Question

The admissions office of a small college is asked to accept deposits from a number of qualified prospective freshmen so that, with probability about 0.95, the size of the freshman class will be less than or equal to 1,100. Suppose the applicants constitute a random sample from a population of applicants, 80% of whom would actually enter the freshman class if accepted. (Use the normal approximation.)

(a)

How many deposits should the admissions counselor accept? (Round your answer up to the nearest integer.)

(b)

If applicants in the number determined in part (a) are accepted, what is the probability that the freshman class size will be less than 1,045? (Round your answer to four decimal places.)

Answer 1

Solution:-

Pr(Any accepted student will actually appear and enter the first year) = 0.80

Maximum number of student accepted = 1100

Let say there are X if the number of students which must be asked to accept the deposits to make the probability of students entering in the first year equal or less than 1100 shall be greater than 0.95

that means statistically,

where expected number of first year student entering in the class = 0.8 X

standard deviation of the number of students entering in the class

Z - value for p - value = 0.95 is

Z = 1.645

1100 - 0.8x = 0.658√x

so x1 is invalid

so number of maximum student whose application must be accepted = 1344.83 or 1344

(b) so, let say that number of applicants accepted are 1344

Expected number of admissions

standard deviation of admission =

so Pr(X < 1045; 1075.2 ;14.664) = ?

so Pr(Z <-2.05) = 0.0202.