Regression analysis, like most multivariate statistics, allows you to infer that there is a relationship between two or more variables. These relationships are seldom exact because there is variation caused by many variables, not just the variables being studied.
If you say that students who study more make better grades, you are really hypothesizing that there is a positive relationship between one variable, studying, and another variable, grades. You could then complete your inference and test your hypothesis by gathering a sample of (amount studied, grades) data from some students and use regression to see if the relationship in the sample is strong enough to safely infer that there is a relationship in the population. Notice that even if students who study more make better grades, the relationship in the population would not be perfect; the same amount of studying will not result in the same grades for every student (or for one student every time). Some students are taking harder courses, like chemistry or statistics, some are smarter, some will study effectively, some will get lucky and find that the professor has asked them exactly what they understood best. For each level of amount studied, there will be a distribution of grades. If there is a relationship between studying and grades, the location of that distribution of grades will change in an orderly manner as you move from lower to higher levels of studying.
Regression analysis is one of the most used and most powerful multivariate statistical techniques for it infers the existence and form of a functional relationship in a population. Once you learn how to use regression you will be able to estimate the parameters—the slope and intercept—of the function which links two or more variables. With that estimated function, you will be able to infer or forecast things like unit costs, interest rates, or sales over a wide range of conditions. Though the simplest regression techniques seem limited in their applications, statisticians have developed a number of variations on regression which greatly expand the usefulness of the technique. In this chapter, the basics will be discussed. In later chapters a few of the variations on, and problems with, regression will be covered. Once again, the t-distribution and F-distribution will be used to test hypotheses.
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