BHStat

August 31, 1999

Variation amongst Samples: Homework

We started this part of the course asking a question:

How small can randomly chosen samples be so that we are still satisfied that samples of this size provide an acceptable representation of the population?

We looked at samples from a population of students who chose their favorite singers from a list, and then we made an "eyeball comparison" of the percents from the samples with the percents from the population.

Omega came up with this observation:

When we compare a population percent (such as the percent of the population who are Dave Matthews fans) with the equivalent sample percent (such as the percent of a sample drawn from that population who are Dave Matthews fans), larger samples tend to be more accurate than smaller samples.

We gave a very special meaning to the phrase "tend to be more accurate." We said it means that the percents calculated from larger samples tend to vary less, over the long run, from the actual population percent than do the percents calculated from smaller samples.

In class today, we devised ways to compare "variation over the long run" for samples of various sizes.

Your assignment is to write a short essay that responds to this question:

We are going to take a sample from a population having approximately 40,000 individuals in it. How small can the sample be so that, over the long run, samples of this size accurately reflect the population’s composition?

Your response is important, but the major part of this assignment is that you justify your response. Your justification must be based on the data presented in the attached sheets.

In your justification, you should make clear that there are two competing motives. One motive is that we want the sample to be very small. This will make it easier to actually collect it. The other is that we want the sample size to be such that there is relatively little variation, amongst samples of this size, from the actual population percents. So, your justification should make explicit reference to variation among samples of sizes larger than the one you picked, and variation among samples of sizes smaller than the one you picked.