To construct a 95% confidence interval for the population mean [tex]\(\mu\)[/tex] given a sample, we need to follow these steps:
1. Identify Given Information:
- Sample size ([tex]\(n\)[/tex]) = 14
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 8.5
- Sample standard deviation ([tex]\(s\)[/tex]) = 2.6
- Confidence level = 95%
2. Determine the t-critical value:
- Since the sample size is small (less than 30), we use the t-distribution.
- Degrees of freedom ([tex]\(df\)[/tex]) = [tex]\(n - 1 = 14 - 1 = 13\)[/tex].
- For a 95% confidence level, we look up the t-critical value ([tex]\(t^*\)[/tex]) corresponding to [tex]\(\frac{0.95 + 1}{2} = 0.975\)[/tex] in the t-distribution table for [tex]\(df = 13\)[/tex].
- This value is approximately 2.160.
3. Calculate the Standard Error of the Mean (SEM):
[tex]\[
\text{SEM} = \frac{s}{\sqrt{n}} = \frac{2.6}{\sqrt{14}} \approx 0.695
\][/tex]
4. Calculate the Margin of Error (ME):
[tex]\[
\text{ME} = t^* \times \text{SEM} = 2.160 \times 0.695 \approx 1.501
\][/tex]
5. Construct the Confidence Interval:
- Lower limit:
[tex]\[
\bar{x} - \text{ME} = 8.5 - 1.501 \approx 6.999
\][/tex]
- Upper limit:
[tex]\[
\bar{x} + \text{ME} = 8.5 + 1.501 \approx 10.001
\][/tex]
Therefore, the 95% confidence interval for the population mean [tex]\(\mu\)[/tex] is approximately:
[tex]\[
6.9988 < \mu < 10.0012
\][/tex]
Comparing with the options provided, the closest interval is:
[tex]\[ \boxed{7.0 < \mu < 10.0} \][/tex]
