An investigation on central limit theorems under finite population sampling

INVESTIGATION OF CENTRAL LIMIT THEOREM UNDER FINITE POPULATION USING SIMPLE RANDOM SAMPLING
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Abstract
This statistical project shall apply R software to demonstrate the central limit theorem of a finite population. A significantly large sample will be generated from an arbitrary generated population from R using simple random sampling technique and investigations in relation to central theorem be conducted. The procedure shall be replicated in multiple ways to observe the changes of the sample mean as sample numbers increases in relation to the population mean. A precise explanation on the understanding of central limit theorem shall be brought into light in relation to simple random sampling.
Introduction
Central limit theorem mathematics and statistics application has been described to produce most remarkable results worldwide (Kothari 2004). The theory, in the world of mathematics and statistics, has been identified to be one of the oldest as compared to others (Hudson 2002). Additionally, the theory is not only useful but also a critical pillar in the probability theory (Beth and Robert 2005). The theorem also plays a unique role in statistical inference. Assuming that the variance of the population σ2 and the mean of the population µ when a sample is randomly drawn from the population then it is observed that the mean of the same sample as compared to that of the population follows a normal distribution.
With a sufficiently large sample size (n>=30) the theorem holds true if the population is normal even for a sample smaller than the stipulated less than thirty. Still, under binomial distribution, the property holds true given that minimum (np,n(1-p))>=5 with the size of the sample as n and p as the success probability. This signifies that the model of probability normal distribution is applied to ascertain the uncertainty in statistical inferencing of the mean of the population given the sample mean. Random samples are drawn with an appropriate unbiased method and computing the mean of the sample means given by the formula:
µӯ=µand sample mean standard deviation given as:
σӯ= σ/√n
Statement of the central limit theorem
Suppose we have a sequence of independent random variables; y1, y2, y3…yneach of the same distribution having a finite population mean µ and a variance of σ2
Then if ӯnis the mean of y1, y2, y3…ynthen the distribution standardized variable Zn=(yn-ӯ)/(σ/√n) that converges to the normal (0,1) as n approaches infinity. The idea of this theorem is that regardless of the population, the mean of a sufficiently large sample will assume almost normal distribution. The sampling theoretically of a binomial distribution has a mean equal to p and a standard deviation equal to √npq. These powerful results and facts may be applied to explain why in many psychological, engineering and physical processes follow the characteristic of near bell-shaped or a perfectly normal distribution.
Amid central limit theorem investigation, all the standard errors of the sample proportions are based on and those of the samples are selected by the application of (...