An elementary school houses a total of 430 students in kindergarten, first, and

Question

An elementary school houses a total of 430 students in kindergarten, first, and second grade. There are 6 classes in each grade and each class has between 20 and 25 students. You are asked to select a sample of between 30 and 50 students from the school using a random process. In our slides and textbook there are 4 primary sampling techniques discussed, simple random sampling, systematic random sampling, cluster sampling, and stratified sampling. Choose 3 out of the 4 techniques and describe, in detail, how each method could be used to collect the sample here. (this will take some time end you should certainly consult your notes, the slides, and the textbook). Attach a separate sheet if needed.

Explanation / Answer

1. Simple random sampling is the most widely-used probability sampling method, probably because it is easy to implement and easy to analyze.

Key Definitions

To understand simple random sampling, you need to first understand a few key definitions.

Our population here would be 430 students. Sample is assumed at 30-50 students. Assuming 40 as it is the mean of 30 and 50.

Simple random sampling refers to a sampling method that has the following properties.

An important benefit of simple random sampling is that it allows researchers to use statistical methods to analyze sample results.

We would do the simple Random sampling here as such:

One way would be the lottery method. Each of the N (430) population members is assigned a unique number. The numbers are placed in a bowl and thoroughly mixed. Then, a blind-folded researcher selects n (30 to 40) numbers. Population members having the selected numbers are included in the sample.

Suppose we use the lottery method described above to select a simple random sample. After we pick a number from the bowl, we can put the number aside or we can put it back into the bowl. If we put the number back in the bowl, it may be selected more than once; if we put it aside, it can selected only one time.

When a population element can be selected more than one time, we are sampling with replacement. When a population element can be selected only one time, we are sampling without replacement.

2. Sytemic random Sampling is a bit like arithmetic progression

In this case, the sampler or the person doing the sample takes any random number which is less than the population or the total number of samples N. Let this number be a. Therefore a<N

Let a = 5. Then we consider an interval i = 8 (for example). Now if we need 30 samples, our samples are going to be the people with the numbers.

5, 13,21,29,37,45,53,61,69,77,85,93,101,109,117,125,133,141,149,157,165,173,181,189,197,205,213, 221, 229, 237

3. Stratified Sampling

Stratified sampling refers to a type of sampling method . With stratified sampling, the researcher divides the population into separate groups, called strata. Then, a probability sample (often a simple random sample ) is drawn from each group.

Stratified sampling has several advantages over simple random sampling. For example, using stratified sampling, it may be possible to reduce the sample size required to achieve a given precision. Or it may be possible to increase the precision with the same sample size.

Here we would first take a random sample of the students who would be in proportion to the population so if we take say 43 students. Now we divide these 43 students based on the class they are studying in or other characteristics such as marks etc;

We then do a random sample on each of these sub groups to create our samples.

Another example here

Assume that we need to estimate average number of votes for each candidate in an election. Assume that country has 3 towns: Town A has 1 million factory workers, Town B has 2 million office workers and Town C has 3 million retirees. We can choose to get a random sample of size 60 over entire population but there is some chance that the random sample turns out to be not well balanced across these towns and hence is biased causing a significant error in estimation. Instead if we choose to take a random sample of 10, 20 and 30 from Town A, B and C respectively then we can produce a smaller error in estimation for the same total size of sample.

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