Measures of Regression and Prediction Intervals 9.3 498 CHAPTER 9 Correlation and Regression What You Should Learn How to interpret the three types of variation about a regression line How to find and interpret the coefficient of determination How to find and interpret the standard error of estimate for a regression line How to construct and interpret a prediction interval for y Variation About a Regression Line The Coefficient of Determination The Standard Error of Estimate Prediction Intervals Variation About a Regression Line In this section, you will study two measures used in correlation and regression studies—the coefficient of determination and the standard error of estimate. You will also learn how to construct a prediction interval for y using a regression equation and a given value of x. Before studying these concepts, you need to understand the three types of variation about a regression line. To find the total variation, the explained variation, and the unexplained variation about a regression line, you must first calculate the total deviation, the explained deviation, and the unexplained deviation for each ordered pair 1xi, yi2 in a data set. These deviations are shown in the figure. Total deviation = yi - y Explained deviation = nyi - y Unexplained deviation = yi - nyi After calculating the deviations for each data point 1xi, yi2, you can find the total variation, the explained variation, and the unexplained variation. The total variation about a regression line is the sum of the squares of the differences between the y@value of each ordered pair and the mean of y. Total variation = Σ1yi - y2 2 The explained variation is the sum of the squares of the differences between each predicted y@value and the mean of y. Explained variation = Σ1nyi - y2 2 The unexplained variation is the sum of the squares of the differences between the y@value of each ordered pair and each corresponding predicted y@value. Unexplained variation = Σ1yi - nyi2 2 The sum of the explained and unexplained variations is equal to the total variation. Total variation = Explained variation + Unexplained variation DEFINITION As its name implies, the explained variation can be explained by the relationship between x and y. The unexplained variation cannot be explained by the relationship between x and y and is due to other factors, such as sampling error, coincidence, or lurking variables. (Recall from Section 9.1 that lurking variables are variables that have an effect on the variables being studied but are not included in the study.) (xi, y) (xi, yi) x Unexplained deviation yi −yi Total deviation yi −y Explained deviation yi −y y x (xi, yi) y
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