A Comparison among SRSWOR, Stratified SR-SWOR, Single Stage Cluster Sampling, and Two-Stage Sampling - Statistics Project Example

A comparison among some basic sampling designs such as SRSWOR, Stratified SR-SWOR, single stage cluster sampling, two-stage sampling, with focus on point and variance estimation and confidence Abstract This paper determined how sample performances can be calculated using the increasing and decreasing functions of p (population proportion) when factors such as finite population are ignored. The project in this paper evaluated clear sample performances of proportion and point estimator and how they relates to normal theory variance and confidence intervals on the mean and proportion of a population by using Simple Random Sample Without Replacement. The tool that was used in this project was the R Program for Simulation. It was found that Stratified SR-SWOR is a better tool in determining a suitable population proportion from two variances. Section 2: Introduction It is important to understand the sampling plan of a given population when deciding on the best sampling technique to use in data analysis. The key sampling plans listed above (cluster and stratified sampling methods) have their advantages and disadvantages in their applications, in addition to their definite mode of operations. Firstly, cluster samples are vital in the construction of a frame for the observation samples that are impossible, expensive, or difficult, for example, when determining the numbers or trends of birds in a location or clients of a store. Other parameters that may be analyzed using clusters are cases where a population is widely distributed in terms of geographical setting. Clusters compares with stratification in that (1) partitioning of the population into clusters or strata-subgroups is applied in both, (2) stratification requires that each group should be sampled, and (3) cluster employs its sampling procedure in a single subset for all the units. The two methods also differ in their precision; for a population sample size-n, SRS are intermediate, cluster technique produces approximations with large variances, and stratification gives smallest variances. SRS (Simple Random Sampling) Simple random sampling refers to a random sampling technique where every population characteristic has an equal opportunity of being chosen in the sample. Major methods of SRS include WR (sampling with replacement) and WOR (sampling without replacement). In SRSWOR, replacement is not needed since the required units are already drawn. SRSWR involves randomly picking one sample in a unit then giving every unit an equal opportunity (chance) 1/N. After assigning equal chances, the selected units are returned or replaced back to the total population. Second sampling procedure follows while the process used in the first sample. This repetition is not allowed in simple random sampling SRSWOR (Simple Random Sampling without Replacement) Several complex sampling techniques base their plans on SRSWOR. The method emerges as the best quality used in sampling against which the other sampling techniques and plans are compared. Often, it happens that sophisticated sampling designs entail a sequence of simple random portions that are linked to in complex fashions. In SRSWOR method, once the units’ frame has been tallied, n-a sample size-is selected without replacing it from the total population units-N. Two-stage Cluster Sampling The principle that applies in the use of the techniques that state that if there is a similarity within cluster units, it is not economical to measure every cluster, and that the alternative lies in taking the units’ SRS in each chosen cluster (psu). In the first stage, the following conditions apply: (a) population is valued at N clusters-psus, and (b) SRS of the clusters should be taken. The second stage has the following conditions: (a) the cluster’s-i-sample size is mi, (b) there is need to draw the SRS for clusters that have been sampled, and (c) in a cluster I, the number of stratified clusters (ssus) is Mi. The sampling techniques described above are crucial in determining factors such as variances, mean, and standard deviation, among others. Technically, they are applicable in evaluating finite sample performances of mean population and proportion population and the related usual confidence intervals. Using SRSWOR and R program, it is easy to determine population parameters and how such parameters can affect the overall outcome of a sampling procedure. Simulation helps in coming up with values that are used in finding point estimator, including the upper and lower limits of 95% confidence interval. The selection of a particular method of estimation is largely dependent on the existence of auxiliary data and the characteristic of the association between the supplementary information and the aimed variable. Accordingly, a post-stratification estimator, regression estimator, or quotient estimator may be employed in the process. Section 3: Technical Details In the stratified simple random sampling without replacement (S-SRSWOR), it is easy to choose a stratified unit (sample) from population once ‘outstrata’ (optimal stratification) has been identified,, and a brand new frame developed by assigning new strata to the old units. The function that is vital in assigning new strata is referred to as ‘updateFrame’. Assuming that a state’s Revenues Department is interested in finding if a large organization has paid the exact sales tax amounts for all its transactions in, there is a possibility that the transactions conducted by the company or organization are quite large. It not an easy task to audit the transactions population at once. Stratifying the transaction size and applying pps allows one to find the larger proportion of the organization’s bigger sales in addition to the access of smaller transactions. The model that is used in estimating population total is the ratio estimator: yi – βxi + Zi Two values for the resulting variance from the basic SRSWOR formula for variance calculation need to be found. The population variance unbiased estimator is found by using the sample variance s2-(n-1)-1∑i€s(yi- ӯ)2. The unbiased variance population-size estimator is found by v (ӯ)-(1-n/N) s2/n. After estimation the population size, the two samples-biased and unbiased-ratio estimates of the total assume that the sampling plan is a SRSWOR (simple random sampling without replacement). R generator is applied to the biased and unbiased ratio estimates to model based variance estimates for the value of ratio estimate. The importance of determining the total population’s total is it also gives the absolute error in the calculation. This is the result of using variance formula based on proportional allocation: In the case above, V(ӯ) denotes SRSWOR’s ӯ variance in a sample size. There are several variance formulas used in getting the three estimation variances after the biased and unbiased values have been determined. However, the best formula depends on the type of allocation given to the stratified samples. In the simulation process, the settings had been identified earlier own. The major tasks in this section deal with further refining the samples to obtain a trend that will be helpful in making proper analysis and recommendations. The performance evaluation or simulation setting gives the general overview of the sample population in relation to the trends seen in computed confidence intervals. Section 4: Simulation Studies and Results from the Simulation Summary Simulator tool is important because it shows the estimates of standard deviation- ẟ2 and variance-ẟ in a statistical work aimed at finding the population proportion and population mean. First there is the need to develop the population size to be used in the simulation. Determining population size helps in understanding whether a proportion of a population is large or small. Simulations and the results The following design helps in determining the population’s ratio estimator: yi – βxi + Zi β is unknown constant, Zi is the variance and mean is set at 0. Zi (variance) equals a given constant multiplied by Xi. The following code generates the population in addition to taking 2 samples from the result. set.seed(13449875) popx Read More