Inferences Concerning Future Response¶

The previous section discussed the distribution of the mean response. In many scenarios, we are interested in the distribution of the actual response \(Y\) at input \(x_{0}\), which takes the noise into account as well. We note

\[ \begin{aligned} Y_{0} &\sim \mathcal{N}(\theta_{0} + \theta_{1}x_{0}, \sigma^{2})\newline \hat{\theta}\_{0} + \hat{\theta}\_{1}x_{0} &\sim \mathcal{N}\bigg(\theta_{0} + \theta_{1}x_{0}, \sigma^{2} \bigg[ \frac{1}{n} + \frac{(x_{0} - \bar{x})^{2}}{S_{xx}} \bigg]\bigg)\newline Y_{0} - \hat{\theta}\_{0} - \hat{\theta}\_{1}x_{0} &\sim \mathcal{N}\bigg(0, \sigma^{2}\bigg( 1 + \frac{1}{n} + \frac{(x_{0} - \bar{x})^{2}}{S_{xx}} \bigg)\bigg) \end{aligned} \]

Now we utilise the distribution of \(SS_{R}\) to eliminate \(\sigma^{2}\) and get to the t-distribution

\[ \begin{aligned} \frac{Y_{0} - \hat{\theta}\_{0} - \hat{\theta}\_{1}x_{0}}{\sigma\sqrt{1 + \frac{1}{n} + \frac{(x_{0} - \bar{x})^{2}}{S_{xx}}}} \div \sqrt{\frac{SS_{R}}{(n-2)\sigma^{2}}} \sim t_{n-2} \end{aligned} \]

and the prediction interval for the response (not mean response is) at \(1-\alpha\) confidence

\[ \begin{aligned} (\hat{\theta}\_{0} + \hat{\theta}\_{1}x_{0}) \pm t_{\alpha/2, n-2} \sqrt{\bigg( 1+ \frac{1}{n} + \frac{(x_{0} - \bar{x})^{2}}{S_{xx}} \bigg) \bigg( \frac{SS_{R}}{n-2}\bigg)} \end{aligned} \]

Note that prediction interval is the interval where we expect the value of a random variable to lie, whereas the confidence interval is the one where the value of a parameter estimate to lie.