Page 176 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

OTE/SPH
OTE/SPH
2:57
August 31, 2006
JWBK119-11
Char Count= 0
Empirical Distribution Function based Approaches 161
For a sample size of n = 40, the critical value for the maximum absolute difference,
D, at the 5 % level of significance for the case where the population mean is known
but the standard deviation is unknown is approximately 0.206. Since the maximum
absolute difference for this sample is 0.081, we do not have sufficient evidence to reject
the null hypothesis that this data set comes from a normal distribution at the 5% level
of significance. It can also be observed from Figure 11.1 that the sample observations
all fall within the acceptance band.
In some GOF tests, neither the population mean, μ, nor the standard deviation, σ,
is specified and they must be estimated from the sample observations. In this case,
a modified Kolmogorov--Smirnov statistic, D * , can be used. The modifications to
the original D statistic and corresponding critical values for this case are given by
15
Stephens. These modifications are derived from points for finite n obtained through
Monte Carlo simulations instead of theoretical results.

11.4.2 Cram´ er--von Mises
The Cram´er--von Mises test statistic for GOF testing belongs to the class of
quadratic measure based EDF statistics, based on the measure of deviation shown in
equation (11.3). In this case the weighting function, ψ(x), is assumed unity. Hence,
from (11.3), the Cram´er--von Mises test statistic is given by:

∞

2
2
W = n [F n (x) − F 0 (x)] dF 0 (x).
−∞
In practical implementations, the following formula based on ordered observations
(x (i) ) can be used instead:

n 2
2 i − 0.5 1
W = F 0 (x (i) ) − + . (11.5)
n 12n
i=1
The residuals data from helicopter flight times shown in Table 11.1 is used to illus-
trate the procedure for conducting a the Cram´er--von Mises GOF test. This test is again
for a null hypothesis that the data comes from a normal distribution with population
mean known but standard deviation unknown.
2
The W statistic computed from equation (11.5) is less than the theoretical critical
2
values of the W statistic at the 5% significance level (the table of critical values for
2
the W statistics is given in Table 11.4(b)). 15 Hence, the specified hypothesis that the
sample data comes from a normal distribution cannot be rejected at the 5% level of
significance. Unlike the GOF test for residuals in this example where the population
mean is known, the population parameters are often unknown in practice and have to
be estimated from the sample data. For these cases, modifications have been proposed
to the original Cram´er--von Mises statistic. 9,15 These modifications are reproduced in
Table 11.5.