Explanation

In simple linear regression, the F-test is used to test the overall significance of the regression model. Specifically:

1. F-test purpose: Tests whether there is a significant linear relationship between the dependent variable (Y) and the independent variable (X).

2. Null hypothesis: H₀: β₁ = 0 (the slope coefficient is zero, meaning no linear relationship)

3. Alternative hypothesis: H₁: β₁ ≠ 0 (the slope coefficient is not zero, meaning a significant linear relationship exists)

4. Why not the other options:

   • Option A: Testing if the intercept is significantly different from zero uses a t-test, not an F-test.
   • Option B: Testing for positive correlation specifically would use a one-tailed t-test for the correlation coefficient or slope, not an F-test.

5. F-statistic calculation:

   $$F = \frac{MSR}{MSE} = \frac{\frac{SSR}{k}}{\frac{SSE}{n-k-1}}$$

   where:

   • SSR = Regression sum of squares
   • SSE = Error sum of squares
   • k = number of independent variables (1 for simple linear regression)
   • n = sample size

6. Interpretation: A significant F-statistic indicates that the regression model explains a significant portion of the variance in the dependent variable, meaning there is a significant linear relationship between X and Y.

Therefore, the F-distributed test statistic is most appropriate for testing option C: whether there is a significant linear relationship between the dependent and independent variables.