Absolutely! Let's break down the exercise step by step: --- ### Given Regression Output A random sample of size \( n = 250 \) gives the regression: \[ \hat{Y} = 5.4 + 3.2X, \quad R^2 = 0.26, \quad SER = 6.2 \] (Standard errors in parentheses: \( (3.1) \) under 5.4, \( (1.5) \) under 3.2) --- #### a. **Test \( H_0: \beta_1 = 0 \) vs. \( H_1: \beta_1 \neq 0 \) at the 5% level** - **Null hypothesis (\(H_0\)):** The coefficient of \( X \) is zero (\( \beta_1 = 0 \)) - **Alternative hypothesis (\(H_1\)):** The coefficient of \( X \) is not zero (\( \beta_1 \neq 0 \)) **Test Statistic:** \[ t = \frac{\text{Estimate} - \text{Null value}}{\text{Standard error}} = \frac{3.2 - 0}{1.5} \approx 2.13 \] - With \( n = 250 \), the degrees of freedom are large, so the critical value for a 5% significance level (two-sided) is about 1.96. **Decision:** - \( |t| = 2.13 > 1.96 \), so we **reject \( H_0 \) at the 5% level**. --- #### b. **Construct a 95% confidence interval for \( \beta_1 \)** The formula is: \[ \text{Estimate} \pm 1.96 \times \text{Standard error} \] \[ 3.2 \pm 1.96 \times 1.5 = 3.2 \pm 2.94 = [0.26, 6.14] \] **So, the 95% confidence interval is:** \[ [0.26, 6.14] \] --- #### c. **Suppose you learned that \( Y_i \) and \( X_i \) were independent. Would you be surprised? Explain.** - If \( Y \) and \( X \) are independent, the true value of \( \beta_1 \) should be **zero** (because changes in \( X \) don't affect \( Y \)). - However, our confidence interval **does not include zero**, and we rejected \( H_0 \) at 5%. This suggests there **is** a relationship in the data. - **Conclusion:** Yes, you should be surprised, because the data suggests a statistically significant relationship, but independence would mean there should be none. --- #### d. **(Theoretical) If \( Y_i \) and \( X_i \) are independent and many samples of size \( n = 250 \) are drawn...** 1. **Fraction of samples in which \( H_0 \) is rejected:** - If \( H_0 \) is true, the **Type I error rate** is the significance level: **5%** of samples would (incorrectly) reject \( H_0 \). 2. **Fraction of samples in which \( \beta_1 = 0 \) is included in the 95% confidence interval:** - By construction, **95%** of the confidence intervals would contain the true value (\( \beta_1 = 0 \)). --- ### Summary Table | Part | Concept | Key Steps/Answers | |------|---------|------------------| | (a) | t-test | t = 2.13, reject \( H_0 \) | | (b) | CI | [0.26, 6.14] | | (c) | Independence | Surprised, since the data suggests a relation | | (d) | Repeated Sampling | 5% reject \( H_0 \); 95% include 0 in CI | Let me know if you want more details on any part!