To find the [tex]\(90 \% \)[/tex] confidence interval for the population mean given the provided data, follow these steps:
1. Identify the given values:
- Sample size ([tex]\( n \)[/tex]) = 90
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 138
- Sample standard deviation ([tex]\( s \)[/tex]) = 34
2. Determine the [tex]\( z \)[/tex]-score for a 90% confidence level:
- From the provided table, [tex]\( z^* \)[/tex]-score for a 90% confidence level is 1.645.
3. Calculate the margin of error (ME):
[tex]\[
ME = \frac{z^* \cdot s}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
ME = \frac{1.645 \cdot 34}{\sqrt{90}}
\][/tex]
4. Compute the margin of error:
[tex]\[
ME \approx \frac{1.645 \cdot 34}{9.486832980505138} \approx 5.895539651107248
\][/tex]
5. Determine the confidence interval:
- The formula for the confidence interval is:
[tex]\[
\left( \bar{x} - ME, \bar{x} + ME \right)
\][/tex]
- Substituting the values:
[tex]\[
\left( 138 - 5.895539651107248, 138 + 5.895539651107248 \right)
\][/tex]
[tex]\[
\left( 132.10446034889276, 143.89553965110724 \right)
\][/tex]
Therefore, the 90% confidence interval for the population mean is [tex]\( \left( 132.10, 143.90 \right) \)[/tex].
This interval suggests that we are 90% confident that the true population mean lies within the range from approximately 132.10 to 143.90.
