Question

A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let ^ p denote the proportion in the sample who say they support the increase. Suppose that 34% of all adults in Ohio support the increase. The standard deviation of the sampling distribution is

Answer 1

The standard deviation of the sampling distribution is 0.0122 = 1.22%

Explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

A survey asks a random sample of 1500 adults in Ohio

This means that
`n = 1500`

34% of all adults in Ohio support the increase.

This means that
`p = 0.34`

The standard deviation of the sampling distribution is

`s = \sqrt{(0.34*0.66)/(1500)} = 0.0122`

The standard deviation of the sampling distribution is 0.0122 = 1.22%