Answer:
Step-by-step explanation:
To determine if we can conclude that the slope of the regression line (of the population) is less than zero, we will perform a hypothesis test using the given information.
Given:
- Regression equation: \( \hat{y} = 29.29 - 0.96x \)
- Standard error of the slope (\( SE_b \)) = 0.22
- Sample size (n) = 8
- Significance level (\( \alpha \)) = 0.05
### Hypotheses:
- **Null hypothesis (\( H_0 \))**: The slope of the regression line (\( \beta_1 \)) is equal to zero or greater (\( \beta_1 \geq 0 \)).
- **Alternative hypothesis (\( H_1 \))**: The slope of the regression line (\( \beta_1 \)) is less than zero (\( \beta_1 < 0 \)).
### Test Statistic:
To test the hypothesis about the slope, we use the t-statistic:
\[ t = \frac{b_1 - 0}{SE_{b_1}} \]
where:
- \( b_1 \) is the estimated slope from the regression equation (here, \( b_1 = -0.96 \)).
- \( SE_{b_1} \) is the standard error of the slope (\( SE_b = 0.22 \)).
\[ t = \frac{-0.96 - 0}{0.22} = \frac{-0.96}{0.22} \approx -4.36 \]
### Decision Rule:
- **Rejection region**: Since we are testing \( \beta_1 < 0 \), we are interested in the left tail of the t-distribution.
- Critical value for a one-tailed test at \( \alpha = 0.05 \) with \( df = n - 2 = 8 - 2 = 6 \): \( t_{\alpha, df} = t_{0.05, 6} \).
Using a t-table or statistical software, \( t_{0.05, 6} \approx -1.943 \).
### Conclusion:
- **Compare the test statistic to the critical value**: \( t = -4.36 \) is less than \( t_{0.05, 6} = -1.943 \).
- **Decision**: \( t \) falls in the rejection region.
- **Conclusion**: We reject the null hypothesis \( H_0 \) in favor of the alternative hypothesis \( H_1 \).
- **Interpretation**: There is sufficient evidence at the 0.05 significance level to conclude that the slope of the regression line (of the population) is less than zero. Therefore, we can conclude that the slope of the regression line is negative.
