To determine the 90% confidence interval for the population mean, we will 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
- Z-score for 90% confidence level ([tex]\( z^* \)[/tex]) = 1.645 (from the given table)
2. Calculate the standard error of the mean (SE):
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{34}{\sqrt{90}}
\][/tex]
Plugging in the values:
[tex]\[
SE \approx 3.5839
\][/tex]
3. Calculate the margin of error (ME):
[tex]\[
ME = z^* \times SE = 1.645 \times 3.5839
\][/tex]
Plugging in the values:
[tex]\[
ME \approx 5.8955
\][/tex]
4. Calculate the confidence interval:
- Lower bound = Sample mean - Margin of error
[tex]\[
Lower\ Bound = 138 - 5.8955 \approx 132.1045
\][/tex]
- Upper bound = Sample mean + Margin of error
[tex]\[
Upper\ Bound = 138 + 5.8955 \approx 143.8955
\][/tex]
Therefore, the 90% confidence interval for the population mean is approximately [tex]\( (132.10, 143.90) \)[/tex].
