# It is found that on average 32

It is found that on average 32% of undergraduate studentsparticipate in a research study during their first year of program.Suppose that 200 freshman are randomly selected and a researcher islooking at the count, X, of how many of those freshman willparticipate in a research study. Find the following.

a. Indicate whether normal distribution can be used toapproximate the binomial distribution

b. The mean and standard deviation of the distribution

c. The probability that at least 50 will participate

d. The probability that 80 of them will participate

e. The probability that less than 59 of them are seniorcitizens

Answer:

Here X has binomial distribution with parameters

n=200 and p = 0.32

(a)

Here we have

np = 200 * 0.32 = 64

n(1-p) = 200 * 0.68 = 136

Since np and n(1-p) are greater than 5 so normal approximationwill be appropriate.

b)

The mean is:

The standard deviation is

c)

Here we to use continuity correction.

The z-score for X = 50 -0.5 = 49.5 is

The probability that at least 50 will participateis

(d)

The z-score for X = 80 -0.5 = 79.5 is

The z-score for X = 80 +0.5 = 80.5 is

The probability that 80 of them will participateis

e)

The z-score for X = 59 -0.5 = 58.5 is

The probability that less than 59 of them are seniorcitizens is