Interval estimates

In the previous page, we showed that the same value,

  =  b0 + b1 x

is used both to estimate the mean response at x and to predict a new individual's response at x. However the errors are different in the two situations — the errors tend to be larger for predicting a new value.

In both situations, it is more informative to give an interval of 'likely' values rather than a single value.

Estimating mean response

A 95% confidence interval for the mean response takes the form

We do not provide a formula for the standard error on the right of this formula — the details are not important and statistical software will tell you the value.

Predicting a new individual's response

For prediction, a similar interval can be found

where the value k is greater than the corresponding standard error for the confidence interval. (Again we do not provide a full formula. Statistical computer software will perform the calculations for you.)

A 95% prediction interval is wider than a 95% confidence interval for the distribution's mean.

Extrapolation

These 95% confidence intervals and 95% prediction intervals are valid within the range of x-values about which we have collected data, but they should not be relied on for extrapolation. Both intervals assume that the normal linear model describes the process, but we have no information about linearity beyond the x-values that have been collected.

From a scatterplot, we can check that there is approximate linearity within the observed range of x-values but there is no way to check linearity beyond the observed data.

Inertia welding experiment.

Manufacturers use inertia welding to join different metals that cannot easily be joined by other means (e.g. aluminium to steel). One part of a workpiece is attached to a flywheel that is rotated at speed and forced into contact with another piece that is restrained from rotating. The heat generated by friction at the interface produces a hot-pressure weld.

The diagram below shows data from an inertia welding experiment. The two variables in the experiment were the velocity (ft per minute) of the rotating workpiece and the breaking strength of the weld.

Drag the slider to display...

Because of uncertainty about linearity of the relationship, the confidence intervals and prediction intervals are unreliable if X is greater than 3.0 or less than 2.0.