In simple random sampling, it is true that each member of the population is equa

ID: 3351896 • Letter: I

Question

In simple random sampling, it is 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. Complete parts (a) and (b) below.

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

This is true if the sample size evenly divides into the population size. In this case, there are no members of the population that will be excluded from being selected.

Your answer is correct.

This is true if there are no cyclical patterns in the population. In this case, every sample of size n is equally likely to be selected.

This is true if the sample size does not evenly divide into the population size. In this case, there are no members of the population that will be excluded from being selected.

This is true if there are no distinct subpopulations. In this case, every sample of size n is equally likely to be selected.

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

When selecting a sample of 3 from a population of 31, the 31st member of the population will never be selected.

When selecting a sample of 5 from a population of 50, the 50th member of the population will never be selected.

When selecting a sample of 6 from a population of 60 in which the first 20 people have some characteristic, these people are more likely to be selected.

When selecting a sample of 4 from a population of 40 in which every 4th person has some characteristic, these people are more likely to be selected.

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Explanation / Answer

under systematic sampling if the population size is N and the sample size be n

then we get a number k as k=N/n and then choose a random number from 1,2,....,k and let that number be r.

then the sample would be consisting of rth element,(r+k)th element,....,(r+(n-1)k)th element

now if k is an integer then there is no problem but if it is not then k is taken to be [N/n] where [x] means largest integer less than or equal to x.

hence the answer of question a is

option A:This is true if the sample size evenly divides into the population size. In this case, there are no members of the population that will be excluded from being selected.

because if N/n is an integer that is k is an integer then we generate a random number from 1,2,...,k with each having probability of getting selected as 1/k=1/(N/n)=n/N which is equivalent to the probability of simple random sampling.

b. the answer is option A:When selecting a sample of 3 from a population of 31, the 31st member of the population will never be selected.

because here N=31 is not a multiple of n=3

so k=[31/3]=10

so first sample element will come from 1st to 10th elements , second sample element will come from 11th to 20th element and third sample element will come from 21st to 30th element.

hence the 31st element will not at all get selected in the sample. hence the assumption of simple random sampling is violated

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In simple random sampling, it is true that each member of the population is equa

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