm < u1-u2 < 7.2bpm. In this confidence interval, women correspond to population 1 and men correspond to population 2.


a. What does the confidence interval suggest about equality of the mean pulse rate of women and the mean pulse rate of men?

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Textbook Question

Friday the 13th Refer to the sample data from Exercise 1.


a. Find the differences d, then find the values of d_bar and sd

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Textbook Question

Smoking Cessation Programs Among 198 smokers who underwent a “sustained care” program, 51 were no longer smoking after six months. Among 199 smokers who underwent a “standard care” program, 30 were no longer smoking after six months (based on data from “Sustained Care Intervention and Postdischarge Smoking Cessation Among Hospitalized Adults,” by Rigotti et al., Journal of the American Medical Association, Vol. 312, No. 7). We want to use a 0.01 significance level to test the claim that the rate of success for smoking cessation is greater with the sustained care program. Test the claim using a hypothesis test.

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