(a) Find the standard error of the mean for each sampling situation (assuming a normal population). Round your answers to 2 decimal places.

| Situation | Standard Error |
|---|---|
| a. [tex]\(\sigma=26, n=4\)[/tex] | |
| b. [tex]\(\sigma=26, n=16\)[/tex] | |
| c. [tex]\(\sigma=26, n=64\)[/tex] | |

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

The standard error is reduced by half.



Answer :

Sure, let's solve this step-by-step.

### (a) Find the standard error of the mean for each sampling situation:
The standard error of the mean (SEM) is calculated using the formula:

[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \][/tex]

where [tex]\(\sigma\)[/tex] is the population standard deviation and [tex]\(n\)[/tex] is the sample size.

Let's calculate the SEM for each given situation:

1. Situation (a): [tex]\(\sigma = 26\)[/tex], [tex]\(n = 4\)[/tex]

[tex]\[ \text{SEM}_{\text{a}} = \frac{26}{\sqrt{4}} = \frac{26}{2} = 13.00 \][/tex]

2. Situation (b): [tex]\(\sigma = 26\)[/tex], [tex]\(n = 16\)[/tex]

[tex]\[ \text{SEM}_{\text{b}} = \frac{26}{\sqrt{16}} = \frac{26}{4} = 6.50 \][/tex]

3. Situation (c): [tex]\(\sigma = 26\)[/tex], [tex]\(n = 64\)[/tex]

[tex]\[ \text{SEM}_{\text{c}} = \frac{26}{\sqrt{64}} = \frac{26}{8} = 3.25 \][/tex]

So, the standard errors are:
- For (a): 13.00
- For (b): 6.50
- For (c): 3.25

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

Let's observe the trend when the sample size is quadrupled:
- From [tex]\(n = 4\)[/tex] to [tex]\(n = 16\)[/tex]: [tex]\( \text{SEM}_{\text{a}} \to \text{SEM}_{\text{b}} \)[/tex]
- From [tex]\(n = 16\)[/tex] to [tex]\(n = 64\)[/tex]: [tex]\( \text{SEM}_{\text{b}} \to \text{SEM}_{\text{c}} \)[/tex]

When the sample size is quadrupled:

[tex]\[ \text{Ratio of SEMs} = \frac{\text{SEM}_{n}}{\text{SEM}_{4n}} \][/tex]

For [tex]\(n = 4\)[/tex] to [tex]\(n = 16\)[/tex]:

[tex]\[ \text{Ratio} = \frac{13.00}{6.50} = 2.0 \][/tex]

For [tex]\(n = 16\)[/tex] to [tex]\(n = 64\)[/tex]:

[tex]\[ \text{Ratio} = \frac{6.50}{3.25} = 2.0 \][/tex]

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