Basic Statistics

172 REGRESSION AND CORRELATION

I X

Figure 12.3 Position of regression line for bivariate normal distribution.

because both the slope and the intercept are used in computing the line. The square

root of the residual mean square is often called the standard error of estimate.

The residual mean square measures the variation of the Y observations around

the regression line. If it is large, the vertical distances from the line are large. If it

is small, the vertical distances are small. In the example using the 10 data points,

s;,~ = 27.49. The variance of the original 10 Y values can be computed from

Table 12.2 as 828.90/9 = 92.1, so the variance about the regression line is much

smaller than the variance of Y in this example. This indicates that Y values tend to be

closer to Y than they are toy. Thus, using the least-squares regression line to predict

systolic blood pressure from weight gives us a closer prediction than does using the

sample mean systolic blood pressure.

12.2.6 Model Underlying Single-Sample Linear Regression

Up to this point in this chapter, we have discussed computing a regression line from

a single sample. To compute confidence limits and to make tests of hypotheses, we

shall make the basic assumption that we have taken a simple random sample from a

population where X and Y follow a bivariate normal distribution. In Chapter 6 we

discussed a normal distribution for a single variable X. For X and Y to be bivariately

normal requires not only that both X and Y be individually normally distributed but

also that the relationship between X and Y has a bivariate normal distribution (see

van Belle et al. [2004]).

One way of determining whether the data points may be from a bivariate nor-

mal distribution is to examine the scatter diagram. If the variables have a bivariate

normal distribution, data points should lie approximately within an ellipse. An el-

lipse is depicted in Figure 12.3. Also, if the data points (X, Y) follow a bivariate

normal distribution, both X and Y separately should be normally distributed. Nor-

mal probability plots can be used to check this. The residuals should be normally