College

Use the given information to construct a [tex]$95 \%$[/tex] confidence interval estimate of the mean of the population.

Given:
[tex]\[ n=62, \sigma=11.4, \bar{x}=102.1 \][/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 are provided with the following information:

- Sample size ([tex]\(n\)[/tex]) = 62
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 11.4
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 102.1

Here's how to determine the confidence interval:

1. Identify the Z-value for a 95% confidence level:
- For a 95% confidence level, the Z-value is approximately 1.96. This is derived from standard normal distribution tables.

2. Calculate the margin of error:
- The margin of error (E) is given by the formula:
[tex]\[
E = Z \times \left(\frac{\sigma}{\sqrt{n}}\right)
\][/tex]
- Plugging in the values, the margin of error is approximately 2.84.

3. Calculate the confidence interval:
- The confidence interval is calculated using the formula:
[tex]\[
\text{Confidence interval} = \left(\bar{x} - E, \, \bar{x} + E\right)
\][/tex]
- Substituting the values, the confidence interval is approximately [tex]\((99.3, 104.9)\)[/tex].

Thus, the 95% confidence interval estimate for the mean of the population is [tex]\(99.3 < \mu < 104.9\)[/tex]. This corresponds to the option: [tex]\(99.3 < \mu < 104.9\)[/tex].