Explanation

This question involves testing a joint null hypothesis using both individual t-tests and an F-test:

Key Information:

• Joint null hypothesis: β₁ = 0 and β₂ = 0
• Confidence level: 95% (α = 0.05)
• Individual p-values:
  • β₁: p-value = 0.07
  • β₂: p-value = 0.06
• F-statistic p-value: 0.045

Analysis:

1. Individual t-tests:

   • For β₁: p-value = 0.07 > 0.05 → Cannot reject H₀: β₁ = 0
   • For β₂: p-value = 0.06 > 0.05 → Cannot reject H₀: β₂ = 0
   • Therefore, neither coefficient is individually significant at the 95% confidence level

2. F-test for joint hypothesis:

   • p-value = 0.045 < 0.05 → Can reject the joint null hypothesis H₀: β₁ = β₂ = 0
   • The F-test examines whether at least one of the coefficients is significantly different from zero

Why Option C is Correct:

• The F-statistic p-value (0.045) is less than the significance level (0.05), so we can reject the joint null hypothesis
• This means that at least one of the coefficients (β₁ or β₂) is significantly different from zero
• The F-test is more powerful for testing joint hypotheses than individual t-tests

Why Other Options are Incorrect:

• A: Incorrect because neither β is individually significant (both p-values > 0.05)
• B: Incorrect because the F-test allows us to reject the joint null hypothesis
• D: Incorrect because the F-statistic IS significant (p-value = 0.045 < 0.05)

This demonstrates the important distinction between individual coefficient tests and joint hypothesis testing in regression analysis.