Page 41 - Mechanical Engineers' Handbook (Volume 2)

30 Instrument Statics

(O E ) 2
j
j
2 exp (62)
E j
When using Eq. (62) at least five items per class interval are normally used, and a continuity
correction must be made if less than three class intervals are used.

Example 9 Use of the Chi-Square Distribution. In a test laboratory the following record
shows the number of failures of a certain type of manufactured part.

Number of Parts
Number of Failures Observed Expected a
0 364 325
1 376 396
2 218 243
3 89 97
4 33 30
5 13 7
6 4 1
7 3 0
8 2 0
9 1 0
1103
a From an assumed statistical model that the failures are random.
At a 95% confidence level, determine whether the failures are attributable to chance
alone or to some real cause. For H , assume failures are random and not related to a cause
0
so that the expected distribution of failures follows the Poisson distribution. For H , assume
1

r
failures are not random and are cause related. The Poisson probability is P e /r !,
r
where x P : i
i
(0) (1) (2) (3) (4) (5)
13
364
376
218
89
33
1103 1103 1103 1103 1103 1103
24 21 16 9

1103 1103 1103 1103
1346
1.22
1103
0.295 and E P n 325. Similarly P 0.360, P 0.220, P 0.088,...
Then P 0 0 0 1 2 3
and, correspondingly, E 396, E 243, E 97,...are tabulated above for the expected
3
1
2
number of parts to have the number of failures listed.
Goodness-of-fit test—H represents random failures and H represents real cause:
1
0
P[ 2 2 ] 0.05
exp Table
2 ( , ) (5,0.05) 11.070
2
2
Table
(0 E) 2
2 exp
E