Applied Statistics and Probability for Engineers

10-4 PAIRED t-TEST 351

3.

4.

5. The test statistic is

6. Reject H 0 if t 0

 t0.025, 82.306 or if t 0 t0.025, 82.306.

7. Computations: The sample average and standard deviation of the differences djare
   0.2736 and sD0.1356, so the test statistic is


8. Conclusions: Since t 0 6.05 2.306, we conclude that the strength prediction
   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 0 6.05 is P0.0002, so the test statistic is well into the
   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 nmeasurements are to be

made on each population, the two-sample t-statistic is

which would be compared to t 2 n 2 , and of course, the paired t-statistic is

which is compared to tn 1. Notice that since

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



2 ^211  2

n

V 1 X 12 V 1 X 22 2 cov 1 X 1 , X 22

V 1 D 2 V 1 X 1 X 2  02

D a

n

j 1

Dj

n a

n

j 1

1 X 1 jX 2 j 2

n ^ a

n

j 1

X 1 j

n a

n

j 1

X 2 j

n X^1 X^2

T 0 

D 0

SD 

1 n

T 0 

X 1 X 2  0

Sp

B

1

n^

1

n

t 0 

d

sD 

1 n



0.2736

0.1356 

19

6.05

d

t 0 

d

sD 

1 n

0.05

H 1 : D 0

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