Use the given information to construct a [tex]95\%[/tex] 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]

To construct a 95% confidence interval estimate of the mean of the population, follow these steps:

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

2. Determine the Critical Z-Value:
For a 95% confidence level, the critical Z-value is 1.96. This value corresponds to the standard normal distribution where 95% of the data falls within this Z-score.

3. Calculate the Standard Error:
The standard error (SE) of the sample mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]$ \sigma $[/tex] is the population standard deviation and [tex]$ n $[/tex] is the sample size. In this case:

[tex]\[
SE = \frac{11.4}{\sqrt{62}} \approx 1.448
\][/tex]

4. Compute the Margin of Error:
The margin of error (ME) is calculated using the formula:
[tex]\[
ME = Z \times SE
\][/tex]
where [tex]$ Z $[/tex] is the critical Z-value. In this example:

[tex]\[
ME = 1.96 \times 1.448 \approx 2.838
\][/tex]

5. Determine the Confidence Interval:
The confidence interval is found by subtracting and adding the margin of error from the sample mean:

- Lower Bound: [tex]$ \bar{x} - ME = 102.1 - 2.838 \approx 99.262 $[/tex]
- Upper Bound: [tex]$ \bar{x} + ME = 102.1 + 2.838 \approx 104.938 $[/tex]

6. Conclusion:
Therefore, the 95% confidence interval for the population mean is approximately [tex]$ (99.3, 104.9) $[/tex].

Among the given options, the correct choice that matches this interval is:
[tex]$ 99.3 < \mu < 104.9 $[/tex]