Answer: We can reject the null hypothesis because the F-statistic is significant at 90% confidence

## Explanation To test the joint hypothesis that all coefficients are simultaneously equal to zero ($\beta_1 = 0$, $\beta_2 = 0$, and $\beta_3 = 0$), we need to use the **F-test**, not individual t-tests. ### Key Points: 1. **Individual t-tests vs. Joint F-test**: - Individual t-tests examine whether each coefficient is individually different from zero - The F-test examines whether **at least one** coefficient is different from zero, considering the joint effects and interactions among variables 2. **P-value interpretation**: - For $\beta_1$: p-value = 0.11 > 0.10 → cannot reject $\beta_1 = 0$ at 90% confidence - For $\beta_2$: p-value = 0.12 > 0.10 → cannot reject $\beta_2 = 0$ at 90% confidence - For F-statistic: p-value = 0.09 < 0.10 → can reject the joint null hypothesis at 90% confidence 3. **Decision rule**: - Reject the null hypothesis when p-value < significance level (α = 0.10 for 90% confidence) - Since F-test p-value (0.09) < 0.10, we reject the joint null hypothesis 4. **Correct interpretation**: - The F-test result means that **at least one** of the coefficients is statistically different from zero - This doesn't mean all coefficients are significant individually - The F-test accounts for multicollinearity and joint effects that individual t-tests don't capture ### Why other options are incorrect: - **A**: Incorrect because the F-test allows us to reject the joint null hypothesis - **B**: Incorrect because individual t-tests don't show each β is different from zero (p-values > 0.10) - **C**: Incorrect because the F-statistic IS significant at 90% confidence (p-value = 0.09 < 0.10) **Answer D is correct**: We can reject the null hypothesis because the F-statistic is significant at 90% confidence level.

Author: Nikitesh Somanthe