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 9.3.5b

Chapter 9, Problem 9.3.5b

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).

b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

Verified step by step guidance

Step 1: Calculate the differences between the paired measured and reported weights. For each pair, subtract the reported weight from the measured weight. For example, for the first pair, the difference is 147.3 - 142.

Step 2: Compute the mean of the differences. Add all the differences calculated in Step 1 and divide by the total number of pairs.

Step 3: Calculate the standard deviation of the differences. Use the formula for standard deviation: \( s = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \), where \( d_i \) are the differences, \( \bar{d} \) is the mean of the differences, and \( n \) is the number of pairs.

Step 4: Determine the margin of error for the confidence interval. Use the formula \( E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \), where \( t_{\alpha/2} \) is the critical t-value from the t-distribution table for the given confidence level and degrees of freedom \( n-1 \).

Step 5: Construct the confidence interval for the mean difference. The confidence interval is given by \( \bar{d} \pm E \). Interpret the confidence interval to determine whether it includes 0, which would indicate no significant difference between measured and reported weights.

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Key Concepts

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

Paired Sample Data

Paired sample data involves two related groups where each subject is measured twice, resulting in pairs of observations. This design is often used to compare two conditions or treatments on the same subjects, allowing for a more controlled analysis of differences. In this context, the measured weights and reported weights of the same subjects are paired, which helps in assessing the accuracy of reported weights.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around a sample mean or difference. In hypothesis testing, if the confidence interval for the difference between paired samples does not include zero, it suggests a statistically significant difference, supporting the conclusions drawn from the hypothesis test.

Normal Distribution Assumption

The normal distribution assumption is crucial in statistics, particularly for hypothesis testing and constructing confidence intervals. It implies that the differences between paired samples should follow a bell-shaped curve, allowing for the application of parametric tests. When this assumption holds, it enhances the validity of the results, as many statistical methods rely on the properties of normality to make inferences about the population from sample data.

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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)

Color and Cognition Researchers from the University of British Columbia conducted a study to investigate the effects of color on cognitive tasks. Words were displayed on a computer screen with background colors of red and blue. Results from scores on a test of word recall are given below. Higher scores correspond to greater word recall.

b. Construct a confidence interval appropriate for the hypothesis test in part (a). What is it about the confidence interval that causes us to reach the same conclusion from part (a)?

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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.

b. The 40 cotinine measurements from the nonsmoking group consist of these values (all in ng/mL): 1, 1, 90, 244, 309, and 35 other values that are all 0. Does this sample appear to be from a normally distributed population? If not, how are the results from part (a) affected?

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.

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.

b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

Textbook Question

Independent Samples Which of the following involve independent samples?

b. Data Set 6 “Births” includes birth weights of a sample of baby boys and a sample of baby girls.

Textbook Question

Readability of Font On a Computer Screen The statistics shown below were obtained from a standard test of readability of fonts on a computer screen (based on data from “Reading on the Computer Screen: Does Font Type Have Effects on Web Text Readability?” by Ali et al., International Education Studies, Vol. 6, No. 3). Reading speed and accuracy were combined into a readability performance score (x), where a higher score represents better font readability.

b. Construct the confidence interval suitable for testing the claim in part (a).

Textbook Question

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).

b. Construct the confidence interval suitable for testing the claim in part (a).