Question

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

a. Indicate whether normal distribution can be used to approximate 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 senior citizens

Answer 1

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 approximation will 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 participate is

(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 participate is

e)

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

The  probability that less than 59 of them are senior citizens is