Question

In simple random sampling, it is also true that each member of the population is equally likely to be selected, the chance for each member being equal to the sample size divided by the population size.

(a) Under what circumstances is that fact also true for systematic random sampling? Explain your answer.

(b) Provide an example in which that fact is not true for systematic random sampling.

Short answer

Part (a) Selection of the first sample in the systematic sample.

Part (b) We have taken the population of six different schools to prove this fact.

Step-by-step solution

Part (a) Step 1. Given information.

The given statement is:

n simple random sampling, it is also true that each member of the population is equally likely to be selected, the chance for each member being equal to the sample size divided by the population size.

Part (a) Step 2. Explanation.

In simple random sampling, each member of the population has an equal chance of being chosen, with each person's probability equal to the sample size (divided by the population size).

This is true for the systematic sample in the first sample selection.

Part (b) Step 1. Explanation.

As we know that all items have the same chance of being selected, systematic sampling is an EPS (equal probability of one selection) approach (in the example given, one in ten). Because various subsets of the same size have varying selection probabilities (e.g., the set (4, 14, 24, 994) has a one in ten likelihood of selection, but the set (4, 13, 24, 34) has zero probability of selection, it is not "simple random sampling." A non-EPS technique can also use systematic sampling.

The disadvantage of systematic sampling is that, even when it is more accurate than SRS, its theoretical properties make it difficult to quantify that accuracy. In that case, we use PPS sampling, which improves accuracy for a given sample size by concentrating the sample on large elements that have the greatest impact on population estimates.

Consider the following six school populations: 150, 180, 200, 22), 260, and 490 pupils (a total of 1500 students), and we wish to use the student population as the basis for a PPS sample of size three. To do so, we may assign numbers 1 to 150 to the first school, 151 to 330 (150+ 180) to the second school, 331 to 530 to the third school, and so on to the last school (1011 to 1500).

Then, using a random start number between 1 and 500 (equivalent to 1500/3), we count through the school populations in multiples of 500. If our random start was 137, we would choose the schools with the numbers 137, 637, and 1137, or the first, fourth, and sixth schools, respectively.