Researchers conducted a study to determine whether there were significant differ

Question

Researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs (such as affirmative action) and medical students admitted through the regular admissions criteria. It was found that the graduation rate was 84% for the medical students admitted through special programs. a. If 15 of the students from the special programs are randomly selected, find the probability that exactly 12 of them graduated. b. If 15 of the students from the special programs are randomly selected, find the probability that fewer than 10 of them graduated.

Explanation / Answer

Binomial Distribution
PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
a.
P( X = 12 ) = ( 15 12 ) * ( 0.84^12) * ( 1 - 0.84 )^3
= 0.23
b.
P( X < 10) = P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 15 9 ) * 0.84^9 * ( 1- 0.84 ) ^6 + ( 15 8 ) * 0.84^8 * ( 1- 0.84 ) ^7 + ( 15 7 ) * 0.84^7 * ( 1- 0.84 ) ^8 + ( 15 6 ) * 0.84^6 * ( 1- 0.84 ) ^9 + ( 15 5 ) * 0.84^5 * ( 1- 0.84 ) ^10 + ( 15 4 ) * 0.84^4 * ( 1- 0.84 ) ^11 + ( 15 3 ) * 0.84^3 * ( 1- 0.84 ) ^12 + ( 15 2 ) * 0.84^2 * ( 1- 0.84 ) ^13 + ( 15 1 ) * 0.84^1 * ( 1- 0.84 ) ^14 + ( 15 0 ) * 0.84^0 * ( 1- 0.84 ) ^15
= 0.0227

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