To find the [tex]\(99\% \)[/tex] confidence interval for the population mean, we'll follow these steps:
1. Identify the sample size ([tex]\(n\)[/tex]): The sample size provided is [tex]\(n = 85\)[/tex].
2. Identify the confidence level: We are given a [tex]\(99\%\)[/tex] confidence level.
3. Determine the appropriate [tex]\(z^\)[/tex]-score: From the provided table, the [tex]\(z^\)[/tex]-score for a [tex]\(99\%\)[/tex] confidence level is [tex]\(2.58\)[/tex].
4. Calculate the margin of error (E):
The margin of error is given by the formula:
[tex]\[
E = z^* \times \frac{s}{\sqrt{n}}
\][/tex]
Here:
- [tex]\(z^* = 2.58\)[/tex]
- [tex]\(s\)[/tex] is the standard deviation of the sample (not provided in the question)
- [tex]\(\sqrt{n}\)[/tex] is the square root of the sample size, here [tex]\(\sqrt{85}\)[/tex]
5. Calculate the confidence interval:
The confidence interval is given by:
[tex]\[
\bar{x} \pm E
\][/tex]
This expands to:
[tex]\[
\left( \bar{x} - E, \bar{x} + E \right)
\][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean (not provided in the question)
- [tex]\(E\)[/tex] is the margin of error calculated above
In summary, to find the [tex]\(99\%\)[/tex] confidence interval for the population mean, you'll need the sample mean ([tex]\(\bar{x}\)[/tex]) and the standard deviation ([tex]\(s\)[/tex]) of the sample. If either of these values is not available, we cannot compute the exact confidence interval.
Since in this specific case the mean ([tex]\(\bar{x}\)[/tex]) and standard deviation ([tex]\(s\)[/tex]) are not provided, you cannot calculate the exact [tex]\(99\%\)[/tex] confidence interval for the population mean.
As a result:
"Mean of the sample ([tex]\(\bar{x}\)[/tex]) and standard deviation ([tex]\(s\)[/tex]) must be provided."
