# Lesson 36: More Regression

### Introduction

In the previous lesson, we learned that one of the primary uses of an estimated regression line:

\[\hat{y}=\hat{\alpha}+\hat{\beta}(x-\bar{x})\]

is to determine whether or not a linear relationship exists between the predictor *x* and the response *y*. In that lesson, we learned how to calculate a confidence interval for the slope parameter *β *as a way of determining whether a linear relationship does exist. In this lesson, we'll learn learn two other primary uses of an estimated regression line:

(1) If we are interested in knowing the value of the mean response *E*(*Y*) = *μ _{Y}* for a given value

*x*of the predictor, we'll learn how to calculate a

**confidence interval for the mean**

*E*(*Y*) =*.*

**μ**_{Y}(2) If we are interested in knowing the value of a new observation *Y*_{n+1} for a given value *x* of the predictor, we'll learn how to calculate a **prediction interval for the new observation **** Y_{n+1}**.