1. **Problem (d): Compute the 95% confidence interval for $\beta_1$** To compute the confidence interval for $\beta_1$, we use the formula: $$ CI = \hat{\beta}_1 \pm t_{\alpha/2, n-2} \times SE(\hat{\beta}_1) $$ where: - $\hat{\beta}_1$ is the estimated slope coefficient - $SE(\hat{\beta}_1)$ is the standard error of the slope - $t_{\alpha/2, n-2}$ is the critical value from the t-distribution with $n-2$ degrees of freedom at $\alpha=0.05$ (for 95% confidence) 2. **Problem (e): Is the impact of the interest rate on the inflation rate statistically significant?** We perform the hypothesis test: - Null hypothesis $H_0: \beta_1 = 0$ - Alternative hypothesis $H_1: \beta_1 \neq 0$ Calculate the t-statistic: $$ t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} $$ Compare $|t|$ to the critical value $t_{\alpha/2, n-2}$. If $|t| > t_{\alpha/2, n-2}$, reject $H_0$ and conclude significance. 3. **Problem (f): Interpret the OLS estimate of $\beta_1$** The OLS estimate $\hat{\beta}_1$ represents the expected change in the inflation rate for a one-unit increase in the interest rate, holding other factors constant. A positive $\hat{\beta}_1$ means inflation increases as interest rate increases; a negative value means the opposite. 4. **Problem (g): Effect of increasing sample size to 1000 on the standard error estimate** Standard error of $\hat{\beta}_1$ typically decreases as sample size increases because: $$ SE(\hat{\beta}_1) \propto \frac{1}{\sqrt{n}} $$ Increasing $n$ to 1000 reduces the standard error, improving precision of the estimate, assuming variance and other conditions remain unchanged.