Answer: [1.1165; 1.3295]

## Explanation The 90% confidence interval for the slope coefficient is calculated using the formula: $$ \text{CI} = \hat{b} \pm t_{\frac{\alpha}{2}, n-2} \times s_{\hat{b}} $$ Where: - $\hat{b} = 1.223$ (point estimate of slope coefficient) - $s_{\hat{b}} = 0.063$ (standard error of slope coefficient) - $n = 36$ (number of observations) - Degrees of freedom = $n - 2 = 36 - 2 = 34$ - $\alpha = 0.10$ (for 90% confidence level) - $\alpha/2 = 0.05$ - $t_{0.05, 34} = 1.69$ (from t-distribution table) **Calculation:** Lower bound: $1.223 - (1.69 \times 0.063) = 1.223 - 0.10647 = 1.11653 \approx 1.1165$ Upper bound: $1.223 + (1.69 \times 0.063) = 1.223 + 0.10647 = 1.32947 \approx 1.3295$ Therefore, the 90% confidence interval is [1.1165; 1.3295]. **Why other options are incorrect:** - **Option B** [1.223; 1.3295]: Only includes the upper bound correctly but uses the point estimate as the lower bound instead of calculating the proper lower bound. - **Option C** [0.002; 1.223]: Uses the intercept coefficient (0.002) as the lower bound and the slope coefficient as the upper bound, which is incorrect. - **Option D** [0.063; 1.223]: Uses the standard error (0.063) as the lower bound and the slope coefficient as the upper bound, which is incorrect. This question tests understanding of confidence interval construction for regression coefficients using t-distribution with appropriate degrees of freedom.

Author: Nikitesh Somanthe