An economist estimates the following model: y = β₀ + β₁x + ε. She would like to construct interval estimates for y when x equals 2. She estimates a modified model where y is the response variable and the explanatory variable is now defined as x⁺ = x - 2. A portion of the regression results is shown in the accompanying table. According to the modified model, which of the following is a 95% confidence interval for E(y) when x equals 2? (Note that t_0.025,10 = 2.228.)

Question

Quizplus · Accepted Answer

To construct a 95% confidence interval for E(y) when x equals 2, we need to use the modified model and the estimated coefficients from the regression results. The estimated regression equation is: y = 23.29 + 6.53(x⁺) - 0.85(x⁺)^2 + ε Substituting x = 2, we get: y = 23.29 + 6.53(2⁺) - 0.85(2⁺)^2 + ε y = 23.29 + 13.06 - 10.2 + ε y = 26.15 + ε The standard error of the mean is SE(ŷ) = SE(26.15) = 1.29 Using the t-distribution with n - k - 1 = 10 degrees of freedom (where k = 2 is the number of parameters estimated), we find the critical value of t for a 95% confidence interval to be 2.228. The 95% confidence interval for E(y) is: E(y) ± tα/2*SE(ŷ) = 26.15 ± 2.228(1.29) = [18.63, 30.93] Therefore, the best choice is C.

An Economist Estimates the Following Model: Y = β0

An Economist Estimates the Following Model: Y = β₀