a study was conducting to determine whether there were significant differences between medical students admitted through special programs (such as affirmative action) and medical students admitted with regular admissions criteria. it was found that the graduation rate was 94% for the medical students admitted through special programs (based on data from the journal of the american medical association)

a- if 10 of the students from the special program are randomly selected , find the probability that at least 9 of them graduated

b- would it be unusual to randomly select 10 students from the special programs and get only 7 that are graduate ? why or why not ?

a) Let X be the total amount of graduated students in the selection. X is a random variable with Binomial distribution B(10,0.94). The probability of X to be at least 9 is

[tex]P(X \geq 9) = P(X = 9) + P(X = 10) = {10 \choose 9}0.94^9*0.06+{10 \choose 10}0.94^{10} = 0.8824[/tex]

Thus, the probability is 0.8824

b) Lets compute the probability of X being at most 7 (because it would be unusual to get few students). Note that

P(X ≤ 7) = 1-P(X > 7) = 1- (P(X=8) + P(X ≥9))

P(X≥9) was computed before, and

[tex] P(X=8) = {10 \choose 8} 0.94^8*0.06^2 = 0.0986 [/tex]

Thus,

P(X ≤ 7) = 1-(0.0986+0.8824) = 0.019 < 0.05

The probability to select 7 or less graduated students is 0.019, less than 2% of the cases. We can say that it is unusual.