Statistical Analysis for Education and Psychology Researchers

Confidence Interval Limits for the Predicted Mean Response

The 95 per cent CI limits for the predicted mean response with a fixed value for the
explanatory variable, in this example, 8, is given by:
Ŷ−[t 1 −α/2 SE Ŷ] to Ŷ+[t 1 −α/2 SE Ŷ]
CI
for
mea
n
resp
onse
—8.9

where Ŷ, the predicted mean value obtained from the regression equation when xi=8 is:

The 95 per cent CI is therefore 124.285−[2.306×2.2245] to 124.285+

[2.306−2.2245]=119.155 to 129.415

When this calculation is made for all of the observed values of xi in the sample, these
confidence intervals can be plotted (see the output from PROC GPLOT in Figure 8.7b).

Interpretation

We can predict that the population mean for the response variable, SMATHS, will lie
within the range 119 to 129 given a value of 8 for the explantory variable. Note that the
population mean is a fixed parameter and it is the sample estimates which will vary from
sample to sample. Any inference based on results would only be valid for the target
population from which the sample was selected. The large width of the confidence
interval here is probably attributable to the small sample size.

Confidence Intervals for an Individual Response Score (Prediction

Interval)

A computational procedure similar to that in the previous worked example is followed
when a prediction interval is estimated. A 95 per cent CI for the prediction interval is
given by,
Ŷ−[t 1 −α/2Spred] to Ŷ+[t 1 −α/2Spred]
CI
for
indiv
idual
resp
onse

Inferences involving continuous data 267