Introduction to Probability and Statistics for Engineers and Scientists

352 Chapter 9: Regression

theregression coefficients, and must usually be estimated from a set of data. A regression
equation containing a single independent variable — that is, one in whichr=1—is
called asimple regression equation, whereas one containing many independent variables is
called amultiple regression equation.
Thus, a simple linear regression model supposes a linear relationship between the mean
response and the value of a single independent variable. It can be expressed as

Y=α+βx+e

wherex is the value of the independent variable, also called the input level,Yis the
response, ande, representing the random error, is a random variable having mean 0.

EXAMPLE 9.1a Consider the following 10 data pairs (xi,yi),i=1,..., 10, relatingy, the
percent yield of a laboratory experiment, tox, the temperature at which the experiment
was run.

ixi yi ixi yi

1 100 45 6 150 68

2 110 52 7 160 75

3 120 54 8 170 76

4 130 63 9 180 92

5 140 62 10 190 88

A plot ofyiversusxi— called ascatter diagram— is given in Figure 9.1. As this scatter
diagram appears to reflect, subject to random error, a linear relation betweenyandx,it
seems that a simple linear regression model would be appropriate. ■

100

90

80

70

60

50

40100 120 140 160 180 200 x

y

FIGURE 9.1 Scatter plot.