Question

Suppose the true proportion of high school juniors who skateboard is 0.18. If many random samples of 250 high school juniors are taken, by how much would their sample proportions typically vary from the true proportion?
0

0.0006

0.024

0.148

6.07

Answer 1

Answer:

0.024

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

Suppose the true proportion of high school juniors who skateboard is 0.18.

This means that $p = 0.18$

Samples of 250 high school juniors are taken

This means that $n = 250$

By how much would their sample proportions typically vary from the true proportion?

This is the standard error, so:

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

$s = \sqrt{\frac{0.18*0.82}{250}}$

$s = 0.024$

So 0.024 is the answer.