Answer: there is a significant linear relationship between the dependent and independent variables.

## 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.

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