Multiple Regression

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University of Windsor *

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8320

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Statistics

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Feb 20, 2024

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3

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Multiple Regression 1. Consider the following data. a. The estimated multiple regression equation for these data has b 0 = 63.4, b 1 = − 3.33, and b 2 = 0.417. What is the predicted value of y when x 1 = 9 and x 2 = 8? b. Find the predicted values of y for the other pairs of x 1 and x 2 values in the table. c. Find the residuals e i . d. Calculate the residual standard deviation, s . 2. Refer to exercise 1. a. Calculate SSE. b. The results indicate that SST = 783.33. Find SSR, MSR and MSE. c. Calculate the F -statistic to test the overall significance of the multiple regression model. d. At α = .05, is the multiple regression model significant overall? Explain.
3. Refer to the example in exercises 1 and 2. a. Interpret the meaning of b 1 , and b 2 in the estimated multiple regression equation. b. The results indicate that SE( b 1 ) = .731. Construct a 95% confidence interval for β 1 . Interpret.c. The results indicate that SE( b 2 ) = .715. Construct a 95% confidence interval for β 2 . Interpret.d. Would you recommend dropping either independent variable from the model? Explain. 4. A data set of 12 observations for y, x 1 , x 2 and x 3 results in the estimated multiple regression equation y = 88.9 + 0.34 x 1 – 2.49 x 2 – 0.137 x 3 with SST = 2198.92 and SSR = 1552.55. a. Find SSE, MSR and MSE. b. Calculate the F -statistic to test the overall significance of the multiple regression model. c. State the null and alternative hypotheses. d. At α = .05, is the multiple regression model significant overall? Explain. 5. Refer to exercise 4. a. Compute R 2 . b. Compute the adjusted R 2 . c. Interpret their meaning. d. At α = .05, can we conclude that R 2 is significantly different from zero. Explain. 6. For the example in exercises 4 and 5, the results also show that SE( b 1 ) = 2.374, SE( b 2 ) = 0.628 and SE( b 3 ) = 0.140.
a. Calculate the t -ratio to test if β 1 = 0. b. Calculate the t -ratio to test if β 2 = 0. c. Calculate the t -ratio to test if β 3 = 0. d. At α = .05, would you recommend dropping any of the independent variables from the model? Explain.
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