7.4 Interval estimation

• Everything is estimated with some degree of error
• Confidence intervals for the mean of \(Y\) at some given value of \(X\) (say, \(X^*\))

\[ \hat{Y} \pm tS \bigg[ \frac{1}{N} + \sqrt{\frac{(X^* - \bar{X})^{2}}{\sum(X - \bar{X})^{2}}} \quad \bigg] \]

• Prediction intervals for an individual \(Y\)

\[ \hat{Y} \pm tS \bigg[ 1+ \frac{1}{N} + \sqrt{\frac{(X^* - \bar{X})^{2}}{\sum(X - \bar{X})^{2}}} \quad \bigg] \]

Which one is wider? Why?

How does this relate to the standard deviation of individual \(x\)’s, and the standard deviation of \(\bar{x}\)’s?