10-4 Multiple Regression 563 17. Testing Hypotheses About Regression Coefficients If the coefficient b1 has a nonzero value, then it is helpful in predicting the value of the response variable. If b1 = 0, it is not helpful in predicting the value of the response variable and can be eliminated from the regression equation. To test the claim that b1 = 0 use the test statistic t = 1b1 - 02>sb 1 . Critical values or P-values can be found using the t distribution with n - 1k + 12 degrees of freedom, where k is the number of predictor 1x2 variables and n is the number of observations in the sample. The standard error sb 1 is often provided by software. For example, see the accompanying StatCrunch display for Example 1, which shows that sb 1 = 0.071141412 (found in the column with the heading of “Std. Err.” and the row corresponding to the first predictor variable of height). Use the sample data in Data Set 1 “Body Data” and the StatCrunch display to test the claim that b1 = 0. Also test the claim that b2 = 0. What do the results imply about the regression equation? 18.Confidence Intervals for a Regression Coefficients A confidence interval for the regression coefficient b1 is expressed as b1 - E 6 b1 6 b1 + E where E = ta>2sb 1 The critical t score is found using n - 1k + 12 degrees of freedom, where k, n, and sb 1 are described in Exercise 17. Using the sample data from Example 1, n = 153 and k = 2, so df = 150 and the critical t scores are {1.976 for a 95% confidence level. Use the sample data for Example 1, the Statdisk display in Example 1 on page 554, and the StatCrunch display in Exercise 17 to construct 95% confidence interval estimates of b1 (the coefficient for the variable representing height) and b2 (the coefficient for the variable representing waist circumference). Does either confidence interval include 0, suggesting that the variable be eliminated from the regression equation? 19.Dummy Variable Refer to Data Set 18 “Bear Measurements” in Appendix B and use the sex, age, and weight of the bears. For sex, let 0 represent female and let 1 represent male. Letting the response 1y2 variable represent weight, use the variable of age and the dummy variable of sex to find the multiple regression equation. Use the equation to find the predicted weight of a bear with the characteristics given below. Does sex appear to have much of an effect on the weight of a bear? a. Female bear that is 20 years of age b. Male bear that is 20 years of age 10-4 Beyond the Basics 16.Full IQ Score Refer to Data Set 11 “IQ and Lead” in Appendix B and find the best regression equation with IQ FULL (full IQ score) as the response 1y2 variable. Use predictor variables of IQ VERB (verbal IQ score) and IQ PERF (performance IQ score). Why is this equation best? Based on these results, can we predict someone’s full IQ score if we know their verbal IQ score and their performance IQ score? Is such a prediction likely to be very accurate?
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