Introduction to Probability and Statistics for Engineers and Scientists

354 Chapter 9: Regression

Substituting this value ofAinto the second normal equation yields

∑

i

xiYi=(Y−Bx)nx+B

∑

i

x^2 i

or

B

(

∑

i

xi^2 −nx^2

)

=

∑

i

xiYi−nxY

or

B=

∑

i

xiYi−nxY

∑

i

xi^2 −nx^2

Hence, using Equation 9.2.2 and the fact thatnY =

∑n
i= 1 Yi, we have proven the
following proposition.

PROPOSITION 9.2.1 The least squares estimators ofβandαcorresponding to the data set
xi,Yi,i=1,...,nare, respectively,

B=

∑n

i= 1

xiYi−x

∑n

i= 1

Yi

∑n

i= 1

xi^2 −nx^2

A=Y−Bx

The straight lineA+Bxis called the estimated regression line.
Program 9.2 computes the least squares estimatorsAandB. It also gives the user the
option of computing some other statistics whose values will be needed in the following
sections.

EXAMPLE 9.2a The raw material used in the production of a certain synthetic fiber is stored
in a location without a humidity control. Measurements of the relative humidity in the
storage location and the moisture content of a sample of the raw material were taken over
15 days with the following data (in percentages) resulting.

Relative
humidity 46 53 29 61 36 39 47 49 52 38 55 32 57 54 44

Moisture

content 1215 7 171011111214 9 16 8 181412