TABLE 17-2 One of the most common questions of prospective house buyers pertains to the cost of heating in dollars (Y).To provide its customers with information on that matter,a large real estate firm used the following 4 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit (X 1 ),the amount of insulation in inches (X 2 ),the number of windows in the house (X 3 ),and the age of the furnace in years (X 4 ).Given below are the EXCEL outputs of two regression models. -Referring to Table 17-2 and allowing for a 1% probability of committing a type I error,what is the decision and conclusion for the test H₀ : β₁ = β₂ = β₃ = β₄ = 0 vs.H₁ : At least one β_j ≠ 0,j = 1,2,^...,4 using Model 1? A)Do not reject H₀ and conclude that the 4 independent variables have significant individual linear effects on heating costs. B)Reject H₀ and conclude that the 4 independent variables taken as a group have significant linear effects on heating costs. C)Do not reject H₀ and conclude that the 4 independent variables taken as a group do not have significant linear effects on heating costs. D)Reject H₀ and conclude that the 4 independent variables taken as a group do not have significant linear effects on heating costs.

Question

Quizplus · Accepted Answer

The null hypothesis H₀ states that all four independent variables have no significant linear effects on heating costs, while the alternative hypothesis H₁ states that at least one of the independent variables has a significant linear effect on heating costs. We can use the F-test to compare the significance of the regression and the residual sum of squares. If the regression sum of squares is significant, then at least one of the independent variables has a significant linear effect. From the ANOVA table in Model 1, we can see that the regression sum of squares is 20,872.862 with 4 degrees of freedom and the residual sum of squares is 70,379.789 with 293 degrees of freedom. The mean square regression (MSR) is 5,218.2155, and the mean square error (MSE) is 240.284. The F-statistic is the ratio of MSR to MSE, which is 21.73. To test the null hypothesis, we compare the F-statistic to the critical value of F with 4 and 293 degrees of freedom at a 1% significance level. The critical value of F is 2.54. Since the F-statistic is greater than the critical value of F, we reject the null hypothesis and conclude that at least one of the independent variables has a significant linear effect on heating costs. Therefore, the best choice is B, which states that we reject H₀ and conclude that the 4 independent variables taken as a group have significant linear effects on heating costs.

TABLE 17-2 One of the Most Common Questions of Prospective House Buyers