Single acceptance sampling plans based on truncated lifetime tests for two-parameter Xgamma distribution with real data application

: Recently, the two-parameter Xgamma distribution (TPXGD) is suggested as a new lifetime distribution for modeling some real data. The TPXGD is investigated in different areas and generalized to other forms by many of the researchers. The acceptance sampling plans are one of the main important statistical tools in production and engineering fields. In this paper, modified acceptance sampling plans for the TPXGD are proposed with the assumption that the lifetime is truncated at a predetermined level. The mean of the TPXGD model is utilized as a quality parameter. The variables of the acceptance sampling plans including the acceptance numbers, the minimum sample sizes, operating characteristic function and the producer’s risk are investigated for various values of the model parameters. Numerical examples are offered to illustrate the process of the proposed plans. Also, a real data is fitted to the TPXGD and an application based on the suggested acceptance sampling plans is considered for explanation.


Introduction
The single acceptance sampling plans (SASP) are very important in the production sector to assert the acceptability of a lot based on its lifetime. The manufacturers are interested in producing a good

The TPXGD
This section describes the TPXGD which is proposed by [24] as a modification to the well-known Xgamma distribution by adding a new parameter to the base distribution. The distribution function of the TPXGD is given by 2 with corresponding probability density function 2 2 ( ) Plots of the distribution pdf are presented in Figure 1 which shows that the model is positively skewed.
and 2 Figure 2 consists of the hazard and survival functions of the TPXGD for some parameters.
The rth moment the distribution is given by

The suggested ASP
This section describes the single acceptance sampling plans suggested in the current study proposing that the lifetime tests follows the TPXGD. Here, the minimum sample size (MSS), operating characteristic function (OC), and the producer's risk (PR) are introduced. The method for implementing the SASP to get at a decision about the product can be explained as follows: 1) Draw a sample of size m randomly from the lot collected from the supplier or the final production.
2) The sample size m that is drawn from the lot to be tested and distinguish it to good or defective.
3) The test duration time, t. 4) An acceptance number of defective items, c such that if c or less failures occurred within the test time t, the lot is not rejected. 5) The minimum ratio To explain the process, let the consumer's risk is preassigned to be at most * 1 P  . That is, the probability of the actual average lifetime of the quality parameter  is not larger than where     (2) is the probability that the lifetime does not more than t for the true mean 0 .  The above equation is based on the assumption that the size of the lot is large as possible to use the binomial distribution theory. When the number of failures up to the time t is found to be c or less, then based on (7) is known as the incomplete beta function.
For the offered ASP with a given value of the PR, say  , it is essential to find the value of the ratio 0 / ,   that keeps the PR at most  . Since satisfies the For the suggested plan    Table 3.

Applications of real data
In this section, a real data set is analyzed to demonstrate the practicality of the suggested ASP in practical situations. The data is given by [25] in which 20 items are tested till failure are discussed.  Table 4. It is clear that the data is nonsymmetrical distributed where its positively skewed.

Conclusions
In this paper, new single ASP under truncated lifetime tests for the two parameters Xgamma distribution are offered. For different options of the distribution parameters, sample plans have been built. The essential tables of the minimum sample size required to affirm an assured mean lifetime of the test units are presented. The OC function values as well as the corresponding producer's risks are obtained for various plan parameters. A real data is fitted to the TPXGD distribution and an application to this data is discussed to illustrate the usefulness of the proposed ASP. It is indicated that the results encourage the researchers to use the advised ASP when the life duration of components follow the the TPXGD. The current study can be extended using neutrosophic statistics as future research.