Question

a hotel rewards club wants to randomly select 100 of its 5,000 members to participate in a survey. the club wants to determine if peoples opinions differ based on age. there are 2,200 members between the ages of 25 and 50, and there are 2,800 who are age 51 or older. the club decides to randomly select 44 members between the ages of 25 and 50, and 56 members ages 51 and older. is the sample of 100 members a simple random sample?

yes, because the hotel rewards members were randomly selected for the sample.

yes, because each hotel rewards member has the same chance of being selected.

no, because the sample of 100 is too small a sample to represent the population.

no, because each sample of 100 does not have the same chance of being selected.

Explanation:

To determine if it's a simple random sample, recall the definition: every possible sample of the same size must have an equal chance of being selected, and every individual must have an equal chance of being selected.

• For option A: Random selection of members doesn't guarantee it's a simple random sample. A simple random sample requires every possible sample of size 100 to have the same chance. Here, the population has subgroups (ages 25 - 50 and 51+). The number of members in each subgroup is different (2200 and 2800). If we just randomly pick 100, a sample with all 100 from 25 - 50 or all from 51+ is possible, but the chance of such a sample might not be equal to a sample with a mix, because the sizes of the subgroups differ. So random selection of individuals isn't enough for a simple random sample. Eliminate A.
• For option B: Each member has a chance, but a simple random sample requires every sample of size 100 to have the same chance. Since the population is split into two groups with different sizes, a sample with, say, 100 from the 2200 - member group and a sample with 100 from the 2800 - member group have different probabilities (because the number of ways to choose 100 from 2200 is different from choosing 100 from 2800). So not every sample of size 100 has the same chance. Thus, B is incorrect.
• For option C: The size of the sample (100) being small isn't the reason it's not a simple random sample. The issue is about the sampling method and equal chance of all samples, not sample size. Eliminate C.
• For option D: As explained, because the population has two subgroups with different sizes, a sample of 100 from one subgroup has a different probability than a sample of 100 from the other or a mixed sample. So not every sample of 100 has the same chance of being selected, so it's not a simple random sample. This is correct.

Answer:

D. No, because each sample of 100 does not have the same chance of being selected.