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

Answer:

Standard error of: 2.47%

Step-by-step 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 $s = \frac{\sigma}{\sqrt{n}}$.

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 $s = \sqrt{\frac{p(1-p)}{n}}$

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:

$s = \sqrt{\frac{0.18*0.82}{242}} = 0.0247$

0.0247*100% = 2.47%

Standard error of: 2.47%