Question

According to a​ magazine, as of January​ 2018, 9​% of chief executive officers were women. Complete parts a through d based on a random sample of 12 CEOs.

A. What is the probability that one corporate officer was​ female? The probability is ________. ​(Round to four decimal places as​ needed.)

b. What is the probability that fewer than four corporate officers were​ female? The probability is ________. ​(Round to four decimal places as​ needed.)

c. What is the probability that more than two corporate officers were​ female? The probability is________. ​(Round to four decimal places as​ needed.)

d. What are the mean and standard deviation for this​ distribution? The mean number of females is _______. ​(Type an integer or a decimal. Do not​ round.)

The standard deviation of the number of females is _________..​(Round to four decimal places as​ needed.)

Answer 1

a)a)

Here, n = 12, p = 0.09, (1 - p) = 0.91 and x = 1
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 1)
P(X = 1) = 12C1 * 0.09^1 * 0.91^11
P(X = 1) = 0.3827

b)

Here, n = 12, p = 0.09, (1 - p) = 0.91 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X < 4).
P(X < 4) = (12C0 * 0.09^0 * 0.91^12) + (12C1 * 0.09^1 * 0.91^11) + (12C2 * 0.09^2 * 0.91^10) + (12C3 * 0.09^3 * 0.91^9)
P(X < 4) = 0.3225 + 0.3827 + 0.2082 + 0.0686
P(X < 4) = 0.9820

c)

n = 12 , p = 0.09

Here, μ = n*p = 1.08,

σ = sqrt(np(1-p)) = 0.9914