In a multiple regression model involving 50 observations, the following estimated regression equation was obtained: = 20 + 5x₁ - 4x₂ + 8x₃ + 8x₄ For this model, SSR = 700 and SSE = 100. At the 5% level, A) there is no evidence that the model is significant. B) it can be concluded that the model is significant. C) the conclusion is that the slope of x₁ is significant. D) there is evidence that the slope of x₂ is significant.

Question

Quizplus · Accepted Answer

In order to determine if the model is significant at the 5% level, we can conduct an F-test with the following hypotheses:
H0: β1 = β2 = β3 = β4 = 0
Ha: At least one βi is not equal to 0
Using the given SSR and SSE values, we can calculate MSResidual and MSTotal as follows: 
MSResidual = SSE / (n - k - 1) = 100 / (50 - 4 - 1) = 2
MSTotal = SSR / (k) = 700 / 4 = 175
where k is the number of predictors in the model (in this case, 4).
The F-statistic for the test is then:
F = (MSTotal - MSResidual) / MSResidual = (175 - 2) / 2 = 86.5
Using an F-distribution with (4, 45) degrees of freedom (numerator degrees of freedom = k, denominator degrees of freedom = n - k - 1), we find that the p-value for this test is less than 0.05 (p-value << 0.001). Therefore, we reject the null hypothesis and conclude that the model is significant at the 5% level. This means that at least one of the predictors is a significant contributor to the model. We cannot determine which specific predictor(s) are significant without conducting additional tests.

In a Multiple Regression Model Involving 50 Observations, the Following