Page 39 - Mechanical Engineers' Handbook (Volume 2)

28 Instrument Statics

2
where 2 1 and 2 2 are each estimates of the population variance . A better estimate of 2
2
is the combined variance , and it replaces both 2 1 and 2 2 in Eq. (55). The combined
c
variance is determined by weighting the individual estimates of variance based on their
degrees of freedom according to the relation
ˆ v ˆ v
2
2
2
11 22 (56)
c
v v 2
1
Then
ˆ 2 ˆ 2 1
1
2
c c 2 c (57)
d
n 1 n 2 n 1 n 2
Under the hypothesis H that (no effect due to treatment), the resulting probabilistic
1
0
2
statement is
P t 1 1 1
1
1
c
2
n n x x t c n n (58)
1 2 1 2
If the variances of the items being compared are not equal (homogeneous), a modified t- (or
d-) statistic is used, 9,15 where d depends on confidence level , degrees of freedom v, and a
parameter that depends on the ratio of standard deviations according to
ˆ / n 1
1
tan (59)
ˆ / n 2
2
The procedure for using the d-statistic is the same as described for the t-statistic.
Example 8 Testing for Homogeneous Means. A part manufacturer has the following
data:
Sample Number of Mean Lifetime Variance
Number Parts (h) (h)
1 15 2530 490
2 11 2850 360

Determine if the lifetime of the parts is due to chance at a 10% significance level (90%
confidence level).
Check variance first (H —homogeneous variances and H —nonhomogeneous vari-
0 1
ances):
ˆ 490
15
1 2 14 525
ˆ 360
11
2
2
10 396
(14)(525) (10)(396)
471
2
c
24