Question

Past records show that at a given college, 20% of the students who began as economics majors either changed their major or dropped out of school. An incoming class has 110 beginning economics majors. What is the probability that at most 30% of these students will leave the economics program?

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then

probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n …………………………….........……..…..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST...(1a)

Normal approximation

X ~ B(n, p) np ≥ 5 and np(1 - p) ≥ 5, (X – np)/√{np(1 - p)} ~ N(0, 1) [approximately] ……..........….. (2)

Also, (pcap – p)/√{p(1 - p)/n} ~ N(0, 1) [approximately]

Where pcap = X/n = sample proportion  

Continuity Correction …………………………………………………………………..............………….. (3)

If X ~ B(n, p), P(X ≤ k) can be approximated by the Normal probability by

P[Z ≤ (k + 0.5 – np)/√{np(1 - p)}], where Z ~ N(0, 1)

Adding 0.5 to the k value is referred as Continuity Correction

Now to work out the solution,

Let X = number of students who began as economics majors either changed their major or dropped out of school in a sample of 110 beginning economics majors students. Then, X ~ B(110, 0.2), …….. (4)

Where 0.2 = given proportion of students who began as economics majors either changed their major or dropped out of school

Probability that at most 30% of these 110 students will leave the economics program

= P(X ≤ 33) [30% of 110 = 33]

= P[Z ≤ {(33 + 0.5 – 22)/√(22 x 0.8)}] [vide (2), (3) and (4)]

= P(Z ≤ 2.7412)

= 0.9969 [Using Excel Function: Statistical NORMSDIST] Answer

DONE