To determine the 90% confidence interval for the population mean when a simple random sample is drawn from a normally distributed population, follow these steps:
1. Identify the Given Values:
- Sample size [tex]\( n = 85 \)[/tex]
- Sample mean [tex]\( \bar{x} = 22 \)[/tex]
- Sample standard deviation [tex]\( s = 13 \)[/tex]
- [tex]\( z^* \)[/tex] value for a 90% confidence level, which from the table is [tex]\( 1.645 \)[/tex]
2. Calculate the Margin of Error:
The margin of error (E) is given by:
[tex]\[
E = z^* \cdot \left(\frac{s}{\sqrt{n}}\right)
\][/tex]
Plugging in the given values:
[tex]\[
E = 1.645 \cdot \left(\frac{13}{\sqrt{85}}\right) \approx 2.3195289202259812
\][/tex]
3. Determine the Confidence Interval:
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean.
- Lower bound:
[tex]\[
\text{Lower bound} = \bar{x} - E = 22 - 2.3195289202259812 \approx 19.68047107977402
\][/tex]
- Upper bound:
[tex]\[
\text{Upper bound} = \bar{x} + E = 22 + 2.3195289202259812 \approx 24.31952892022598
\][/tex]
4. State the Confidence Interval:
Therefore, the 90% confidence interval for the population mean is approximately:
[tex]\[
(19.68047107977402, 24.31952892022598)
\][/tex]
This interval means that we are 90% confident that the true population mean falls within the range [tex]\( 19.68047107977402 \)[/tex] to [tex]\( 24.31952892022598 \)[/tex].
