Answer: (-5.97%, 9.37%)

## Explanation ### Step 1: Calculate the Predicted Value Using the regression equation: $$ \overline{\text{WPO}} = -3.2\% + 0.49 \times 10\% = 1.7\% $$ ### Step 2: Determine the Critical t-value - Sample size: n = 30 - Degrees of freedom = n - 2 = 28 - For 95% confidence interval, α = 0.05, α/2 = 0.025 - t-value for 28 degrees of freedom at 0.025 significance level: t₀.₀₂₅,₂₈ = 2.04 ### Step 3: Calculate the Margin of Error Margin of error = t-value × Standard error of forecast $$ \text{Margin of Error} = 2.04 \times 3.76\% = 7.67\% $$ ### Step 4: Construct the Confidence Interval $$ \text{CI}_{95\%} = \overline{\text{WPO}} \pm \text{Margin of Error} = 1.7\% \pm 7.67\% $$ $$ \text{Lower Bound} = 1.7\% - 7.67\% = -5.97\% $$ $$ \text{Upper Bound} = 1.7\% + 7.67\% = 9.37\% $$ Therefore, the 95% confidence interval is **(-5.97%, 9.37%)**. ### Key Points - The confidence interval is centered around the predicted value (1.7%) - The width of the interval reflects the uncertainty in the prediction - The interval includes negative values, indicating that WPO could potentially decrease even when S&P 500 has a positive excess return - The standard error of forecast accounts for both the uncertainty in the regression coefficients and the variability of individual observations around the regression line

Author: Tanishq Prabhu