8.3.2.6.2 ChiSquare
GoodnessofFit Test
The chisquare goodnessoffit test may be used to test the
validity of any assumed distribution, discrete or continuous. The test may be
summarized as follows for a continuous distribution.
(a) 

Determine the underlying distribution to be
tested.

(b) 

Determine a level of significance, a, which is defined as
the risk of rejecting the underlying distribution if it is, in fact,
the real distribution.

(c) 

Divide the continuous scale into k
intervals. For reliability analysis, this scale is usually
time.

(d) 

Determine the number of sample
observations falling within each interval.

(e) 

Using the assumed underlying
distribution, determine the expected number of observations in each
interval. Combining of intervals may be required because the expected
number of observations in an interval must be at least 5.0. This
determination may require an estimation of the distribution parameters
from the sample data (w is the number of estimated
parameters).

(f) 

Compute
where: O_{i} = number of sample observations
in the ith interval E_{i} = expected number of
observations in the ith
interval k = number of
intervals

(g) 

Let w be the number of parameters
estimated from the data and let c^{2} _{a,kw1} be the value found in
Table
8.311.

(h) 

Compare the calculated c
2 statistic with the tabled c 2
value for the discrete level of
the signature
reject the distribution under test. Otherwise, we do not
have sufficient evidence to reject the assumed underlying
distribution. 
1. When to Use
When failure times are available from a relatively large sample
and it is desired to determine the underlying distribution of failure
times.
2. Conditions for Use
(a) 

In the statistical analysis of failure data it is common
practice to assume that failure times follow a given failure
distribution family. This assumption can be based on historical data
or on engineering judgment. This test for goodnessoffit is used to
determine if the empirical data disproves the hypothesis of fit to the
assumed distribution.

(b) 

The c^{2} test for
goodnessoffit is “distributionfree” and can therefore be used
regardless of the failure distribution that the data are assumed to
follow.

(c) 

This test is not directly dependent on
sample size but on the number of intervals into which the scale of
failure times is divided with the restriction that no interval should
be so narrow that there are not at least 5 theoretical failures within
the interval. Therefore, the test is only useful if a relatively large
number of failures has been observed.

(d) 

A table of c^{2} percentage points is required (see Table
8.312). 
3. Method (Example
Using Exponential Distribution)
Consider the data in Figure 8.319 indicating the failure times
obtained from testing a sample of 100 fuel systems. Using a significance level
of a = 0.05, test
whether the assumption of an exponential distribution is reasonable. The
sample mean was found to be 8.9 hours.
(a) 

Figure 8.320 is used as a means of computing
"1"> 
(b) 

The expected frequency, Ei, is found
by multiplying the sample size by the probability of falling within
the i^{th}
interval if the assumed
(exponential) distribution is true.
"1">
Interval (Hours) 
Frequency 
0  5.05 5.05  10.05 10.05 
15.05 15.05  20.05 20.05  25.05 25.05 
30.05 30.05  35.05 35.05  40.05 40.05 
45.05 45.05  50.05 50.05  55.05 
48 22 11 7 3 5 2 0 1 0 1
100 
FIGURE 8.319: FUEL SYSTEM FAILURE
TIMES 


where U_{i}
and L_{i} are
the upper and lower limits of the i^{th} interval, U_{i} =
L_{i} + 5, and q
= 8.9
hours.

(c) 

Some of the original intervals were combined to satisfy the
requirement that no E_{i}
value be less than 2.5.
"1">
(See
Table 8.311)

we do not have sufficient
evidence to reject the exponential distribution as a model for these failure
times.