Acceptance Sampling Plans from Life Tests Based on Percentiles of Half Normal Distribution

The design of acceptance sampling plans is developed under truncated life testing based on the percentiles of half normal distribution. The minimum sample size necessary to ensure the specified life 
percentile is obtained under a given consumer’s risk. The operating characteristic values of the sampling plans as well as the producer’s risk are presented. The results are illustrated by examples.


Introduction
Acceptance sampling is concerned with inspection and decision making regarding lots of products and constitutes one of the oldest techniques in quality control.If the lifetime of the product represents the quality characteristics of interest, the acceptance sampling is as follows: a company receives a shipment of product from a vendor.This product is often a component or raw material used in the company's manufacturing process.A sample is taken from the lot and the relevant quality characteristic of the units in the sample is inspected.On the basis of the information in the sample, a decision is made regarding lot disposition.Traditionally, when the life test indicates that the mean life of products exceeds the specified one, the lot of products is accepted, otherwise it is rejected.Accepted lots are put into production, while rejected lots may be returned to the vendor or may be subjected to some other lot disposition actions.For the purpose of reducing the test time and cost, a truncated life test may be conducted to determine the sample size to ensure a certain mean life of products when the life test is terminated at a time  0 , and the number of failures does not exceed a given acceptance number .
A common practice in life testing is to terminate the life test by a predetermined time  0 and note the number of failures.One of the objectives of these experiments is to set a lower confidence limit on the mean life.It is then to establish a specified mean life with a given probability of at least  * which provides protection to consumers.The test may be terminated before the time is reached or when the number of failures exceeds the acceptance number  in which case the decision is to reject the lot.
Normal distribution is the most preferred distribution in statistical studies.But for life test models it is not suitable distribution because of its domain [−∞, +∞].Half normal distribution is an increasing failure rate (IFR) model which is most useful in reliability studies.Because of this IFR nature with domain [0, +∞] we are motivated to study this distribution.

Acceptance Sampling Plans
The probability density function (pdf) of a half normal distribution is given by Its cumulative distribution function (cdf) is where erf is an error function and  is a parameter.Assume that the life time of a product follows half normal distribution with  as scale parameter.Its cumulative distribution function (⋅) is given by Given 0 <  < 1, the 100th percentile is given by Substituting  in (3) in the scaled form we get where  0 = /  .Here erf −1 is simulated by A common practice in life testing is to terminate the life test by a predetermined time , to require the probability of rejecting a bad lot to be at least  * , and to have the maximum permissible number of bad items to accept the lot equal to .The acceptance sampling plan for percentiles under a truncated life test is to set up the minimum sample size  for a given acceptance number  such that the consumer's risk, the probability of accepting a bad lot, does not exceed 1 −  * .A bad lot means that the true 100th percentile,   , is below a specified percentile,  0  .Thus the probability  * is a confidence level in the sense that the chance of rejecting a bad lot with   <  0  is at least equal to  * .Therefore, for given  * , the proposed acceptance sampling plan can be characterized by the triplet (, ,  0 ) = (, , / 0  ), where  = / 0  .We consider large sized lots so that the binomial distribution can be applied.The problem is to determine for given values of  * ,  0  , and  the smallest positive integer  required to assert that   >  0  must satisfy the relation where  0 =   ( 0 ) is the probability of a failure during the time  =  0  for the specified 100qth percentile of lifetime  0  .The value  0 depends only on  0 = / 0  .Since   ()/ > 0, it is an increasing function of .Accordingly, we have or equivalently The smallest sample size  satisfying (9) can be obtained for any given , / 0  , or  * .In contrast, only the input values /,  * are needed to calculate the smallest sample size .In particular, if we take  = 0.50, the value of  would become "the smallest sample size required to test that the population median life exceeds a given specified value." Half normal distribution is a skewed distribution; for our present skewed population the median is a more appropriate average for decision making about the quality of the life than population mean.Thus we may conclude that population median based sampling plans of half normal distribution model are more economical than those based on population mean with respect to sample size.To save space, only the results of small sample sizes for  = 0.50;  * = 0.75, 0.90, 0.95, 0.99;  = 0(1)10;  = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0 are displayed in Table 1.

Operating Characteristic of the Sampling Plan and Producer's Risk
The operating characteristic (OC) function  of the sampling plan (, , / 0  ) is the probability of accepting a lot as a function of  =   () with  0 = /  .It is given as Therefore, we have where   =   / 0  .Using (10) the OC values can be obtained for any sampling plan (, , / 0 0.5 ) (Table 5).To save space we present the OC values for sampling plans with  = 0.50;  * = 0.75, 0.90, 0.95, 0.99;  = 0.1,0.2,0.4,0.6,0.8,1.0,1.5,2.0,2.5,3.0; 0.5 / 0 0.5 = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0; for  = 1 and  = 5 which are given in Table 2 and Table 3, respectively.The producer's risk is defined as the probability of rejecting the lot when   >  0  .For a given value of the producer's risk, say , we are interested in knowing the value of   to ensure that the producer's risk is less than or equal to  if a sampling plan (, , /  ) is developed at a specified confidence level  * .Thus, one needs to find the smallest value   according to (10) as where To save space, based on the sampling plans (, , / 0  ) given in Table 1 the minimum ratios of  0.5 for the acceptability of a lot at the producer's risk of  = 0.05 are presented in Table 4.

Illustrative Examples
In this section, we consider two examples with real data sets to illustrate the proposed acceptance sampling plans.

Example 1.
The first data refers to software reliability presented by Wood [20].The data set was reported in hours as 519, 968, 1430, 1893, 2490, 3058, 3625, 4422, and 5218.The confidence level is assumed for this acceptance sampling plan only if the lifetimes are from half normal distribution with shape parameter .So in order to apply this example for our tables we have to confirm the goodness of fit of half normal distribution to the data in the example.This is done through the well-known QQ-plot method, and the value we get is  = 0.9287.Hence we conclude that the half normal distribution provides a reasonable goodness of fit for the data.
Suppose that the experimenter would like to establish the unknown 50th percentile life time for the software mentioned above to be at least 300 h and the life test would be ended at 600 h which should have led to the ratio / 0 0.5 = 2.0.Thus, with  = 1 and  * = 0.75, the experimenter may take from Table 1 the sample size  which must be at least 3; thus the sampling plan to the data is given by Since there is one item with a failure time less than or equal to 600 h in the given sample of 9 observations, the lot is accepted as the result indicates that the 50th percentile life time  0.5 is at least 300 h with a confidence level of  * = 0.75.
The OC values for acceptance sampling plan in ( 14) and confidence  * = 0.75, for a half normal distribution, may be taken from Table 2.
The producer's risk is almost equal to 0.3683 (= 1 − 0.6317), when the true percentile is greater than or equal to 2.0 times the specified 50th percentile.From Table 4, the experimenter could get the values of  0.5 for different choices of  and / 0 0.5 in order to assert that the producer's risk is less than 0.05.In this example, the value of  0.5 should be 7.9120 for  = 1,  0.5 / 0 0.5 = 2.0, and  * = 0.75.This means that the product can have a life of 7.9120 times the required 50th percentile life time in order that under the above acceptance sampling plan the product is accepted with probability at least 0.75.

Summary and Conclusions
This paper provides the minimum sample size required to decide upon accepting/rejecting a lot based on its specified 50th percentile (median) when the data follows half normal distribution.Assuming that the size of the given sample is minimum, we can use the tables in this paper to pick up acceptance number  and the upper bound in order to accept/reject the lot for a given probability of acceptance, say  * .This paper also provides the sensitivity of the sampling plans in terms of the OC.We advise the industrial practitioner and the experimenter to adopt this plan in order to save the cost and time of the experiment.This plan can be further studied for many other distributions and various quality and reliability characteristics as future research.

Table 1 :
Minimum sample sizes necessary to assert the median life of a product.

Table 2 :
OC values of sampling plans.

Table 3 :
OC values of sampling plans.

Table 4 :
Minimum ratio for the acceptability of a lot with producer's risk 0.05.