Predicting Responses

For the new value $ x$ we can predict the response by

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

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 $ x$ 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}}}

The data set consists of

  1. explanatory variable for $ x_i$'s;
  2. dependent variable for $ Y_i$'s.
Provide the explanatory value $ x$ on the column 'New.value.' Then it produces the corresponding fitted value $ \hat{y}$ along with the intervals of choice in the following table.

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