Inference for Regression

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Introduction

In this chapter, we continue our study of relationships between variables and describe methods for inference when there is a quantitative response variable and a single quantitative explanatory variable. The descriptive tools we learned in Chapter 2—scatterplots, least-squares regression, and correlation—are essential preliminaries to these methods and also provide a foundation for confidence intervals and significance tests.

We first met the sample mean in Chapter 1 as a measure of the center of a collection of observations. Later, we learned that when the data are a random sample from a population, the sample mean is an unbiased estimate of the population mean μ. In Chapters 6 and 7, we used as the basis for confidence intervals and significance tests for inference about μ.

Now we take this same approach for the problem of fitting straight lines to data. In Chapter 2, we met the least-squares regression line as a description of a straight-line relationship between a response variable y and an explanatory variable x. At that point, however, we did not distinguish between sample and population. In this chapter, we will now think of the least-squares line computed from the sample as an estimate of the true regression line for the population.

Following the common practice of using Greek letters for population parameters, we write the population line as . This notation reminds us that the intercept of the fitted line b₀ estimates the intercept of the population line β₀, and the fitted slope b₁ estimates the slope of the population line β₁.

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The methods detailed in this chapter will help us answer questions such as