قرآن مجید میں دعوی تضاد کا علمی محاسبہ

One of the main arguments that Allah has made in the Quran about the authenticity of this last book is that the Quran is free from all kinds of contradictions and differences. Whoever interprets the Quran, the authenticity of the Quran has become clearer on it. Different forms of language and literature are adopted in the Quran. If one is not familiar with the Quranic verses or does not have access to the truth of the words or is unfamiliar with the reality of the ayah, it may be possible to feel the contradiction in some places, when in reality it is not.

A Class of Generalized Estimators of Population Means Using Auxiliary Variables in Single and Two Phase Sampling With Properties of the Estimators

In this study generalized exponential ratio type estimators have been proposed for estimating the finite population mean, using the information from single and two auxiliary variables, under sampling design such as single and two phase sampling. For each proposed estimator bias of order two and mean square error of order one are derived using single and two phase equal probability sampling with single and two auxiliary variables. A class of these estimators have been developed and their bias of order one and mean square errors of order one have been derived. Mathematical comparison has been furnished of the generalized proposed estimators with the existing estimators. Shrinkage estimators of the proposed estimators have been defined along with their mean square errors for single and two phase sampling. To put side by side the proposed estimators with existing estimators, numerical study has been carried out with single and two phase equal probability sampling. It is experiential that the proposed estimators are more efficient than simple random sampling, Ratio, Bhal and Tuteja (1991), Samiuddin and Hanif (2006), Sisodia and Dwivedi (1981), Upadhyaya (1999), Singh and Tailor (2003) and Noor-ul-Amin and Hanif (2012) and others. In case of unequal probability sampling the proposed estimator is also more efficient than the simple random sampling, Ratio, Bhal and Tuteja (1991), Sisodia and Dwivedi (1981) and Singh and Tailor (2003). To study the properties of the estimators, simulation has been carried out and it is evident that the proposed estimators are consistent estimators of population mean. It also shows that normal distribution is the most significant fitted distribution on the proposed estimators.