Answer:
The probability that fewer than 420 of the admitted students will enroll at this university is 0.9015
Step-by-step explanation:
The binomial distribution is a discrete distribution in which n trials can produce a success or a failure. We call the probability of success p and the probability of failure will be q = (1 - p).
A binomial distribution B (n, p) can be approximated by a normal distribution, provided that n is large and p is not very close to 0 or 1. The approximation consists of using a normal distribution with the same mean and standard deviation as the distribution binomial.
In practice, the approximation is used when n≥ 30, np ≥ 5 and n (1 - p) ≥ 5 and also p is close to 0.5.
The mean and standard deviation of the normal distribution are obtained by the expressions:
μ= n*p
σ= √n*p*q=√n*p*(1-p)
In this case, you know that n=1000 and p=0.4. Then:
μ= 1000*0.4= 400
σ= √1000*0.4*(1-0.4)= √240= 15.5
With X being the number of admitted students who enroll in this university and z=(X-μ)÷σ, then:
P(X≤420)=P(z≤(420-400)/15.5)= P(z≤ 1.29)= 0.9015
The probability that fewer than 420 of the admitted students will enroll at this university is 0.9015
