Question

Consider the multiple regression model \[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon, \] where \( \epsilon \) is normally distributed with mean 0 and variance \( \sigma^2>0 \), and \( \beta_0, \beta_1, \beta_2, \beta_3 \) are unknown parameters. Suppose 52 observations of \( (Y, X_1, X_2, X_3) \) yield sum of squares due to regression as 18.6 and total sum of squares as 79.23. Then, for testing the null hypothesis \( H_0: \beta_1 = \beta_2 = \beta_3 = 0 \) against the alternative hypothesis \( H_1: \beta_i \neq 0 \) for some \( i = 1, 2, 3 \), the value of the test statistic (rounded off to three decimal places), based on one way analysis of variance, is ________

Hint:

The F-test in multiple regression checks the overall significance of the model using \( F = \frac{MSR}{MSE} \).

Answer 1

Correct Answer: 4.907

Given: \[ SSR = 18.6, \quad SST = 79.23, \quad n = 52, \quad \text{number of predictors } k = 3. \] The residual sum of squares (SSE) is: \[ SSE = SST - SSR = 79.23 - 18.6 = 60.63. \] The mean squares are: \[ MSR = \frac{SSR}{k} = \frac{18.6}{3} = 6.2, \] \[ MSE = \frac{SSE}{n - k - 1} = \frac{60.63}{52 - 3 - 1} = \frac{60.63}{48} = 1.2631. \] The test statistic for the overall F-test is: \[ F = \frac{MSR}{MSE} = \frac{6.2}{1.2631} = 4.908. \] Thus, the value of the test statistic is \( \boxed{4.909} \).