Question

Hi, I don't know how to use R Studio to find the solution to this problem. Can someone please help me? It would be greatly appreciated.

In 1974, the University of California-Berkeley was sued for alleged gender discrimination in graduate admissions. Let's teleport back to the fall of 1973. Applications for the upcoming 1974 academic year are in, and you're on the admission review board responsible for overseeing the admission results. You know going into the application review season that Berkeley's long run average rate of admission is 40 percent for both men and women. This year 4,321 women and 8,442 men submitted applications. a) Assuming the departments admit applicants at the historical rate, how many admitted applicants should we expect to see for this upcoming school year? How many of these applicants are men? How many are women? Use R to calculate this number Use the pbinom function to calculate the probability of admitting more than 8,000 men, assuming the applicants perform along historical expectations. (Hint:Os P(X x)S1). Use the dbinom function to calculate the probability of admitting exactly 1,600 female applicants, assuming the applicants perform along historical expectations. Why is the probability so low? b) c)

Answer 1

the probability that a randomly selected applicant is admitted is 0.40 (We assume that this is true for both men and women applicants)

Let X be the number of women admitted out of a total of 4,321 women applicants. X has a Binomial distribution with parameters, number of trial n=4,321 and success probability (probability of getting admitted) p=0.40

The probability of admitting X=x women is

$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}=\binom{4321}{x}0.40^x(1-0.40)^{4321-x}$

Let Y be the number of men admitted out of a total of 8,442 men applicants. Y has a Binomial distribution with parameters, number of trial n=8442 and success probability (probability of getting admitted) p=0.40

The probability of admitting Y=y men is

$P(Y=y)=\binom{n}{y}p^y(1-p)^{n-y}=\binom{8442}{x}0.40^y(1-0.40)^{8442-y}$

a) The expected value of X is (the average number of women applicants admitted is) (using the result for Binomial distribution)

$E(X)=np=4321\times 0.40$

The expected value of Y is (the average number of men applicants admitted is) (using the result for Binomial distribution)

$E(Y)=np=8442\times 0.40$

The expected value of total number of admitted students is

$E(X+Y)=E(X)+E(Y)$

Below is the R code (all statements starting with # are comments in R)

#part a)
#set the probability of success
p<-0.40
#set the number of women applicants
nW<-4321
#set the number of men applicants
nM<-8442
#Expected Number of women admitted is
aW<-nW*p
#Expected Number of men admitted is
aM<-nM*p
#Expected total Number admitted is
aT<-aW+aM
print(paste('The expected number of admitted applicants is',aT))
print(paste('The expected number of admitted women applicants is',aW))
print(paste('The expected number of admitted men applicants is',aM))

##get these output

b) As already stated, Y is the number of men admitted out of a total of 8442 men applicants. The probability of admitting more than 8000 men is

$P(Y>8000)=1-P(Y\le 8000)$

The function pbinom(y,n,p) calculates the probability $P(Y\le y)$

The following is the R code

#part b)
pMen<-1-pbinom(8000,nM,p)
print(paste('The probability of admitting more than 8000 men is',pMen))

#get this

> 휴part b) > pMen<-1-pbinom (8000, nM, p) > print (paste ('The probability of admitting more than 8000 men is' ,pMen)) 1] "The probability of admitting more than 8000 men is 0"

c) As stated earlier

The probability of admitting X=x women is

$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}=\binom{4321}{x}0.40^x(1-0.40)^{4321-x}$

The probability of admitting exactly 1600 women is the probability of X=1600, P(X=1600)

the function dbinom(x,n,p) calculates the probability P(X=x)

R code is below

#part c)
pWomen<-dbinom(1600,nW,p)
print(paste('The probability of admitting exactly 1600 women is',round(pWomen,8)))

#get this

pWomen<-dbinom (1600,nW,p) print (paste ('The probability of admitting exactly 1600 women is',round (pWomen, a))) 1] "The probability of admitting exactly 1600 women is 4.13e-06"

The standard deviation of X is $\sqrt{np(1-p)}=\sqrt{4332\times 0.4\times (1-0.4)}=32.20$ . The mean of X is 1728.4 and hence 1600 is (1728.4-1600)/32.20= 4 standard deviations away from the mean of the distribution.

Hence this probability is low because 1600 is far away from the mean number of admissions for women which is 1728.4.

The probability is also low because we have calculated the probability of admitting an exact number of students. The point probability becomes very small and tends towards zero as the number of trials increase (like in the case of a continuous random variable, where the point probability is zero).  Hence it would make sense to calculate the probability for a range of numbers.