Applied Statistics and Probability for Engineers

374 CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION

of the data in Table 11-1. This is just a graph on which each (xi,yi) pair is represented as a point

plotted in a two-dimensional coordinate system. This scatter diagram was produced by

Minitab, and we selected an option that shows dot diagrams of the xand yvariables along the

top and right margins of the graph, respectively, making it easy to see the distributions of the

individual variables (box plots or histograms could also be selected). Inspection of this scatter

diagram indicates that, although no simple curve will pass exactly through all the points, there

is a strong indication that the points lie scattered randomly around a straight line. Therefore, it

is probably reasonable to assume that the mean of the random variable Yis related to xby the

following straight-line relationship:

where the slope and intercept of the line are called regression coefficients.While the mean

of Yis a linear function of x, the actual observed value ydoes not fall exactly on a straight

line. The appropriate way to generalize this to a probabilistic linear modelis to assume

that the expected value of Yis a linear function of x, but that for a fixed value of xthe actual

value of Yis determined by the mean value function (the linear model) plus a random error

term, say,

(11-1)

where is the random error term. We will call this model the simple linear regression model,

because it has only one independent variable or regressor.Sometimes a model like this will

arise from a theoretical relationship. At other times, we will have no theoretical knowledge of

the relationship between xand y, and the choice of the model is based on inspection of a scat-

ter diagram, such as we did with the oxygen purity data. We then think of the regression model

as an empirical model.

To gain more insight into this model, suppose that we can fix the value of xand observe

the value of the random variable Y. Now if xis fixed, the random component on the right-

hand side of the model in Equation 11-1 determines the properties of Y. Suppose that the mean

and variance of are 0 and ^2 , respectively. Then

Notice that this is the same relationship that we initially wrote down empirically from inspec-

tion of the scatter diagram in Fig. 11-1. The variance of Ygiven xis

Thus, the true regression model is a line of mean values; that is, the height

of the regression line at any value of xis just the expected value of Yfor that x. The slope,

can be interpreted as the change in the mean of Yfor a unit change in x. Furthermore, the vari-

ability of Yat a particular value of xis determined by the error variance ^2. This implies that

there is a distribution of Y-values at each xand that the variance of this distribution is the same

at each x.

For example, suppose that the true regression model relating oxygen purity to hydrocarbon

level is and suppose that the variance is ^2 2. Figure 11-2 illustrates this

situation. Notice that we have used a normal distribution to describe the random variation

in. Since Yis the sum of a constant  0  1 x(the mean) and a normally distributed ran-

dom variable, Yis a normally distributed random variable. The variance ^2 determines the

Y 0 x 75  15 x,

 1 ,

Y 0 x 0  1 x

V 1 Y 0 x 2 V 1  0  1 x 2 V 1  0  1 x 2 V 1  2  0 ^2 ^2

E 1 Y 0 x 2 E 1  0  1 x 2  0  1 xE 1  2  0  1 x

Y 0  1 x

E 1 Y 0 x^2 Y 0 x 0  1 x

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