67Capability Indices

As an alternative approach, a normal approximation could be used with

this expression for the standard error. The critical value would be similar to

the case of the single mean, namely:

Reject H

0

if:

−β

CzSE K

pk

where z

1 − β

= 100*(1 − β) percentile of a standard normal distribution.

Discussion:

As in the case of all the hypothesis tests described in this work, we assume

that the “null” value stated for C

pk

is acceptable, so that we desire a high

probability of rejecting the null hypothesis if the true C

pk

is equal to the null

value. Thus, the critical value is the lower 100βth percentile of the appropri-

ate sampling distribution.

It should be noted that the estimates

and

pU

are biased, in that their

expected values exceed the value of the parameter. The bias is a multiplier

described by:

=

−

Γ

−

Γ

−

bias

n

n

n

1

2

2

2

1

2

where Γ(.) is the gamma function. The bias is relatively small for values of

n≥100. At n = 100, the bias is approximately 1.008, or 0.8 percent too high.

To compensate for bias, multiply the critical value by the bias formula.

Alternatively, divide the usual estimator by the bias, and then use the “unad-

justed” critical value.

The power to reject the null is given by:

C

Tn nK

n

nKPr

ˆ

1, 3

3

|3

pL

0

≥

′

−δ=

δ=

β

and

Tn nK1, 3

0

′

−δ=

β

is the lower 100βth percentile of a noncentral t-distribution with n – 1 degrees

of freedom and noncentrality parameter

. K

a

is the alternate value

of the population C

pL

.