Let's solve each part step by step: --- **Given regression output:** \[ \hat{Y} = 5.4 + 3.2X, \qquad R^2 = 0.26, \qquad SER = 6.2 \] - Standard error of intercept: (3.1) - Standard error of slope: (1.5) - Sample size \( n = 250 \) ### **a. Test \( H_0: \beta_1 = 0 \) vs. \( H_1: \beta_1 \neq 0 \) at 5% level** #### **Step 1: Find test statistic** \[ t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} = \frac{3.2 - 0}{1.5} = \frac{3.2}{1.5} = 2.13 \] #### **Step 2: Find critical value** - Degrees of freedom: \( df = n - 2 = 248 \). For large \( n \), \( t_{0.025, 248} \approx 1.96 \). #### **Step 3: Compare and decide** \[ |t| = 2.13 > 1.96 \] - **Reject \( H_0 \) at the 5% significance level.** --- ### **b. Construct a 95% confidence interval for \( \beta_1 \)** #### **Step 1: Formula** \[ \hat{\beta}_1 \pm t^* \times SE(\hat{\beta}_1) \] - \( t^* \approx 1.96 \) #### **Step 2: Plug in values** \[ 3.2 \pm 1.96 \times 1.5 = 3.2 \pm 2.94 \] \[ \text{Interval: } [3.2 - 2.94,\ 3.2 + 2.94] = [0.26,\ 6.14] \] --- ### **c. Suppose you learned that \( Y_i \) and \( X_i \) were independent. Would you be surprised? Explain.** - If \( Y_i \) and \( X_i \) were truly independent, in the population, \( \beta_1 \) should equal zero. - Here, the sample estimate \( \hat{\beta}_1 = 3.2 \), and the 95% CI barely excludes zero. - However, in a sample of 250, it's not *impossible* to get a coefficient this far from zero just by chance, but it's somewhat unlikely (about 5% chance under the null). - Since we rejected the null at 5%, it would be **somewhat surprising** to learn \( Y_i \) and \( X_i \) are independent, but it's not impossible due to sampling variability. --- ### **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? - This is the **Type I error rate**, which is the significance level (5%). - **Answer:** About 5% of samples. #### - In what fraction of samples would the value \( \beta_1 = 0 \) be included in the confidence interval from (b)? - The true value will be inside the 95% CI **95% of the time**. --- ## **Summary of Answers:** **a.** Reject \( H_0 \) at 5% level; \( t = 2.13 \) is greater than 1.96. **b.** 95% confidence interval for \( \beta_1 \): **[0.26, 6.14]** **c.** Somewhat surprising, but not impossible; would expect this about 5% of the time by chance. **d.** - \( H_0 \) rejected in **5%** of samples (Type I error). - \( \beta_1 = 0 \) included in **95%** of confidence intervals. Let me know if you need more detailed explanation or calculations!