Question

Given a parameterization and some assumptions about the underlying distribution, there are two possible approaches to approximating a solution to the problem.

1) We could generate a single sample of size N from the distribution and do M bootstrap samples. (Method 1). This basically mirrors real life.

2)We could generate M samples of size N directly from the distribution. (Method 2). This basically mirrors what we would like to do in real life.

Compare these approaches with the following problem:

"Within a school district, students were randomly assigned to one of two Math teachers - Mrs. Smith and Mrs. Jones. After the assignment, Mrs. Smith had 30 students, and Mrs. Jones had 25 students. "At the end of the year, each class took the same standardized test. Mrs. Smith's students had an average test score of 78, with a standard deviation of 10; and Mrs. Jones' students had an average test score of 85, with a standard deviation of 15.

"What is the probability that Mrs. Smith and Mrs. Jones are equally effective teachers." Using N as respective class sizes and M = 100:

1) Define what "equally effective might mean".

2)Answer the question using Method 1.

3)Answer the question using Method 2.

4)Comment on any observed similarities or differences between the two results. Why might either (or both) approaches work or not quite work the same way as our binary simulations do?