Question

47.5% of children say that chocolate chip cookie is their favorite kind of cookie. If you randomly select 4 children find the probability that: a) exactly 2 say that chocolate chip cookie is their favorite. c) at least one of the 4 say that chocolate chip cookie is their favorite. d) Find the mean and standard deviation for this sample.

Answer 1

a) 0.3731

b) 0.9241

c) Mean = 1.9

Standard Deviation = 0.9987

Explanation:

We are given the following information:

P(chocolate chip cookie) = 47.5% = 0.475

Then the number of children follows a binomial distribution, where

`P(X=x) = \binom{n}{x}.p^x.(1-p)^(n-x)`

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 4

a) x = 2

We have to evaluate:

`P(x =2)\\= \binom{4}{2}(0.475)^2(1-0.475)^(4-2)\\= 0.3731`

b) x = 1

We have to evaluate

`P(x \geq 1) = 1 - P(x = 0)\\= 1 - \binom{4}{0}(0.475)^0(1-0.475)^(4-0)\\= 1 - 0.0759\\= 0.9241`

c) Mean and standard deviation of distribution

`\text{Mean} = np = 4* 0.475 = 1.9\\\text{Standard deviation} = √(np(1-p)) = √(4* 0.475(1-0.475)) = 0.9987`