Question

A simple random sample of 450 residents in the state of New York is taken to estimate the proportion of people who live within one mile of a hazardous waste site. If 135 of the residents in the sample live within one mile of a hazardous waste site, what are the values of the sample proportion of people who live within one mile of a hazardous waste site and its standard error?

A. 0.3 and 0.01
B. 0.3 and 0.022
C. 0.3 and 0.21
D. 0.7 and 0.022

Answer 1

B. 0.3 and 0.022

Explanation:

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 sample proportion p in a sample of size n, the standard error is
`s = \sqrt{(p(1-p))/(n)}`

135 of the 450 residents sampled live within one mile of a hazardous waste site.

So the sample proportion is
`p = (135)/(450) = 0.3`

Standard error

`s = \sqrt{(0.3*0.7)/(450)} = 0.022`

So the correct answer is:

B. 0.3 and 0.022