Question

One of the benefits of a linear regression model, is that it’s relatively easy to create a confidence interval on the mean response. Imagine you created a linear regression model from a dataset with n = 12, that applies over the range 0.0 ≤ x ≤ 10.0, where the mean value of x = 5.0 , the fitted model is Y = 74 + 15 x, Sxx = 0.75, and σ2. = 1.5 At what value of x does the minimum width of the 95% confidence interval on the mean response occur?

Answer 1

The width of 95% confidence interval on the mean response is 2 * Margin or error = 2 * t * σ * standard error

where t is the t value for given df and 95% confidence interval .

So, the width of 95% confidence interval on the mean response is dependent on standard error, as other variables ( t ,  σ ) are constant.

Hence minimum width of the 95% confidence interval on the mean response occur at minimum value of standard error

Standard error is given as,

S² = (1/n) + (x - )² / Sxx

where n is the sample size.

Given, Sxx = 0.75 and n = 12, the square of standard error is,

S² = (1/12) + (x - )² / 0.75

The minimum value of standard error is 1/12 at x =

because the term (x - )² / 0.75 is always positive and its minimum value is 0 at x =

Thus, for x = = 5 , the minimum width of the 95% confidence interval on the mean response occur.