Question

A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 6% to 7%, with the additional revenue going to education. Let X denote the number in the sample that say they support the increase. Suppose that 40% of all adults in Ohio support the increase.

a. The mean of X is:
b.The standard deviation of X is?

Answer 1

The mean of X is 600 and the standard deviation of X is 18.97.

Step-by-step explanation:

The mean of X, denoted by μ, can be calculated using the formula for the mean of a binomial distribution, which is given by μ = n * p, where n is the sample size and p is the probability of success.

In this case, the sample size is 1500 and the probability of supporting the increase is 0.4. Therefore, the mean of X is 1500 * 0.4 = 600.

The standard deviation of X, denoted by σ, can be calculated using the formula for the standard deviation of a binomial distribution, which is given by σ = √(n * p * (1 - p)).

In this case, the sample size is 1500 and the probability of supporting the increase is 0.4. Therefore, the standard deviation of X is √(1500 * 0.4 * (1 - 0.4)) = √360 = 18.97.

Answer 2

a) 600

b) 18.97

Step-by-step explanation:

For each adult, there are only two possible outcomes. Either they support the increase, or they do not. The probability of an adult supporting the increase is independent of other adults. 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.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Sample of 1500 adults

So
`n = 1500`

Suppose that 40% of all adults in Ohio support the increase.

This means that
`p = 0.4`

a. The mean of X is:

`E(X) = np = 1500*0.4 = 600`

b.The standard deviation of X is?

`√(V(X)) = √(np(1-p)) = √(1500*0.4*0.6) = 18.97`