Answer: 0.4169

## Explanation The standard error of the intercept ($\hat{\beta}_0$) in a linear regression can be calculated using the formula: $$ \text{SEE}_{\beta_0} = \sqrt{\frac{s^2(\hat{\mu}_X^2 + \hat{\sigma}_X^2)}{n\hat{\sigma}_X^2}} $$ Where: - $s^2 = 20.45$ (residual variance) - $\hat{\mu}_X = 0.61$ (mean of the independent variable) - $\hat{\sigma}_X^2 = 18.65$ (variance of the independent variable) - $n = 120$ (number of monthly observations over 10 years: 12 months × 10 years) Substituting the values: $$ \text{SEE}_{\beta_0} = \sqrt{\frac{20.45(0.61^2 + 18.65)}{120 \times 18.65}} $$ First, calculate the numerator: - $0.61^2 = 0.3721$ - $0.3721 + 18.65 = 19.0221$ - $20.45 \times 19.0221 = 388.902045$ Then calculate the denominator: - $120 \times 18.65 = 2238$ Now compute the fraction: - $388.902045 / 2238 = 0.1738$ Finally, take the square root: - $\sqrt{0.1738} = 0.4169$ Therefore, the standard error estimate of $\hat{\beta}_0$ is **0.4169**.

Author: Tanishq Prabhu