Question

The percentage of married couples who own a single family home is 33% for a given population. A financial analyst is interested in the impact that home mortgages have on a couple's finances. As the financial analyst sets up a study, they are curious about the impact of sample size. What is the standard error of the sampling distribution of sample proportions for samples of size n=200, n=300, and n=400? Round all answers to the nearest thousandths if applicable.

Answer 1

Answer:

For samples of size n=200, the standard error is of 0.033.

For samples of size n=300, the standard error is of 0.027.

For samples of size n=400, the standard error is of 0.024.

Step-by-step explanation:

Central Limit Theorem

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 $s = \frac{\sigma}{\sqrt{n}}$.

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

The percentage of married couples who own a single family home is 33% for a given population.

This means that $p = 0.33$

Samples of 200:

$s = \sqrt{\frac{0.33*0.67}{200}} = 0.033$

For samples of size n=200, the standard error is of 0.033.

Samples of 300:

$s = \sqrt{\frac{0.33*0.67}{300}} = 0.027$

For samples of size n=300, the standard error is of 0.027.

Samples of 400:

$s = \sqrt{\frac{0.33*0.67}{400}} = 0.024$

For samples of size n=400, the standard error is of 0.024.