552 CHAPTER 10 Correlation and Regression 21.Confidence Interval for Mean Predicted Value Example 1 in this section illustrated the procedure for finding a prediction interval for an individual value of y. When using a specific value x0 for predicting the mean of all values of y, the confidence interval is as follows: yn - E 6 y 6 yn + E where E = ta>2 # se B1 n + n1x0 - x2 2 n1Σx22 - 1Σx22 The critical value ta>2 is found with n - 2 degrees of freedom. Using the 9 pairs of jackpot>tickets data from Table 10-1 on page 507, find a 95% confidence interval estimate of the mean number of tickets sold when the jackpot is 625 million dollars. 10-3 Beyond the Basics Key Concept So far in this chapter we have discussed the linear correlation between two variables, but this section presents methods for analyzing a linear relationship with more than two variables. We focus on these two key elements: (1) finding the multiple regression equation, and (2) using the value of adjusted R2 and the P-value as measures of how well the multiple regression equation fits the sample data. Because the required calculations are so difficult, manual calculations are impractical and a threat to mental health, so this section emphasizes the use and interpretation of results from technology. 10-4 Multiple Regression PART 1 Basic Concepts of a Multiple Regression Equation As in the preceding sections of this chapter, we will consider linear relationships only. The following multiple regression equation describes linear relationships involving more than two variables. DEFINITION A multiple regression equation expresses a linear relationship between a response variable y and two or more predictor variables (x1, x2, . . . , xk). The general form of a multiple regression equation obtained from sample data is yn = b0 + b1x1 + b2x2 + g+ bkxk The following Key Elements box includes the key components of this section. For notation, see that the coefficients b0, b1, b2, c, bk are sample statistics used to estimate the corresponding population parameters b0, b1, b2, c, bk. Also, note that the multiple regression equation is a natural extension of the format yn = b 0 + b1x1 used in Section 10-2 for regression equations with a single independent variable x1. In Section 10-2, it would have been reasonable to question why we didn’t use the more common and familiar format of y = mx + b, and we can now see that using yn = b 0 + b1x1 allows us to easily extend that format to include additional predictor variables.
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