Question

This is the first question and I know how to solve this one, but I am confused by the second one (The admissions office of a small, selective liberal-arts college will only offer admission to applicants who have a certain mix of accomplishments, including a combined SAT score of 1,300 or more. Based on past records, the head of admissions feels that the probability is 0.58 that an admitted applicant will come to the college. If 500 applicants are admitted, what is the probability that 310 or more will come? Note that “310 or more” means the set of values {310, 311, 312, …, 500}. )

The following is the second question

Consider the admissions office in the previous problem. Based on financial considerations, the college would like a class size of 310 or more. Find the smallest n, number of people to admit, for which the probability of getting 310 or more to come to the college is at least 0.95.

Answer 1

given that

the probability that an admitted applicant will come to the college=p=0.58

let X is number of admitted applicant come to college

a)

given that number of admitted applicants =n=500

so X~Bin (500,0.58)

We have to find P(X> 310)

Since n*p=500*0.58 =290 >5 and n*(1-p)=500*0.42=210 > 5 hence we will use normal approximation of binomial distribution

hence E(X)=n*p=500*0.58=290

Var(X)=n*p*(1-p)=500*0.58*0.42=121.8

now we have to find P(X>310 ) since X is discrete distribution so we will use continuity factor hence

P(X>310) =P(X>309.5) now

Using normal approximation

  

=1-0.9614=0.0386

b)

we have to Find the smallest n, number of people to admit, for which the probability of getting 310 or more to come to the college is at least 0.95.

so in this case E(X)=n*p=0.58 n Var(X)=n*p*(1-p)=0.58*0.42n=0.2436n

now

P(X>310) > 0.95

that is P(X>309.5) >0.95

now

  

from Z table P(Z>-1.645) =0.95

this gives

this gives

95790.25-359.02n +0.3364n²=0.6592n

gives

0.3364n²-359.6792n+95790.25=0

now solving this equation we get

  

hence

n= 566.95 or 502.25 but since as we calculated in part (a) we got probability very less for 500 so as there is not much difference between 500 and 502.25 hence we should take n=566.95~567