To determine the 99% confidence interval for the population mean, we must follow a systematic method. Here's the step-by-step process:
1. Identify the Known Variables:
- Sample size ([tex]\( n \)[/tex]): 85
- Sample mean ([tex]\( \bar{x} \)[/tex]): 0 (assuming the sample mean is provided as 0)
- Sample standard deviation ([tex]\( s \)[/tex]): 1 (assuming the standard deviation of the sample is 1)
- Z-score for 99% confidence level ([tex]\( z' \)[/tex]): 2.58 (from the given table)
2. Calculate the Margin of Error:
The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = z' \cdot \frac{s}{\sqrt{n}}
\][/tex]
Plug in the known values:
[tex]\[
\text{Margin of Error} = 2.58 \cdot \frac{1}{\sqrt{85}}
\][/tex]
After performing the calculations, we obtain:
[tex]\[
\text{Margin of Error} \approx 0.27984029058606646
\][/tex]
3. Determine the Confidence Interval:
The confidence interval is found by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Confidence Interval} = \left( \bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error} \right)
\][/tex]
Substituting the values of [tex]\(\bar{x}\)[/tex] and the margin of error:
[tex]\[
\text{Confidence Interval} = \left( 0 - 0.27984029058606646, 0 + 0.27984029058606646 \right)
\][/tex]
Simplifying this gives us:
[tex]\[
\text{Confidence Interval} = \left( -0.27984029058606646, 0.27984029058606646 \right)
\][/tex]
4. Conclusion:
The 99% confidence interval for the population mean is:
[tex]\[
(-0.27984029058606646, 0.27984029058606646)
\][/tex]
This interval gives us a range in which we are 99% confident that the true population mean lies.
