4. Based on Wikipedia, for the academic year 2010, Columbia University’s student

Question

4. Based on Wikipedia, for the academic year 2010, Columbia University’s student population was over 27,000. Further, 65% of the student population did not identify themselves as minority.

PLEASE SHOW ALL WORK!

a) A researcher is planning to randomly sample 11 students. What is the probability that in his sample at least 3 and at most 6 students will identify themselves as minority?

b) Another researcher is planning to randomly sample 1000 students.

I. What is the probability that exactly 351 students will identify themselves as minority?

II. What is the probability that 351 or more students will identify themselves as minority?

Explanation / Answer

Solution

Let X = Number of students in a sample n who will identify themselves as minority. Then,

X is Binomially distributed with parameters n and p, where p = P(a student will identify as minority) = 0.35 [given 65% of the student population did not identify themselves as minority].

Now,

Part (a) [Note: n = 11 and p = 0.35]

We want P(at least 3 and at most 6 students will identify themselves as minority)

= P(3 X 6) = P(X 6) - P(X 2) = 0.949857 – 0.200129 [can obtained from Standard Binomial Probability Tables or using Excel Function ]

= 0.749728 ANSWER

Part (bi) [Note: n = 1000 and p = 0.35]

We want P(exactly 351out of 1000 will identify themselves as minority)

= P(X = 351) = 0.026367 ANSWER[can obtained from Standard Binomial Probability Tables or using Excel Function ]

Part (bii) [Note: n = 1000 and p = 0.35]

We want P(351 or more out of 1000 will identify themselves as minority)

= P(X 351) = 1 - P(X 350) = 1 – 0.514542 = 0.485458 ANSWER[can obtained from Standard Binomial Probability Tables or using Excel Function ]

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