See below: Is this selection process biased?

Question

An elementary school is having a fund raiser to add new books to the library. To encourage participation, the top twenty sellers will be placed in a drawing from which four will be selected to receive theme park tickets. If all four of the winners are from the 4th grade, does that indicate that there was bias in the selection process and the the selection was not random?

Answer 1

There are two parts to the selection process. The first is the selection for the group of twenty students who will be considered in the raffle. The second is the selection process for the winners of the theme park tickets out of those twenty students.

If all four of the ticket winners were fourth graders, then the results indicate bias somewhere in the selection process. Assuming an even distribution of students K-5, the chances that this would occur in a completely random drawing from the student body is roughly (but not exactly since this is done without replacement) (1/6)^4 ≈ 0.077%, which is far less than 1%, so we may reasonably say that the drawing was not completely random.

Given the description, the source of bias appears to be the first part of the selection process. By only placing the top twenty sellers into consideration for the theme park tickets, the school is biasing the drawing in favor of older students, who likely will have an easier time selling the fundraiser products. As a result, we would expect that the top twenty sellers would primarily come from 4th and 5th grade.