Assignment Task

 

Task
Regression analysis is a method for relating one variable to other explanatory variables. However, the term regression has sometimes been used in a slightly di?erent way, meaning "regression toward the mean." The example often cited is of male heights. If a father is unusually tall, for example, his son will typically be taller than average but not as tall as the father. Similarly, if a father is unusually short, the son will typically be shorter than average but not as short as the father. We say that the son’s height tends to regress toward the mean. This case illustrates how regression toward the mean can occur.

Suppose a company administers an aptitude test to all of its job applicants. If an applicant scores below some value, he or she cannot be hired immediately but is allowed to retake a similar exam at a later time. In the interim the applicant can presumably study to prepare for the second exam. If we focus on the applicants who fail the exam the first time and then take it a second time, we would probably expect them to score better on the second exam. One plausible reason is that they are more familiar with the exam the second time. However, we will rule this out by assuming that the two exams are su?ciently di?erent from one another. A second plausible reason is that the applicants have studied between exams, which has a beneficial e?ect. However, we will argue that even if studying has no beneficial e?ect whatsoever, these applicants will tend to do better the second time around. The reason is regression toward the mean. All of these applicants scored unusually low on the first exam, so they will tend to regress toward the mean on the second exam—that is, they will tend to score higher. You can employ simulation to demonstrate this phenomenon, using the following model. Assume that the scores of all potential applicants are normally distributed with mean and standard deviation . Because we are assuming that any studying between exams has no beneficial e?ect, this distribution of scores is the same on the second exam as on the first. An applicant fails the first exam if his or her score is below some cuto? value L. Now, we would certainly expect scores on the two exams to be positively correlated, with some correlation . (This is the Greek letter "rho," often used for a population correlation.) That is, if everyone took both exams, applicants who scored high on the first would tend to score high on the second, and those who scored low on the first would tend to score low on the second. (This isn’t regression to the mean, but simply that some applicants are better than others.)


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  • Uploaded By : Roman
  • Posted on : July 14th, 2019

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