Question

The Office of Student Services at UNC would like to estimate the proportion of UNC's 28,500 students who are foreign students. In their random sample of 50 students, 4 are foreign students. Unknown to them, the proportion of all UNC students that are foreign students is 0.061. For each student, let x=1 if the student is foreign and let x=0 if the student is from the U.S.Find the mean and the standard deviation of the sampling distribution of the sample proportion for a sample of size 50.

Answer 1

For the sampling distribution of the sample proportion for a sample of size 50, the mean is 0.061 and the standard deviation is 0.034.

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 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)}`

In this question:

`p = 0.061, n = 50`

So

Mean:

`\mu = p = 0.061`

Standard deviation:

`s = \sqrt{(0.061*0.939)/(50)} = 0.034`

For the sampling distribution of the sample proportion for a sample of size 50, the mean is 0.061 and the standard deviation is 0.034.