Find a 95 confidence interval for the true mean value of y

Question

Consider the following summary statistics for data obtained in a simple linear regression analysis.

β^{^}₀ = 38.86

n = 21

MSE = 21.23

β^{^}₁ = -4.318

X = 30.891

S_xx = 6.298

a. Find a 95% confidence interval for the true mean value of Y when x = 31.5. Find the width of the resulting interval.

b. Find a 95% confidence interval for the true mean value of Y when x = 31.9. Find the width of the resulting interval.

c. Explain why the width of the confidence interval in part (b) is greater than the width of the confidence interval in part (a).