Question

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- [2 marks] Suppose that we want to find a regression equation relating systolic blood pressure (v) to weight (x1), age (x2) and smoking status (0 = does not smoke, 1 = smokes less than one pack per day, 2 = smokes one or more packs per day). Use the Minitab outputs below to test whether or not the smoking status variable adds to the predictive value of a model which already contains weight and age, using a = .05. i.e., test the hypothesis Ho:B4 = B5 = 0 vs H1 : at least one of B4, B5 0 in the model y = Bo+B1X1 + B2X2 + B3X3 + $4x4 + B5X5. - - - - - - - - - - - - - Regression Analysis: SYSTOLIC versus WEIGHT, AGE F-Value 62.05 27.33 69.36 Source Regression WEIGHT AGE Error Lack-of-Fit Pure Error Total Adj MS 4842.50 1618.81 4108.44 78.05 59.76 8.00 P-value 0.000 0.000 0.000 Adj ss 9685.0 1618.8 4108.4 6399.8 4195.6 2204.2 16084.8 7.47 0.285 S 8.834 R-sq 60.21 % R-sq(adj) 63.54% R-sq (pred) 61.70% VIF Term Constant WEIGHT AGE Coef 87.90 0.1529 0.5053 SE Coef 3.97 0.0293 0.0607 T-Value 22.11 5.23 8.33 P-Value 0.000 0.000 0.000 1.27 1.27 - - - Regression Analysis: SYSTOLIC versus WEIGHT, AGE, x4, x5 Source Regression WEIGHT AGE Adj ss 10372.4 1572.9 4133.6 6.7 29.3 Adj MS 2593.10 1572.93 4133.55 6.69 29.34 F-Value 36.32 26.16 68.74 0.11 0.49 P-Value 0.000 0.000 0.000 0.739 0.487 *4 *5

Answer 1

To test if we can add some variables in the model we should F-statistic. Where the statistic is given by:

where m = number of independent variables being tested for elimination and SS’_E is the value of SS_E for the model without these variables. Here SS’_E = 6399.8, SSE = 5712.3, MS_E=71.4 and m=2. Thus the test statistic is 4.81

_(SS-SSE)/m MSE

To test the significance we know that the F-statistic follows an F-distribution with 2 and 80 degrees of freedom in numerator and denominator respectively. The degrees of freedom for the denominator is calculated by dividing MSE from SSE. Thus the p-value is 0.01< 0.05. Thus we reject the null hypothesis. Hence, we conclude that the two variables add to the predictive value.