It has been estimated that 25% of all university students switch majors within their first two years of starting classes. If a random sample of 500 third-year students is taken at a city university, what is an estimate of the probability that 20% or more had switched majors within their first two years?
A) 4951
B) 0049
C) 9951
D) 0.5059

Question

Quizplus · Accepted Answer

The sample size n = 500 is big enough for us to use the normal approximation to the binomial distribution. 
The mean for switching majors would be:
(25/100) x 500 = 125
The standard deviation would be:
sqrt(500 x 0.25 x 0.75) = 10.61
Using the continuity correction, we have:
P(X >= 0.20 x 500) = P(X >= 100 - 0.5) = P(Z >= (99.5 - 125)/10.61) = P(Z >= -2.419)
Using a standard normal distribution table or calculator, we find the probability to be approximately 0.008 or 0.80%
The closest answer choice is C: 9951

It Has Been Estimated That 25% of All University Students