To construct a [tex]\(90\%\)[/tex] confidence interval for estimating the population mean [tex]\(\mu\)[/tex], we follow these steps:
1. Identify the known information:
- Sample size ([tex]\(n\)[/tex]) = 50
- Sample mean ([tex]\(\bar{x}\)[/tex]) = \[tex]$68,900
- Standard deviation (\(\sigma\)) = \$[/tex]19,773
- Confidence level = 90%
2. Determine the z-score for the 90% confidence level:
At a 90% confidence level, the z-score, [tex]\(z_{\alpha/2}\)[/tex], is obtained from the standard normal distribution table. For a 90% confidence level, the value of [tex]\(\alpha\)[/tex] is 1 - 0.90 = 0.10. So, [tex]\(\alpha/2 = 0.05\)[/tex].
The z-score corresponding to the 95th percentile (0.5 + 0.45) in a standard normal distribution is approximately 1.645.
3. Calculate the margin of error (E):
The margin of error is given by:
[tex]\[
E = z_{\alpha/2} \times \left(\frac{\sigma}{\sqrt{n}}\right)
\][/tex]
Plugging in the values:
[tex]\[
E = 1.645 \times \left(\frac{19,773}{\sqrt{50}}\right)
\][/tex]
Given the final value from the calculation, we have:
[tex]\[
E = 4599.544
\][/tex]
4. Construct the confidence interval:
The confidence interval is given by:
[tex]\[
\left(\bar{x} - E, \bar{x} + E\right)
\][/tex]
Substituting the values:
[tex]\[
\left(68,900 - 4599.544, 68,900 + 4599.544\right)
\][/tex]
Hence,
[tex]\[
\left(64,300.456, 73,499.544\right)
\][/tex]
5. Rounding to the nearest integer:
[tex]\[
\left(64,300, 73,499\right)
\][/tex]
Therefore, the [tex]\(90\%\)[/tex] confidence interval for estimating the population mean [tex]\(\mu\)[/tex] is:
[tex]\[
\$64,300 < \mu < \$73,499
\][/tex]
