To find the [tex]\(99 \%\)[/tex] confidence interval for the population mean, we'll follow these steps:
### Step 1: Identify the given values
- Sample size: [tex]\( n \)[/tex]
- Sample mean: [tex]\( \bar{x} \)[/tex]
- Sample standard deviation: [tex]\( s \)[/tex]
- Confidence level: [tex]\( 99\% \)[/tex]
- [tex]\( z^ \)[/tex]-score for [tex]\( 99\% \)[/tex] confidence level from the provided table: [tex]\( 2.58 \)[/tex]
### Step 2: Identify the Margin of Error (MOE)
The margin of error for a confidence interval (CI) is calculated using the formula:
[tex]\[ \text{Margin of Error} = z^ \cdot \left( \frac{s}{\sqrt{n}} \right) \][/tex]
For a [tex]\( 99\% \)[/tex] CI:
[tex]\[ z^* = 2.58 \][/tex]
Hence,
[tex]\[ \text{Margin of Error} = 2.58 \cdot \left( \frac{s}{\sqrt{n}} \right) \][/tex]
### Step 3: Calculate the Confidence Interval
The general form of a confidence interval for a population mean is:
[tex]\[ \bar{x} \pm \text{Margin of Error} \][/tex]
Substituting the margin of error we found:
[tex]\[ \bar{x} \pm 2.58 \cdot \left( \frac{s}{\sqrt{n}} \right) \][/tex]
### Step 4: Conclusion
The [tex]\(99 \%\)[/tex] confidence interval for the population mean is:
[tex]\[ \bar{x} \pm \frac{2.58 \cdot s}{\sqrt{n}} \][/tex]
Thus, the correct answer, based on the options provided, is:
[tex]\[ \boxed{\bar{x} \pm \frac{2.58 \cdot s}{\sqrt{n}}} \][/tex]
