Page 407 - Applied Statistics And Probability For Engineers

c10.qxd 5/16/02 1:31 PM Page 351 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

10-4 PAIRED t-TEST 351

3. H : 0
D
1
4. 0.05
5. The test statistic is
d
t
0
s D
1n
6. Reject H if t
t 0.025,8 2.306 or if t t 0.025,8 2.306.
0
0
0
7. Computations: The sample average and standard deviation of the differences d are
j
d 0.2736 and s 0.1356, so the test statistic is
D
d 0.2736
t 6.05
0
s D
1n 0.1356
19
8. Conclusions: Since t 6.05
2.306, we conclude that the strength prediction
0
methods yield different results. Specifically, the data indicate that the Karlsruhe
method produces, on the average, higher strength predictions than does the Lehigh
method. The P-value for t 6.05 is P 0.0002, so the test statistic is well into the
0
critical region.

Paired Versus Unpaired Comparisons
In performing a comparative experiment, the investigator can sometimes choose between the
paired experiment and the two-sample (or unpaired) experiment. If n measurements are to be
made on each population, the two-sample t-statistic is
X X 0
2
1
T
0
1 1
p
S B n n
which would be compared to t 2n 2 , and of course, the paired t-statistic is

D 0
T 0
S D
1n

. Notice that since
which is compared to t n 1
n D j n 1X X 2 n X 1j n X 2j
2j
1j
D a n a n a n a n X 1 X 2

j 1 j 1 j 1 j 1
the numerators of both statistics are identical. However, the denominator of the two-sample
t-test is based on the assumption that X 1 and X 2 are independent. In many paired experiments,
a strong positive correlation exists between X 1 and X 2 . Then it can be shown that
V 1D2 V1X 1 X 2 0 2
V1X 1 2 V1X 2 2 2 cov 1X 1 , X 2 2
2
2 11 2

n