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 $$\begin{array}{rl}{Y}_0& \sim \mathcal{N}({\theta }_0+{\theta }_1{x}_0,{\sigma }^2)\\ {\hat{\theta }}_0+{\hat{\theta }}_1{x}_0& \sim \mathcal{N}({\theta }_0+{\theta }_1{x}_0,{\sigma }^2[\frac{1}{n}+\frac{(x_0-\overline{x}{)}^2}{S_{xx}}])\\ Y_0-{\hat{\theta }}_0-{\hat{\theta }}_1{x}_0& \sim \mathcal{N}(0,{\sigma }^2(1+\frac{1}{n}+\frac{(x_0-\overline{x}{)}^2}{S_{xx}}))\end{array}$$

Now we utilise the distribution of $S{S}_R$ to eliminate ${\sigma }^2$ and get to the t-distribution $$\begin{array}{r}\frac{Y_0-{\hat{\theta }}_0-{\hat{\theta }}_1{x}_0}{\sigma \sqrt{1+\frac{1}{n}+\frac{(x_0-\overline{x}{)}^2}{S_{xx}}}}÷\sqrt{\frac{S{S}_R}{(n-2){\sigma }^2}}\sim t_{n-2}\end{array}$$ and the prediction interval for the response (not mean response is) at $1-\alpha$ confidence $$\begin{array}{r}({\hat{\theta }}_0+{\hat{\theta }}_1{x}_0)\pm t_{\alpha /2,n-2}\sqrt{(1+\frac{1}{n}+\frac{(x_0-\overline{x}{)}^2}{S_{xx}})(\frac{S{S}_R}{n-2})}\end{array}$$

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.