Question

All applicants at a large university are required to take a special entrance exam before they are admitted. The exam scores

are known to be normally distributed with a mean of 800 and a standard deviation of 100. Applicants must score 700 or more on the exam before they are admitted.

a. What proportion of all applicants taking the exam is granted admission?
b. What proportion of all applicants will score 1000 or higher on the exam?
c. For the coming academic year, 2500 applicants have registered to take the exam. How many do we expect to be
qualied for admission to the university

Answer 1

(a) The proportion of all applicants granted admission who took the exam is 0.8413.

(b) The proportion of all applicants who scored 1000 or higher is 0.0228.

(c) The expected number of candidates that will be granted admission is 2104.

Explanation:

Let X = score of an applicant.

The random variable X follows a Normal distribution with mean, μ = 800 and σ = 100.

It is provided that applicants must score 700 or more on the exam before they are admitted.

(a)

Compute the probability of a score 700 or more as follows:

`P(X\geq 700)=P((X-\mu)/(\sigma)\geq (700-800)/(100))\\=P(Z\geq -1)\\=P(Z<1)\\=0.8413`

*Use the z-table for the probability.

The proportion of all applicants granted admission who took the exam is 0.8413.

(b)

Compute the probability of a score 1000 or more as follows:

`P(X\geq 1000)=P((X-\mu)/(\sigma)\geq (1000-800)/(100))\\=P(Z\geq 2)\\=1-P(Z<2)\\=1-0.9772\\=0.0228`

*Use the z-table for the probability.

The proportion of all applicants who scored 1000 or higher is 0.0228.

(c)

The number of candidates applying for the exam is, n = 2500.

The probability that a candidate will be granted admission is, p = 0.8413.

The expected number of candidates that will be granted admission is:

E (No. of candidates granted admission) = n × p

`=2500*0.8413\\=2103.25\\\approx2104`

Thus, the expected number of candidates that will be granted admission is 2104.