Let's solve the given problem step-by-step.
### Part a: Calculating the Standard Error of the Mean
The standard error of the mean (SEM) is calculated using the following formula:
[tex]\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Where:
- [tex]\(\sigma\)[/tex] is the population standard deviation
- [tex]\(n\)[/tex] is the sample size
Given:
- [tex]\(\sigma = 1\)[/tex]
- [tex]\(n = 10\)[/tex]
Let's plug in these values into the formula:
[tex]\[
\sigma_{\bar{x}} = \frac{1}{\sqrt{10}}
\][/tex]
Calculate [tex]\(\sqrt{10}\)[/tex]:
[tex]\[
\sqrt{10} \approx 3.162
\][/tex]
Then the standard error:
[tex]\[
\sigma_{\bar{x}} = \frac{1}{3.162} \approx 0.32
\][/tex]
So, the standard error of the mean is approximately:
[tex]\[
0.32
\][/tex]
### Part b: Calculating the Margin of Error at 95% Confidence
The margin of error (ME) at a given confidence level is calculated using the formula:
[tex]\[
ME = z \times \sigma_{\bar{x}}
\][/tex]
Where:
- [tex]\(z\)[/tex] is the z-value corresponding to the desired confidence level
- [tex]\(\sigma_{\bar{x}}\)[/tex] is the standard error of the mean
For a 95% confidence level, the z-value is approximately 1.96.
Given:
- [tex]\(\sigma_{\bar{x}} = 0.32\)[/tex]
- [tex]\(z = 1.96\)[/tex]
Let's plug in these values into the formula:
[tex]\[
ME = 1.96 \times 0.32
\][/tex]
Calculate the margin of error:
[tex]\[
ME \approx 1.96 \times 0.32 \approx 0.63
\][/tex]
So, the margin of error at 95% confidence is approximately:
[tex]\[
0.63
\][/tex]
### Summary of Results
a. The standard error of the mean, [tex]\(\sigma_{\bar{x}}\)[/tex], is [tex]\(0.32\)[/tex].
b. The margin of error at 95% confidence is [tex]\(0.63\)[/tex].
