Chapter 13, Problem 56PS

Based on a sample of $n=20,$ the latest-squares method was used to develop the following prediction line: ${\hat{Y}}_{\text{i}}=5+3{\text{X}}_{\text{i}}.$ In addition,

$S_{\text{YX}}=1.0\bar{X}=2{\displaystyle \underset{i=i}{\overset{n}{\sum }}{\left(X_{\text{i}}-\bar{X}\right)}^{\text{2}}=20}$

a. Construct a $95\%$ confidence interval estimate of the population mean response for $X=4.$

b. Construct a $95\%$ prediction interval of an individual response for $X=4.$

c. Compare the result of (a) and (b) with those of problem 13.55 (a) and (b). Which intervals are wider? Why?