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.
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.
