Biostatistics

240 CHAPTER 7 HYPOTHESIS TESTING
We now compute
126:1 133:1
t ¼ ð rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Þ 0
756:04
þ 756:04
¼ :569
10 10
8. Statistical decision. We fail to reject H 0 , since 1:7341 < :569; that
is, :569 falls in the nonrejection region.
9. Conclusion. On the basis of these data, we cannot conclude that the
population mean pressure is less for healthy subjects than for SCI
subjects.
10. p value. For this test, p >:10 using Table E, or .5764 using a computer
since 1:330 < :569. &
Population Variances Unequal When two independent simple random
samples have been drawn from normally distributed populations with unknown and
unequal variances, the test statistic for testing H 0 : m 1 ¼ m 2 is
t 0 ¼ 1 x 2 ð
s m ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m Þ 2 0
(7.3.3)
s 2 1
þ s2 2
n 1 n 2
The critical value of t 0 for an a level of significance and a two-sided test is approximately
t1 0 ða=2Þ ¼ w 1t 1 þ w 2 t 2
(7.3.4)
w 1 þ w 2
where w 1 ¼ s 2 1 =n 1; w 2 ¼ s 2 2 =n 2; t 1 ¼ t 1 ða=2Þ
for n 1 1 degrees of freedom, and t 2 ¼
t 1 ða=2Þ
for n 2 1 degrees of freedom. The critical value of t 0 for a one-sided test is
found by computing t 0 1 a by Equation 7.3.4, using t 1 ¼ t 1 a for n 1 1 degrees of freedom
and t 2 ¼ t 1 a for n 2 1 degrees of freedom.
For a two-sided test, reject H 0 if the computed value of t 0 is either greater than or
equal to the critical value given by Equation 7.3.4 or less than or equal to the negative of
that value.
For a one-sided test with the rejection region in the right tail of the sampling distribution,
reject H 0 if the computed t 0 is equal to or greater than the critical t 0 . For a one-sided test with a
left-tail rejection region, reject H 0 if the computed value of t 0 is equal to or smaller than the
negative of the critical t 0 computed by the indicated adaptation of Equation 7.3.4.
EXAMPLE 7.3.3
Dernellis and Panaretou (A-7) examined subjects with hypertension and healthy control
subjects. One of the variables of interest was the aortic stiffness index. Measures of this
variable were calculated from the aortic diameter evaluated by M-mode echocardiography
and blood pressure measured by a sphygmomanometer. Generally, physicians wish to