Applied Statistics and Probability for Engineers

11-2 SIMPLE LINEAR REGRESSION 375

variability in the observations Yon oxygen purity. Thus, when ^2 is small, the observed

values of Ywill fall close to the line, and when ^2 is large, the observed values of Ymay de-

viate considerably from the line. Because ^2 is constant, the variability in Yat any value of

xis the same.

The regression model describes the relationship between oxygen purity Yand hydrocar-

bon level x. Thus, for any value of hydrocarbon level, oxygen purity has a normal distribution

with mean 75 15 xand variance 2. For example, if x1.25, Yhas mean value Yx 75 

15(1.25) 93.75 and variance 2.

In most real-world problems, the values of the intercept and slope ( 0 ,  1 ) and the error

variance ^2 will not be known, and they must be estimated from sample data. Then this fitted

regression equation or model is typically used in prediction of future observations of Y, or for

estimating the mean response at a particular level of x. To illustrate, a chemical engineer might

be interested in estimating the mean purity of oxygen produced when the hydrocarbon level is

x1.25%. This chapter discusses such procedures and applications for the simple linear re-

gression model. Chapter 12 will discuss multiple linear regression models that involve more

than one regressor.

11-2 SIMPLE LINEAR REGRESSION

The case of simple linear regressionconsiders a single regressoror predictorxand a de-

pendent or response variableY. Suppose that the true relationship between Yand xis a

straight line and that the observation Yat each level of xis a random variable. As noted previ-

ously, the expected value of Yfor each value of xis

where the intercept  0 and the slope  1 are unknown regression coefficients. We assume that

each observation, Y, can be described by the model

(11-2)

where is a random error with mean zero and (unknown) variance ^2. The random errors

corresponding to different observations are also assumed to be uncorrelated random

variables.

Y 0  1 x

E 1 Y 0 x 2  0  1 x

0 + 1 (1.25)

x = 100 x = 1.25

β β

β 0 + β 1 (1.00)

True regression line

(^) Yx =^0 +^1 x
= 75 + 15x
μ β β
y
(Oxygen
purity)
x (Hydrocarbon level)
Figure 11-2 The distribution of Yfor a given value of xfor the
oxygen purity-hydrocarbon data.
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