This test provides for the demonstration of maintainability when the requirement is stated in terms of both a required mean value (µ1) and a design goal value (µ0) (or when the requirement is stated in terms of a required mean value (µ1) and a design goal value (µ0) is chosen by the contractor). The test plan is subdivided into two basic procedures, identified herein as Test Plan A and Test Plan B. Test A makes use of the lognormal assumption for determining the sample size, whereas Test B does not. Both tests are fixed sample tests, (minimum sample size of 30), which employ the Central Limit Theorem and the asymptotic normality of the sample mean for their development.
Test A -Maintenance times can be adequately described by a lognormal distribution. The variance, 2, of the logarithms of the maintenance times is known from prior information or reasonably precise estimates can be obtained.
Test B -No specific assumption concerning the distribution of maintenance times are necessary. The variance d2 of the maintenance times is known from prior information or reasonably precise estimates can be obtained.
|H0 : Mean = µ0
|H1 : Mean = µ1, ( µ1 > µ0 )
||H0 : µ0 =
||H1 : µ1 =
Note that µ0 is normally the specified maintainability index value, and that µ1 is typically the maximum acceptable value of the specified index.
SAMPLE SIZE - For a test with producer's risk and consumer's risk , the sample size for Test A is given by:
where 2 is a prior estimate of the variance of the maintenance times and Z a and Z are standardized normal deviates. The sample size for Test B is given by:
where 2 is a prior estimate of the variance of the maintenance times. Z and Z are standardized normal deviates.
Decision Procedure - Obtain a random sample of n maintenance times, X1, X2, . . . , Xn, and compute the sample mean,
and the sample variance
|Test A: Accept if
|Test B: Accept if
Discussion - By the central limit theorem, the sample mean is approximately normal for large n with mean E(X) and variance Var (). In Test A, under the log-normal assumption Var = d2 where d2 = e( 2 + )( e2 -1 ) = µ2 (
e2 -1 ) Thus the sample size, N, can be computed using a prior estimate of 2. In Test B, a a prior estimate of d2 is assumed to be available to calculate the sample size. A Critical value C is chosen such that µ0 + Z = C = µ1 - Z . If µ = µ0 , then P( > C) = and if µ = µ1, then P( C) = .
Example - It is desired to test the hypothesis that the mean corrective maintenance time is equal to30 minutes against the alternate hypothesis that the mean is 45 minutes with = = 0.05.
||H0 : µ0 = 30 minutes.|
||H1 : µ1 =
Test A: Under the log-normal assumption with prior estimate of 2 = 0.6, the sample size using equation B-4 is:
Test B: Under the distribution-free case with a prior estimate of 2 = 900, (or = 30), the sample size using equation B-5 is:
Operating Characteristic (OC) Curve -The OC curve for Test B for this example is given in Figure B-4. It gives the probability of acceptance for values of the mean maintenance time from 20 to 60 minutes. The OC curve for Test A for this example is given in Figure B-3. It gives the probability of acceptance for various values of the mean maintenance time. Thus, if the true value of m is 40 minutes, then the probability that a demonstration will end in acceptance is 0.21 as seen from Figure B-3.
Figure B-3 - OC Curve for Test A
Figure B-4 - OC Curve for Test B