Let's work through each part step-by-step using the information given: ### Given Regression Output \[ \hat{Y} = 5.4 + 3.2 X, \quad R^2 = 0.26, \quad SER = 6.2 \] Standard errors (in parentheses) for the intercept and slope: - Intercept: 5.4 (**SE = 3.1**) - Slope: 3.2 (**SE = 1.5**) Sample size: \( n = 250 \) --- ## **a. Test \( H_0: \beta_1 = 0 \) vs. \( H_1: \beta_1 \neq 0 \) at the 5% level** **Step 1: State the hypotheses** - Null: \( H_0: \beta_1 = 0 \) - Alternative: \( H_1: \beta_1 \neq 0 \) **Step 2: Compute the t-statistic** \[ t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} = \frac{3.2 - 0}{1.5} = 2.13 \] **Step 3: Find the critical value** - For large \( n \), use the standard normal critical value for a two-sided test at 5%: \( z_{0.025} \approx 1.96 \) **Step 4: Compare** - \( |t| = 2.13 > 1.96 \) - **Reject \( H_0 \) at the 5% level.** --- ## **b. Construct a 95% confidence interval for \( \beta_1 \)** \[ \text{CI} = \hat{\beta}_1 \pm z_{0.025} \times SE(\hat{\beta}_1) \] \[ = 3.2 \pm 1.96 \times 1.5 \] \[ = 3.2 \pm 2.94 \] \[ = [3.2 - 2.94, 3.2 + 2.94] = [0.26, 6.14] \] **Final 95% CI:** \[ \boxed{[0.26,\ 6.14]} \] --- ## **c. Suppose you learned that \( Y_i \) and \( X_i \) were independent. Would you be surprised? Explain.** **Explanation:** - If \( Y_i \) and \( X_i \) are independent, the true \( \beta_1 = 0 \). - In part (a), we rejected \( H_0 \) and in (b), the 95% CI does **not** include 0; both indicate a statistically significant relationship. - **Therefore, yes, you should be surprised** if \( Y_i \) and \( X_i \) are actually independent but your regression suggests otherwise. This could be due to random sampling variation or a violation of regression assumptions. --- ## **d. Suppose \( Y_i \) and \( X_i \) are independent and many samples of size \( n = 250 \) are drawn, regressions estimated, and (a) and (b) answered. In what fraction of the samples would \( H_0 \) from (a) be rejected? In what fraction of samples would \( \beta_1 = 0 \) be included in the confidence interval from (b)?** **Explanation:** - If \( Y_i \) and \( X_i \) are truly independent, the true \(\beta_1 = 0\). - The test from (a) is a 5% level test, so you'd expect to **reject \( H_0 \) in about 5% of samples** (Type I error rate). - The 95% CI from (b) should contain the true value (\(0\)) in **about 95% of samples** by construction. --- ### **Summary of Answers** **a.** Test statistic is 2.13; reject \( H_0 \) at 5% level. **b.** 95% CI is \(\boxed{[0.26,\ 6.14]}\) **c.** Yes, you would be surprised since independence implies \(\beta_1 = 0\), but regression suggests otherwise. **d.** About 5% of samples would reject \( H_0 \); about 95% of samples' 95% CIs would include \( \beta_1 = 0 \). Let me know if you need any step in more detail!