Each year the admissions committee at a top business school receives a large number of applications for admission to the MBA program and they have to decide on the number of offers to make. Since some of the admitted students may decide to pursue other opportunities, the committee typically admits more students than the ideal class size of 720 students. You were asked to help the admission committee estimate the appropriate number of people who should be offered admission. It is estimated that in the coming year the number of people who will not accept the admission is normally distributed with mean 50 and standard deviation 21. Suppose for now that the school does not maintain a waiting list, that is, all students arc accepted or rejected.

a. Suppose 750 students are admitted. What is the probability that the class size will be at least 720 students?

b. It is hard to associate a monetary value with admitting too many students or admitting too few. However, there is a mutual agreement that it is about two times more expensive to have a student in excess of the ideal 720 than to have fewer students in the class. What is the appropriate number of students to admit?

c. A waiting list mitigates the problem of having too few students since at the very last moment there is an opportunity to admit some students from the waiting list. Hence, the admissions committee revises its estimate: It claims that it is five times more expensive to have a student in excess of 720 than to have fewer students accept among the initial group of admitted students. What is your revised suggestion?

a. The probability that the class size will be at least 720 students is 17.11%

b. The appropriate number of students to admit is 761

c. The appropriate number of students to admit is 750

Explanation:

a. According to the given data we can define X to be a random variable representing the students not taking admission such that X ~ Normal(50, 21), hence Prob(Class size >= 720) = Prob(X <= 30), therefore, to calculate the probability that the class size will be at least 720 students we would have to use the following formula:

= Prob{(X - 50)/21 <= (30 - 50)/21}

= Prob(Z <= -0.95) where Z ~ Normal(0,1)

= Prob(Z >= 0.95) from symmetry

= 1 - Prob(Z <= 0.95) = 1 - 0.8289 (from table) = 0.1711

The probability that the class size will be at least 720 students is 17.11%

b. To calculate the appropriate number of students to admit we would have to use the following formula:

Critical factor = Cu / (Cu + Co) = Cu / (Cu + 2*Cu) = 1/3 = 0.333

Where Cu = cost of one student less 'ordered' than 'demand'

and Co = cost of one student more 'ordered' than 'demand' = 2*Cu

Hence, For optimal excess admission (Q), F(Q) = 0.333

From the standard normal table, we know that at F(Z) = 0.333, Z = -0.43

So, Q = 50 - 0.43*21 = 41

So, the appropriate number of students to admit = 720+Q = 720+41 = 761

c. To calculate the revised suggestion we use the following formula:

Critical factor = Cu / (Cu + Co) = Cu / (Cu + 5*Cu) = 1/6 = 0.167

Where Co = 5*Cu

For optimal excess admission (Q), F(Q) = 0.167

From the standard normal table, we know that at F(Z) = 0.167, Z = -0.97

So, X = 50 - 0.97*21 = 30

So, the appropriate number of students to admit = 720+30 = 750

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