e-Statistics

Predicting Responses

For the new value

we can predict the response by

$\displaystyle \hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x$

The data set consists of

explanatory variable for 's;
dependent variable for 's.

The $(1-\alpha) =$ confidence interval for the expected value $y = \beta_0 + \beta_1 x$ is given by

$\displaystyle \hat{y}\pm t_{\alpha/2,n-2}\hat{\sigma}\sqrt{\frac{1}{n}+\frac{(x - \bar{X})^2}{S_{xx}}}$

The $(1-\alpha)$ -level prediction interval for an unobserved response at the new value

is given by

$\displaystyle \hat{y}\pm t_{\alpha/2,n-2}\hat{\sigma}\sqrt{1+\frac{1}{n}+\frac{(x - \bar{X})^2}{S_{xx}}}$

By providing the explanatory value on the column 'X', the above table produces the corresponding fitted value $\hat{y}$ along with the interval (Lower,Upper) of choice explained above.

The predicted values (solid line) together with the interval of choice (dashed line or blue cross) suggests how well the new values can be predicted.