Question

A poll commissioned by Friends of the Earth and conducted by the Mellman Group found that 72% of American voters are in favor a carbon tax. Suppose that six voters in the United States are randomly sampled and asked whether they favor a carbon tax. Determine the probability that the number answering in the affirmative is a. Exactly four. b. At least two

Answer 1

a

`P(X = 4) = 0.31599`

b

`P(X  \ge 2) = 0.9921`

Explanation:

From the question we are told that

The proportion in favor of carbon tax is
`p = 0.72`

The sample size is n = 6

Generally the distribution of poll follows a binomial distribution

i.e

`X  \~ \ \ \  B(n , p)`

and the probability distribution function for binomial distribution is

`P(X = x) =  ^(n)C_x *  p^x *  (1- p)^(n-x)`

Here C stands for combination hence we are going to be making use of the combination function in our calculators

Generally the probability that the number answering in the affirmative is 4 is mathematically represented as

`P(X = 4) =  ^(6)C_4 *  (0.72)^4 *  (1- 0.72)^(6-4)`

=>
`P(X = 4) = 0.31599`

Generally the probability that the number answering in the affirmative is At least two is mathematically represented as

`P(X  \ge 2) =  1 -  P( X <  2 )`

`P(X  \ge 2) = 1- [ P(X =  0 ) + P( X = 1)]`

=>
`P(X  \ge 2) = 1- [ [ 1 *  1 * 0.00048 ]+ [ 6 * 0.72 * 0.00172]`

=>
`P(X  \ge 2) = 1- [ 0.00791 ]`

=>
`P(X  \ge 2) = 0.9921`