To construct the confidence interval for the population mean, we can follow these steps:
1. Identify the given information:
- Sample mean, [tex]\( \bar{x} = 81.5 \)[/tex]
- Population standard deviation, [tex]\( \sigma = 10.2 \)[/tex]
- Sample size, [tex]\( n = 40 \)[/tex]
- Confidence level, [tex]\( c = 0.90 \)[/tex]
2. Determine the critical value (z-score) for the desired confidence level:
- For a 90% confidence level, the z-score can be found using standard z-tables or statistical tools, resulting in approximately [tex]\( z = 1.645 \)[/tex].
3. Calculate the standard error of the mean (SE):
- The standard error is given by:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
SE = \frac{10.2}{\sqrt{40}} \approx 1.613
\][/tex]
4. Compute the margin of error (ME):
- The margin of error is found by multiplying the z-score and the standard error:
[tex]\[
ME = z \times SE
\][/tex]
[tex]\[
ME = 1.645 \times 1.613 \approx 2.6528
\][/tex]
5. Calculate the confidence interval:
- The lower bound of the confidence interval is:
[tex]\[
\bar{x} - ME = 81.5 - 2.6528 \approx 78.847
\][/tex]
- The upper bound of the confidence interval is:
[tex]\[
\bar{x} + ME = 81.5 + 2.6528 \approx 84.153
\][/tex]
Based on these calculations, the confidence interval for the population mean is approximately [tex]\( (78.8, 84.2) \)[/tex].
Therefore, the correct answer is:
A. [tex]\( (78.8, 84.2) \)[/tex]
