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 \(\beta\) as a way of determining whether a linear relationship does exist. In this lesson, we'll learn two other primary uses of an estimated regression line:

1. If we are interested in knowing the value of the mean response \(E(Y)=\mu_Y\) for a given value \(x\) of the predictor, we'll learn how to calculate a confidence interval for the mean \(E(Y)=\mu_Y\).

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 \(Y_{n+1}\).