Question

Calculate using R, (Please share your R Script if possible):

In 1974, the University of California-Berkeley was sued for alleged gender discrimination in graduate admissions. Let’s go 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 percentfor both men and women. This year 4,321 women and 8,442 men submitted applications.

1. 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.

2. b) Use the pbinom function to calculate the probability of admitting more than 8,000 men,assuming the applicants perform along historical expectations. (Hint: 0 ≤ P(X=x) ≤ 1).

3. c) 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?

Answer 1

First define the number of men and women using the following code -

> nm = 8442
> nw = 4321
Then define the probability of success for men and women. As both men and women have equal probability of getting selected, so let's just define the probability of success as -

> p = 0.4
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a)

Remember that the expected value of a binomial distribution with parameter 'n' and 'p' is = np.

As both men and women have same probability of success, so the expected number of applicants this year would be -

> E = (nm + nw)*p
You can get the output using the code -

> E
[1] 5105.2

So, approximately 5106 applicants can be expected this year to get admitted.

The number of men and women applicants can be calculated separately as -

> Em = nm*p
> Ew = nw*p
> Em
[1] 3376.8
> Ew
[1] 1728.4

So, approximately 3377 of them are men and 1728 are women

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b)

Note that pbinom gives the cumulative probability.

As we need the probability of admitting more than 8000 men, we would have to rewrite it as -

P(X > 8000) = 1 - P(X $\leq$ 8000)

For this, use the following code -

> 1 - pbinom(8000, (nm+nw), 0.4)
[1] 0
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c)

Use the following code -

> dbinom(1600, nw, 0.4)
[1] 4.131132e-06
The probability is so low because the sample size is very large. And thus probability of any individual element in the PMF would be very small.