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350 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

2
and variance D , so testing hypotheses about the difference between and can be
2
1
accomplished by performing a one-sample t-test on . Specifically, testing H :
0
2
D
1
against H : is equivalent to testing
0
0
1
1
2
H : 0
0
D
: (10-21)
H 1 D 0
The test statistic is given below.
The Paired
t-Test Null hypothesis: H 0 : D 0
D 0
Test statistic: T (10-22)
0
S D
1n
Alternative Hypothesis Rejection Region
H : 0 t
t
2, n 1 or t t
2, n 1
D
1
0
0
H :
0 t
t , n 1
0
D
1
H : 0 t t , n 1
1
D
0
In Equation 10-22, D is the sample average of the n differences D , D , p , D , and S is the
2
D
1
n
sample standard deviation of these differences.
EXAMPLE 10-9 An article in the Journal of Strain Analysis (1983, Vol. 18, No. 2) compares several methods
for predicting the shear strength for steel plate girders. Data for two of these methods, the
Karlsruhe and Lehigh procedures, when applied to nine specific girders, are shown in Table
10-2. We wish to determine whether there is any difference (on the average) between the two
methods.
The eight-step procedure is applied as follows:
1. The parameter of interest is the difference in mean shear strength between the two
methods, say, 0.
1
D
2
2. H : 0
0
D
Table 10-2 Strength Predictions for Nine Steel Plate Girders
(Predicted Load/Observed Load)
Girder Karlsruhe Method Lehigh Method Difference d j
S1 1 1.186 1.061 0.119
S2 1 1.151 0.992 0.159
S3 1 1.322 1.063 0.259
S4 1 1.339 1.062 0.277
S5 1 1.200 1.065 0.138
S2 1 1.402 1.178 0.224
S2 2 1.365 1.037 0.328
S2 3 1.537 1.086 0.451
S2 4 1.559 1.052 0.507