Question

38â% of women consider themselves fans of professional baseball. You randomly select six women and ask each if she considers herself a fan of professional baseball. Complete partsâ (a) throughâ (d) below.(a) Find the mean of the binomial distribution.

μequals= ( ) (Round to the nearest tenth asâ needed.) â
(b) Find the variance of the binomial distribution.
sigmasquared= ( ) â(Round to the nearest tenth asâ needed.)
â(c) Find the standard deviation of the binomial distribution.
sigma = ( ) (Round to the nearest tenth asâ needed.) â
(d) Interpret the results in the context of theâ real-life situation.
Onâ average ( ) out of 6 women would consider themselves baseball fans. The standard deviation is ( ) âwomen, so in most samples of 6â women, the number of women who consider themselves baseball fans would differ from the mean by no more than ( ).â(Type integers or decimals rounded to the nearest tenth asâneeded.)

Answer 1

a) 2.3

b) 1.4

c) 1.2

d) On average, 2.3 out of 6 women would consider themselves baseball fans. The standard deviation is 1.2 women, so in most samples of 6 women, the number of women who consider themselves baseball fans would differ from the mean by no more than 1.2.

Explanation:

For each woman, there are only two possible outcoes. Either they are a fan of professional baseball, or they are not. The prbability of a woman being a fan of professional baseball is independent of other woman. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The variance of the binomial distribution is:

`V(X) = np(1-p)`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

38% of women consider themselves fans of professional baseball.

This means that
`p = 0.38`

Six women are sampled:

This means that
`n = 6`

(a) Find the mean of the binomial distribution.

`E(X) = np = 6*0.38 = 2.3`

(b) Find the variance of the binomial distribution

`V(X) = np(1-p) = 6*0.38*0.62 = 1.4`

(c) Find the standard deviation of the binomial distribution.

`√(V(X)) = √(np(1-p)) = √(6*0.38*0.62) = 1.2`

(d) Interpret the results in the context of theâ real-life situation.

On average, 2.3 out of 6 women would consider themselves baseball fans. The standard deviation is 1.2 women, so in most samples of 6 women, the number of women who consider themselves baseball fans would differ from the mean by no more than 1.2.