Report on Simple Random Sampling with or without replacement

One of my friend has a problem on simple random sampling. Can someone provide a complete Report on Simple Random Sampling with or without replacement?

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Abstract:

This project investigates selected sampling methods and their performance is tested under artificial population. Using a finite population size of 500 and a sample size of 30, estimators of first moment of the true population are compared and discussed. It is found that simple sampling method performed better in terms of precision in the analysis.

Introduction:

This project aims at comparing various sampling methods used in survey and experimental design. This project will proceed as follows. Two sampling methods are chosen to be compared in this project. Sampling procedures and descriptions of these two selected sampling methods are discussed in the next section of the project. The third section discussed the estimation theory on the population mean in the sample, with and without knowing the priori information in the sample. The third section applies the sampling method numerically. Point estimates on first and second moments of will be provided in this section. The forth section discusses the bias of the sampling methods based on the assumption that the true population mean is known beforehand.

Sampling Methods:

The sampling methods used in this project are (1) Simple Random Sampling Without Replacement and (2) Simple Random Sampling With Replacement.

One of the simplest probability sampling designs (plans) to select a sample of fixed size n with equal probability, i.e. 0 otherwise. One way to select such a sample is use Simple Random Sampling Without Replacement (SRSWOR): select the 1st element from U = {1, 2, • • • ,N} with probability 1/N; select the 2nd element from the remaining N −1 elements with probability 1/(N −1); and continue this until n elements are selected. Let {yi, i 2 s} be the sample data. It can be shown that under SRSWOR, otherwise. In practice, the scheme can be carried out using a table of random numbers or computer generated random numbers.

The simple random sampling with replacement can be described as below. First, the 1st element is selected from the sample with size N where the sample is labeled as {1,2,…,N} with equal likelihood, then we repeat this step n times to draw the required n samples from the sample. It should be noted that some elements in the population can be drawn more than once. Using the definition in the previous part, the mean estimates of the population using the sample can be computed as:

The strengths of both SRSWOR and SRSWR are (1) they are both simple sampling methods; (2) they are easy to be implemented for large sample size without huge computation power. However, compared to other sampling methods, these two methods suffer from the precision of the estimates. Comparing with other sampling methods, these two methods require additional sample size in order to give similar precision as other methods. Therefore, if the sample size is small in the sample, the precision of estimates will be affected.

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