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Answer: The analyst can reject the joint null hypothesis because the F-statistic is significant at the 95% confidence level.
## Explanation To test the joint null hypothesis that β1 = 0 and β2 = 0 simultaneously, we use the **F-statistic** rather than individual t-statistics. ### Key Points: - **Individual t-tests**: - p-value for β1 = 0.07 > 0.05 → Cannot reject H0: β1 = 0 individually - p-value for β2 = 0.06 > 0.05 → Cannot reject H0: β2 = 0 individually - **Joint F-test**: - p-value for F-statistic = 0.045 < 0.05 → Can reject the joint null hypothesis H0: β1 = β2 = 0 ### Why F-test is appropriate: - The F-test examines whether **at least one** of the coefficients is significantly different from zero - Even though neither coefficient is individually significant at 5% level, the combination of both coefficients together may be significant - The F-statistic p-value of 0.045 indicates that the overall regression model is significant at the 95% confidence level ### Conclusion: The analyst can reject the joint null hypothesis because the F-statistic is significant at the 95% confidence level (p-value = 0.045 < 0.05). This means that collectively, the independent variables have explanatory power, even if individually they may not be significant.
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For a sample of 400 firms, the relationship between house price (Yi) and the area of each house (Xi) is modeled as follows: Y_i = β_1 + β_2 × X_i + ε_i. An analyst wants to test the joint null hypothesis that β1 = 0 and β2 = 0 at the 95% confidence level. The p-value for the t-statistic for β1 is 0.07, and the p-value for the t-statistic for β2 is 0.06. The p-value for the F-statistic for the regression is 0.045. Which of the following statements is correct?
A
The analyst cannot reject the joint null hypothesis because the F-statistic is not significant at the 95% confidence level.
B
The analyst can reject the joint null hypothesis because each β is different from 0 at the 95% confidence level.
C
The analyst cannot reject the joint null hypothesis because neither β is different from 0 at the 95% confidence level.
D
The analyst can reject the joint null hypothesis because the F-statistic is significant at the 95% confidence level.
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