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 (SRSWOR) simple random sampling without replacement. Besides, simple random sampling without replacement is applied in drawing random samples from a finite population N. In all the cases categorically, the sample sizes/ size will always be smaller than the population size unless when carrying out census i.e. involving every unit in the sampling survey. in sampling it's a rule that more than 5% of the total population N should adhere to meaning so that the proportion n/N>0.05. Additionally, at the process of calculating the standard error of the mean and also the standard error of the proportion, the finite correction factor FPC is applied in order to scale up the accuracy. FPC constant is calculated as shown from the formula below:
FPC=√((N-n)/(N-1))
Where N is the population size and n is the sample size.
The standard deviation of the mean for finite populations is given by σ y= σ/√n*√((N-n)/(N-1))
Similarly, to get the FPC of the proportion is calculated using this formula: σp=√(p(1-p)/n*(N-n)/(N-1))
In the examination of the finite correction factor, the denominator is always smaller than the numerator resulting to the finite correction factor becoming less than unit in all the practical cases. As a result of multiplication with the finite correction factor of the finite population by the standard deviation, therefore, the standard error eventually becomes smaller gradually when corrected. For this reason, therefore, accurate estimates are realized when the correction factor is applied during calculation. Below is an illustration of the application of the finite population correction factor.
Example
A sample with 30 cereal selected from a filling process where 4000 boxes are filled daily. Applying the finite correction factor formula calculate the probability of obtaining a sample given the mean is 400grams and a standard deviation of 15.
Solution
Standard deviation(σ)=15, n=30, N=4000 so;
σ y= σ/√n*√((N-n)/(N-1))
σ y= 15/√30*√((4000-30)/(4000-1))
=2.7*√ (3970/3999)
=2.7
From the statistical table the probability obtaining the mean between 400 grams and 405 is 0.167.
Simulation of simple random sampling using R to investigate central limit theorem
This project is oriented to fulfill the following objectives towards the conclusion of the simulation process using R software.
* Choosing an arbitrary sample by simple random sampling from a finite population to investigate central limit theorem.
* Finding the percentile quantile using R’s qnorm command under the probability density function.
Choosing a finite population of size 100 using simple Random sampling with R to compare the mean of the sample(s) chosen with the mean of the pop...