Question

At a university of 25,000 students, 18% are older than 25. The registrar will draw a simple random sample of 242 of the students. The percentage of students older than 25 in the sample has an expected value of 18% and a standard error of:______.

Answer 1

Standard error of: 2.47%

Explanation:

Central Limit Theorem

The Central Limit Theorem establishes 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)}`

18% are older than 25.

This means that
`p = 0.18`

Simple random sample of 242 of the students.

This means that
`n = 242`

Standard error:

By the Central Limit Theorem:

`s = \sqrt{(0.18*0.82)/(242)} = 0.0247`

0.0247*100% = 2.47%

Standard error of: 2.47%