Question

Dartmouth College would like to have 1050 freshmen. This college cannot
accommodate more than 1060. Assume that each applicant accepts with
probability .6 and that the acceptances can be modelled by Bernoulli trials. If
the college accepts 1700, what is the probability that it will have too many
acceptances?

Answer 1

If it accepts 1700 students, the expected number of students who matriculate is

$$np=1700(0.6)=1020.$$

The standard deviation for the number that accept is 

$$\sqrt{npq}=\sqrt{1700(0.6)(1-0.6)}\approx20.2$$

By the central limit theorem, the normalized standardized sum is approximately normal. That means that 

$$S_n^*=\dfrac{S_n-np}{\sqrt{npq}}$$

is approximately a standard normal distribution.

We want to estimate the probability

$$P(S_{1700}>1060)=P(S_{1700}\geq1061)$$

$$=P(S_{1700}^*\geq\dfrac{1060.5-1020}{20.2})\approx P(S_{1700}^*\geq\dfrac{1060.5-1020}{20.2})$$

$$\approx P(S_{1700}^*\geq2.005)\approx0.0225$$

Thus, the college is fairly safe using this admission policy.