Question

Consider the class average in an exam in a few different settings. In all cases, assume that we have a large class consisting of equally well-prepared students. Think about the assumptions behind the central limit theorem, and choose the most appropriate response under the given description of the different settings.

1. Consider the class average in an exam of a fixed difficulty.

a) The class average is approximately normal

b) The class average is not approximately normal because the student scores are strongly dependent

c) The class average is not approximately normal because the student scores are not identically distributed

2. Consider the class average in an exam that is equally likely to be very easy or very hard.

a) The class average is approximately normal

b) The class average is not approximately normal because the student scores are strongly dependent

c) The class average is not approximately normal because the student scores are not identically distributed

3. Consider the class average if the class is split into two equal-size sections. One section gets an easy exam and the other section gets a hard exam.

a) The class average is approximately normal

b) The class average is not approximately normal because the student scores are strongly dependent

c) The class average is not approximately normal because the student scores are not identically distributed

4. Consider the class average if every student is (randomly and independently) given either an easy or a hard exam.

a) The class average is approximately normal

b) The class average is not approximately normal because the student scores are strongly dependent

c) The class average is not approximately normal because the student scores are not identically distributed

Answer 1

Answer 1. The class average is approximately normal

Here we are given:

1. Since students are equally well-prepared

2. The difficulty level is fixed

So, the only chance of error or randomness in scores may arise due to silly or accidental mistakes of the students. This can be thus attributed as Random Error so we assume that each student’s score will be an independent random variable drawn from the same distribution, and hence, Central Limit Theorem applies and The class average is approximately normal.

Answer 2. The class average is not approximately normal because the student scores are strongly dependent

Here, the class average in an exam is equally likely to be very easy or very hard i.e. the score of each student depends strongly on the difficulty level of the exam, which is random but applies to all students. Therefore, there exists strong dependence between the scores and they cannot be called independent.Hence, Central Limit Theorem does not apply.

Answer 3. The class average is approximately normal

Here, the scores of the different students are not identically distributed.

But if we assume Xi to be the score of the ith student from the first section and let Yi be the score of the ith student in the second section. The class average is the average of the random variables (Xi + Yi)/2. Therefore, Xi and Yi  are independent and identically distributed within themselves. Hence, Central Limit Theorem applies.

Answer 4. The class average is approximately normal

Here the student scores would be independent and identically distributed and Central Limit Theorem applies.