Fictional Job Advancement Program Test

This type of scenario happens in our real world. No testing instrument or criteria we have is perfectly fair. This
activity will help you explore at least one way that we could still achieve parity or fairness in admissions, given
both that the testing criteria we have is unfair and that we don't want to lower the standard for admission.
Suppose a test is used to select applicants for a job advancement program. Assume that there are two types of
applicants, for simplicity, we'll call them "red" and "blue".

The mean score on the test for the red applicants is 52 and the mean for the blue applicants is 50. That is, on
average, red applicants perform better on this test than blue applicants.
The test score distributions for both red and blue are nearly normal, and the standard deviation for each
distribution is 10.
The difference in means is 52 - 50 = 2 and is therefore only 2/10 = 0.2 standard deviations. Although this
calculation is slightly different than the one we do for a hypothesis test of a single mean, we can think of this
number as the effect size.
By our classification standards, an effect size of 0.2 would be considered ‘small,’ not visible to the naked eye.
Just looking at these distributions, then, it doesn't seem like the difference between the groups is very large.
The question we want to investigate here is, "When does a small effect size make a big difference?"
Consider the following scenario: At one large company there are two programs that use this test to screen
applicants.
Program 1 is less selective and accepts many applicants. Managers want you to choose a value between 40
and 45 for this program, meaning that any applicant who scores higher than the number you choose will be
accepted into the program.
Program 2 is very selective and accepts fewer applicants. Managers want you to choose a value between 65
and 70 for this program, meaning that any applicant who scores higher than the number you choose will be
accepted into the program.
You can use the Program Comparison Desmos Activity (highly recommended) or your own calculations (less
highly recommended) to help you answer the following questions. (The questions referred to in the Desmos
activity are the questions given below.)
You may round your answers to whole numbers of people for clarity in your explanations.
Task 1

1. [1] State the number that you chose as a cutoff for Program 1.
2. Suppose 500 red applicants and 500 blue applicants apply for Program 1.
   a. [1] What proportion of the total number of accepted applicants are red?
   b. [1] What proportion of the total number of accepted applicants are blue?
3. [2] Do you think this test appears to unfairly privilege red applicants over blue applicants for Program 1?
   State yes or no and explain your reasons.
4. [1] State the number that you chose as a cutoff for Program 2.
5. Suppose 500 red applicants and 500 blue applicants apply for Program 2.
   a. [1] What proportion of the total number of accepted applicants are red?
   b. [1] What proportion of the total number of accepted applicants are blue?
6. [2] Do you think this test appears to unfairly privilege red applicants over blue applicants for Program 2?
   State yes or no and explain your reasons.

Sample Solution