Basic Statistics

174 REGRESSION AND CORRELATION

Table 12.3 Parameters and Statistics for the Single-Sample Case

Population

Parameters Sample Statistics Description

PX X Mean of X

CLY Y Mean of Y

Variance of X

4 4 Variance of Y

g?/.z 2 s;.x Variance of the residuals

0 b Slope of line

0 a Intercept of line

P r Correlation coefficient

gz^2 sf

12.2.7 Confidence Intervals in Single-Sample Linear Regression

If the required assumptions stated in Section 12.2.6 are made, confidence intervals

can be made for any of the parameters given in Table 12.3. The most widely used

confidence interval is the one for the slope of the regression line. When computing

confidence intervals for a mean in Chapter 7, we used the sample mean plus or minus

a t value from Table A.3 times the standard error of the mean. We follow the same

procedure in computing confidence intervals for b, the slope of the line. The standard

error of b is given by

se(b) =

dgF+

Note that in computing se(b), we divide the standard error of the estimate, s~.~, by

the square root of c(X - F)z. Hence, the se(b) becomes smaller as the sample

size increases. Further, the more spread out X’s are around their mean, the smaller

se(b) becomes.

The 95% confidence interval is given by

b & t[.975][se(b)]

For example, for the 10 males we have already computed the square root of the residual

mean-square error, or sy = 5.24. FromTable 12.2 we have c(X -x)’ = 7224.1.

So the se(b) is

5.24

= .062

5.24

se(b) = ~ - -

J7ETi 84.99

The 95% confidence interval is

.29 & 2.306(.062) = .29 =k .14

That is, the confidence interval for the population slope is