Question

Census data show that 3?% of students in a certain country went to college or a trade school within a year of graduating from high school. A random sample of 453 high school graduates is selected. Use the normal approximation to the binomial distribution to complete parts a through e below.

a. What are the mean and standard deviation of this? distribution?

b. What is the probability that 16 or fewer students will go to? college?

c. What is the probability that exactly 16 students will go to? college?

d. What is the probability that 20 or more students will go to? college?

e. What is the probability that between 16 and 20 ?students, inclusive, will go to? college?

Answer 1

Here, n = 453 , p = 0.03

a)
mean = np
= 453 * 0.03
= 13.59

std.deviation = sqrt(npq)
= sqrt(453 * 0.03 * 0.97)
= 3.6307

b)

P(x < 16)
= P(X < 15.5)
= P(z < (15.5 - 13.59)/3.6307)
= P(z < 0.5261)
= 0.7006

c)
P(X = 16)
= P(15.5 < X < 16.5)
= P((15.5 - 13.59)/3.6307 < X < (16.5 - 13.59)/3.6307)
= P(0.5261 < X < 0.8015)
= P(X < 0.8015) - P(X < 0.5261)
= 0.7886 - 0.7006
= 0.0880

d)
P(X >= 20) = P(X > 19.5) .. continuity correction
= P(z > (19.5 - 13.59)/3.6307)
= P(z > 1.6278)
= 0.0518

e)
P(16 <= X <= 20)
= P(15.5 < X < 20.5)
= P((15.5 - 13.59)/3.6307 < X < (20.5 - 13.59)/3.6307)
= P(0.5261 < X < 1.9032)
= P(X < 1.9032) - P(X < 0.5261)
= 0.9715 - 0.7006
= 0.2709