the probability that few than 285 grauated within 4 years

Question

A study about the students graduating within 4 years of their entrance to the universities indicated that 62% of the students do not graduate within 4 years. Suppose a random sample of 500 students was taken. The sample considered the students after 4 years of their college entrance. What is the probability that fewer than 285 graduated within 4 years?

Answer 1

285/500 = .57 less than .62

.62 = 62% of students graduate within 4 years

what is probability less than 285 of the 500 sample graduated within 4 years

find z score for 285, then find area under normal curve, use binomial approximation to normal

.62(500) = 310 = mean = np, p = .62, q=.38 n =500, Var= npq = 310(.38)=117.8

z = (285-310)/sd/sqrn = -25sqr500)/117.8 = -25(sqr5)/117.8= about - .4745

corresponding area = 0.31757 to the left of z=-.4745 found by z tables

= 31.57 % probability of less than 285 graduate within 4 years