Introduction to Probability and Statistics for Engineers and Scientists

9.4Statistical Inferences About the Regression Parameters 365

To compute the value of the test statistic, we first compute the values ofSxx,SYY, andSxY.
A hand calculation yields that

Sxx=700, SYY=21.757, SxY=− 119

Using Equation 9.3.4 gives

SSR=[SxxSYY−SxY^2 ]/Sxx

=[700(21.757)−(119)^2 ]/700=1.527

Because

B=SxY/Sxx=−119/700=−.17

the value of the test statistic is

TS=

√

5(700)/1.527(.17)=8.139

Since, from Table A2 of the Appendix,t.005,5=4.032, it follows that the hypothesis
β=0 is rejected at the 1 percent level of significance. Thus, the claim that the mileage
does not depend on the speed at which the car is driven is rejected; there is strong evidence
that increased speeds lead to decreased mileages. ■

A confidence interval estimator forβis easily obtained from Equation 9.4.2. Indeed,
it follows from Equation 9.4.2 that for anya,0<a<1,

P







−ta/2,n− 2 <

√

(n−2)Sxx

SSR

(B−β)<ta/2,n− 2







= 1 −a

or, equivalently,

P

{

B−

√

SSR

(n−2)Sxx

ta/2,n− 2 <β<B+

√

SSR

(n−2)Sxx

ta/2,n− 2

}

= 1 −a

which yields the following.

Confidence Interval forβ

A 100(1−a) percent confidence interval estimator ofβis

(

B−

√

SSR

(n−2)Sxx

ta/2,n− 2 ,B+

√

SSR

(n−2)Sxx

ta/2,n− 2

)