(a) Finding the Standard Error of the Mean

The standard error of the mean (SE) is calculated using the formula:

$SE = \frac{\sigma}{\sqrt{n}}$

where:

• $\sigma$ is the population standard deviation.
• $n$ is the sample size.

Let's calculate the standard error for each situation:

1. For $\sigma = 44$, $n = 16$:

$SE = \frac{44}{\sqrt{16}} = \frac{44}{4} = 11.00$

2. For $\sigma = 44$, $n = 64$:

$SE = \frac{44}{\sqrt{64}} = \frac{44}{8} = 5.50$

3. For $\sigma = 44$, $n = 256$:

$SE = \frac{44}{\sqrt{256}} = \frac{44}{16} = 2.75$

(b) What happens to the standard error each time you quadruple the sample size?

If you compare the standard errors calculated:

• When the sample size $n$ increased from 16 to 64 (quadrupled), the standard error decreased from 11.00 to 5.50.
• When the sample size $n$ increased from 64 to 256 (quadrupled again), the standard error decreased from 5.50 to 2.75.

Thus, the standard error is reduced by half each time you quadruple the sample size.

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. How does the sample size affect the confidence interval?
2. Why is it important to understand the concept of standard error in statistics?
3. How does increasing the sample size impact the accuracy of the mean estimate?
4. Can the standard error be larger than the standard deviation? Why or why not?
5. How would the standard error change if the sample size was doubled instead of quadrupled?

Tip: Remember that the standard error of the mean decreases as the sample size increases, leading to a more precise estimate of the population mean.