Use the given information to construct a 95% confidence interval estimate of the mean of the population.

Given:
- [tex]n = 62[/tex]
- [tex]\sigma = 11.4[/tex]
- [tex]\bar{x} = 102.1[/tex]

Choose the correct confidence interval:

A. [tex]100.7 \ < \ \mu \ < \ 103.5[/tex]
B. [tex]101.7 \ < \ \mu \ < \ 102.5[/tex]
C. [tex]99.3 \ < \ \mu \ < \ 104.9[/tex]
D. [tex]66.1 \ < \ \mu \ < \ 138.1[/tex]

Answer :

To construct a 95% confidence interval estimate of the mean of the population, we will follow these steps using the provided data:

1. Identify the Given Information:
- Sample size ([tex]\(n\)[/tex]) = 62
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 11.4
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 102.1
- Confidence level = 95%

2. Determine the Z-Score:
For a 95% confidence interval, we need the z-score that corresponds to the area of 0.975 (since it is a two-tailed test, we take the central 95%, leaving 2.5% in each tail). This z-score is approximately 1.96.

3. Calculate the Standard Error of the Mean (SEM):
The standard error is calculated using the formula:
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
\text{SEM} = \frac{11.4}{\sqrt{62}} \approx 1.45
\][/tex]

4. Calculate the Margin of Error (ME):
The margin of error can be calculated using:
[tex]\[
\text{ME} = z \times \text{SEM}
\][/tex]
[tex]\[
\text{ME} = 1.96 \times 1.45 \approx 2.842
\][/tex]

5. Find the Confidence Interval:
Finally, the confidence interval is given by:
[tex]\[
\bar{x} - \text{ME} < \mu < \bar{x} + \text{ME}
\][/tex]
Plugging in the numbers:
[tex]\[
102.1 - 2.842 < \mu < 102.1 + 2.842
\][/tex]
[tex]\[
99.262 < \mu < 104.938
\][/tex]

Therefore, the 95% confidence interval estimate of the mean of the population is between approximately 99.3 and 104.9. This matches the choice [tex]$99.3 < \mu < 104.9$[/tex] from your options.