Statistical Methods for Psychology

the arrays of X—what we now know as conditional normality of Y. For the regression

model, there is no assumption of normality of the conditional distribution of Xor of the mar-

ginal distributions.)

9.9 Confidence Limits on Y

Although the standard error of estimate is useful as an overall measure of error, it is not a

good estimate of the error associated with any single prediction. When we wish to predict

a value of Yfor a given subject, the error in our estimate will be smaller when Xis near

than when Xis far from. (For an intuitive understanding of this, consider what would

happen to the predictions for different values of Xif we rotated the regression line slightly

around the point ,. There would be negligible changes near the means, but there would

be substantial changes in the extremes.) If we wish to predict Yon the basis of Xfor a new

member of the population (someone who was not included in the original sample), the

standard error of our prediction is given by

where is the deviation of the individual’s Xscore from the mean of X. This leads to

the following confidence limits on :

This equation will lead to elliptical confidence limits around the regression line, which are

narrowest for X 5 and become wider as |X 2 | increases.

To take a specific example, assume that we wanted to set confidence limits on the num-

ber of symptoms (Y) experienced by a student with a stress score of 10—a fairly low level

of stress. We know that

N= 107

t.025=1.984

YN=0.0086(10) 1 4.31=4.386

X=21.290

s^2 X=156.05

sY#X=0.173

X X

CI(Y)=YN 6 (ta> 2 )(s¿Y#X)

YN

Xi 2 X

s¿Y#X=sY#X

B

11

1

N

1

(Xi 2 X)^2

(N 2 1)s^2 X

XY

X

X

266 Chapter 9 Correlation and Regression

Figure 9.5 Bivariate normal distribution with r 5 .90