Exploring the Role of Female Successor “Amrah Bint Abd Al-Raḥmān” in Narration of Prophetic Traditions

Although several studies have investigated the contribution of Muslim women about their lives and works in large corpus of the ḥadīth sciences until now, but much are needed to be explored. Doubtlessly 'Amrah bint 'Abd al-Raḥmān had contributed to the large corpus of the ḥadīth sciences and also the core of Islamic knowledge parallel to her male counterparts. By highlighting her efforts for learning and her struggle for transmission of fundamental text, reveals how a female successor acquired exemplary reputations in her era. Particularly, her rich insight, deep knowledge in ḥadīth sciences and exemplary piety attracted students and placed parallel to her male counterparts in Islamic knowledge of her time. In sum, this study not merely sheds light upon the prolific contribution of an eminent female successor 'Amrah bint 'Abd al-Raḥmān in ḥadīth narrations, but argues that these scholarly females should be given more credit for being interacting educators and not just transmitters. Doubtlewssly, the study would be helpful to revise our understandings of Islamic knowledge during early period of Islam regarding gendered dynamics and their great contribution to dissemination and transmission of classical text and Islamic knowledge.

a Study on Skip-R Sampling Plan Using Aql, Lql, Cpk and Some Life Time Distributions

Acceptance sampling plans have been widely used in industries to make sure the high quality of incoming and outgoing products. These sampling plans are very useful to the manufacturer and the buyer for the acceptance of good products and rejection of bad products. Among many acceptance sampling schemes, the skip lot sampling scheme is used for reducing the amount of inspection of a product that has an exceptional quality history. Therefore the objective of this study is to propose a skip-lot sampling plan which provides the minimum average sample number as compared to existing sampling plans. To achieve this goal, three approaches have been used. In the first approach we present a designing methodology for batch inspection from a continuous process called the skip-lot sampling plan of type R (SkSP-R). The concept of resampling has been used in our proposed plan. We formulate the optimization problem of minimizing the average sample number to determine the plan parameters of the proposed SkSP-R sampling plan. The constraints are to satisfy the producer‘s risk at the acceptable quality level and the consumer‘s risk at the limiting quality level. The tables are presented and the results are explained with the help of an example. To check the statistical difference between the proposed plan and the existing plans, the Friedman test is applied. The efficiency of the SkSP-R sampling plan is also shown in terms of probability of acceptance and average sample number. The proposed plan is found more efficient than the single sampling plan, SkSP-2 sampling plan and SkSP-V sampling plan in terms of the average sample number. In the second approach we present an optimal designing methodology of the proposed plan based on the most widely used process capability index Cpk . For designing the optimal parameters, we consider both symmetric and asymmetric fraction non-conforming cases. Tables are constructed for each case. Advantages of the proposed plan over the existing plan are also discussed. Application of the plan is explained with the help of a real life example. Thirdly we present the designing of the proposed plan for time truncated life test. We consider the three most important life time distributions such as Weibull distribution, Exponentiated Weibull (EW) distribution and Birnbaum- Saunders (BS) distribution. The plan parameters of the SkSP-R plan under these three distributions are determined through a nonlinear optimization problem. Tables are constructed for these distributions using different shape parameters. The efficiency of the proposed plan over the existing plans is also discussed. A comparison is also made between three distributions in terms of ASNs and it has been observed that Birnbaum-Saunders distribution provides minimum ASN as compared to the other two distributions.