Question

According to national data, about 15% of American college students earn a graduate degree. Using this estimate, what is the probability that exactly 25 undergraduates in a random sample of 200 students will earn a college degree? Hint: Use the normal approximation to the binomial distribution, where

p = 0.15

and

q = 0.85.

Answer 1

Binomial distribution is a discrete probability distribution of successes over number of repeated trials. In binomial distribution, probability of success remain constant throughout the experiment and each outcome is independent on its own. Normal distribution to the binomial is used when sample size is very large.

The probability of getting exactly 24 undergraduate students will earn collage degree can be approximated using binomial distribution. Binomial probability mass function is given by the following formula:

p(X = x) = nCx * p^x * (1-p)^(n-x)

here, p = 0.15

x = 25

n = 200

sample size is very large and probability is very small, we can use normal approximation to the binomial:

z= x−μ / σ

μ=np

=200×0.15

= 30

σx=√np(1−p)

=√200×0.15(1−0.15)

=5.049

z= 25 - 30 / 5.049

= -0.99

Locate -0.90 on the left most column of z score tables and move across that row up to column with 0.09 and get z value as 0.1611

P(x=25)=0.1611

The probability of getting e exactly 25 undergraduate student that will earn collage degree is 16.11%.