Biostatistics

438 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION
A Confidence Interval for b 1 Once we determine that it is unlikely, in light of
sample evidence, that b 1 is zero, we may be interested in obtaining an interval estimate
of b 1 . The general formula for a confidence interval,
estimator ðreliability factorÞðstandard error of the estimateÞ
may be used. When obtaining a confidence interval for b 1 , the estimator is ^b 1 , the
reliability factor is some value of z or t (depending on whether or not s 2 yx j
is known), and
the standard error of the estimator is
s ^b 1
¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
xi
s 2 yjx
ð x Þ 2
When s 2 yjx is unknown, s b is estimated by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s^b 1
where s 2 yjx ¼ MSE
In most practical situations our 100ð1
¼
P
xi
s 2 yjx
ð x Þ 2
aÞ percent confidence interval for b is
^b 1 t ð1 a=2Þ
s^b 1
(9.4.10)
For our illustrative example we construct the following 95 percent confidence
interval for b:
3:4589 1:9826 ð:2347Þ
ð2:99; 3:92Þ
We interpret this interval in the usual manner. From the probabilistic point of view we say
that in repeated sampling 95 percent of the intervals constructed in this way will include b 1 .
The practical interpretation is that we are 95 percent confident that the single interval
constructed includes b 1 .
Using the Confidence Interval to Test H 0 : b 1 ¼ 0 It is instructive to
note that the confidence interval we constructed does not include zero, so that zero is not a
candidate for the parameter being estimated. We feel, then, that it is unlikely that b 1 ¼ 0.
This is compatible with the results of our hypothesis test in which we rejected the null
hypothesis that b 1 ¼ 0. Actually, we can always test H 0 : b 1 ¼ 0 at the a significance level
by constructing the 100ð1
aÞpercent confidence interval for b 1 , and we can reject or fail
to reject the hypothesis on the basis of whether or not the interval includes zero. If the
interval contains zero, the null hypothesis is not rejected; and if zero is not contained in the
interval, we reject the null hypothesis.
Interpreting the Results It must be emphasized that failure to reject the null
hypothesis that b 1 ¼ 0 does not mean that X and Y are not related. Not only is it possible
that a type II error may have been committed but it may be true that X and Y are related in
some nonlinear manner. On the other hand, when we reject the null hypothesis that b 1 ¼ 0,
we cannot conclude that the true relationship between X and Y is linear. Again, it may be