To determine the 95% confidence interval for the population mean, we need to use the given sample mean and the margin of error. Here's a detailed step-by-step solution:
### Step-by-Step Solution:
1. Understand the Given Data:
- Sample Mean ([tex]\(\bar{x}\)[/tex]): 18.7
- Margin of Error (E): 5.9
- Confidence Level: 95%
2. Confidence Interval Formula:
The formula for the confidence interval for the population mean ([tex]\(\mu\)[/tex]) is:
[tex]\[
\bar{x} \pm E
\][/tex]
where [tex]\(\bar{x}\)[/tex] is the sample mean and [tex]\(E\)[/tex] is the margin of error.
3. Calculate the Confidence Interval Limits:
- Lower Limit: [tex]\(\bar{x} - E = 18.7 - 5.9 = 12.8\)[/tex]
- Upper Limit: [tex]\(\bar{x} + E = 18.7 + 5.9 = 24.6\)[/tex]
4. Write the Confidence Interval:
Therefore, the 95% confidence interval for the population mean is:
[tex]\[
(12.8, 24.6)
\][/tex]
5. Verify the Answer Options:
Among the given options, the correct one must match the interval we calculated.
- [tex]\(18.7 \pm 5.9\)[/tex] results in [tex]\((12.8, 24.6)\)[/tex].
- [tex]\(18.7 \pm 9.7\)[/tex] results in [tex]\((9, 28.4)\)[/tex].
- [tex]\(18.7 \pm 11.6\)[/tex] results in [tex]\((7.1, 30.3)\)[/tex].
- [tex]\(18.7 \pm 15.2\)[/tex] results in [tex]\((3.5, 34.4)\)[/tex].
The only correct option is [tex]\(18.7 \pm 5.9\)[/tex].
### Conclusion:
The 95% confidence interval for the population mean is [tex]\(18.7 \pm 5.9\)[/tex], which corresponds to the interval (12.8, 24.6).
