Book cover for Stats Data and Models

Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

ISBN #9780321986498

4th Edition

1,406 Questions

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16,980 Students Helped

Homework Questions

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Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This section emphasizes the importance of recognizing paired data and using the appropriate paired t-test methods to analyze such data. It highlights that pairing – whether resulting from matching in experiments or natural pairing in observational studies – helps control for extra variability by focusing on differences. Key steps include computing individual differences, checking for independence and Normality of these differences, and then applying one-sample t-test techniques. Inference about the mean difference, through hypothesis tests or confidence intervals, provides clearer insights than comparing grouped data separately. Finally, the concept of blocking is reinforced as a powerful design strategy to reduce variability.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Analysis of Variance (ANOVA)

A collection of statistical techniques used to compare the means of three or more groups by partitioning total variability into components due to differences among groups (between-group) and within groups.

Example Problems

Example 1

Which of the following scenarios should be analyzed as paired data? a) A group of college freshmen and a group of sophomores are asked about the quality of the university cafeteria. Do students' opinions change during their time at school? b) Spouses are asked about the number of hours of sleep they get each night. We want to see if husbands get more sleep than wives. c) 50 insomnia patients are given a placebo and 50 are given a mild sedative. Which subjects sleep more hours?

Example 2

Which of the following scenarios should be analyzed as paired data? a) Students take an MCAT prep course. Their before and after scores are compared. b) 20 male and 20 female students in a class take a midterm. Their scores are compared. c) A group of college freshmen are asked about the quality of the university cafeteria. A year later, the same students are asked about the cafeteria again. Do students' opinions change during their time at school?

Example 3

We have data on the city and highway fuel efficiency of 316 cars and 316 trucks. a) Would it be appropriate to use paired $t$ methods to compare the cars and the trucks? b) Would it be appropriate to use paired $t$ methods to compare the city and highway fuel efficiencies of these vehicles? c) A histogram of the differences (highway - city) is shown. Are the conditions for inference satisfied?

Example 4

One kind of scale for weighing trucks can measure their weight as they drive across a plate. Is this method consistent with the traditional method of static weighing? Are the conditions for matched pairs inference satisfied? Weights are in $1000 \mathrm{~s}$ of pounds $$\begin{array}{|c|c|c|}\hline \text { Weight-in-Motion } & \text { Static Weight } & \text { Diff (Static - Motion) } \\\hline 26.0 & 27.9 & -1.9 \\29.9 & 29.1 & 0.8 \\39.5 & 38.0 & 1.5 \\25.1 & 27.0 & -1.9 \\31.6 & 30.3 & 1.3 \\36.2 & 34.5 & 1.7 \\25.1 & 27.8 & -2.7 \\31.0 & 29.6 & 1.4 \\35.6 & 33.1 & 2.5 \\40.2 & 35.5 & 4.7 \\\hline\end{array}$$

Example 5

In Exercise $3,$ after deleting an outlying value of $-27,$ the mean difference in fuel efficiencies for the 632 vehicles was $7.37 \mathrm{mpg}$ with a standard deviation of $2.52 \mathrm{mpg} .$ Find a $95 \%$ confidence interval for this difference and interpret it in context.

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Common Mistakes

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