7.3 Simple Linear Regression and Least Squares Estimation 103- Standard for Y
^
given X = x :SE Y()ˆˆˆ=+−∑−Syx. n−^12 (x X) (Xi X)^2
.A 100(1 − α )% confi dence interval for predicted value of Y givenX = x is then [ Y ^ − t n − 2 ( α ) SE ( Y ^ ), Y ^ + t n (^) − 2 ( α ) SE ( Y ^ )], where t n (^) − 2 ( α ) is
the 100(1 − α /2) percentile of a t - distribution with n − 2 degrees of
freedom.
A confi dence interval for a prediction of Y is sometimes called a
prediction interval. Now let ’ s go through the steps above to get a pre-
diction interval for Y given X = 110 for the example in Table 7.2.
SSE = 49.56 (see table). Then Syx. ==(. /) .49 56 8 2 73. So,
(^) SE Y() .(^ =+−2 73 10−^12 (110 154 1.)/( .) ..108 9=0 56
Hence, a 95% prediction interval for Y given X = 110 is
[. .(.),. .(.)][.,.].6202 2306 056 6202 2306 056−+=6073 6331
To test the hypothesis H 0 β = 0, where β is the slope parameter of
the regression equation, we use the test statistic t = ( b − β )/ SE ( b ) = b / SE ( b ),
since the hypothesized value for β = 0. Now SE b()=∑−Syx. ⎣⎡ (Xi Xˆ)^2 ⎤⎦.
For our example, SE b(). /==2 73 9108 9. 0 0286.. So, t = 0.77. We
refer to a t - distribution with 8 degrees of freedom to determine the p -
value, and we cannot conclude that β is signifi cantly different from 0.
Since ρ is a simple multiple of β , we also cannot conclude that ρ is
signifi cantly different from 0. In this case, the p - value is greater than
0.2, since it is two - sided, and the 90th percentile of a t - distribution with
8 degrees of freedom is 1.397 and t = 0.77 << 1.397.
One important point about simple linear regression is the term
linear. Linear here means that the relationship is linear in the parameters
and not necessarily in the independent variable. So although we often
think of the linear regression equation as Y = β X + α , this is both linear
in the independent variable X , as well as the parameters α and β.
However, the equations Y = β X 2 +^ α , or Y = β ln( X ) + α , also fi t the
simple linear regression model, although they involve nonlinear func-
tions of the independent variable X.
The term nonlinear regression is defi ned as a form of regression
where the equation for Y is nonlinear in the parameters that we wish to
estimate. So, for example,