To construct a 90% confidence interval for the sample mean, we can use the formula:
\[ \bar{x} \pm Z_{\alpha/2} \times \left( \frac{\sigma}{\sqrt{n}} \right) \]
Given:
- Sample mean (\( \bar{x} \)): 145
- Sample size (\( n \)): 90
- Population standard deviation (\( \sigma \)): 3.8
- Confidence level: 90%
First, we need to find the critical value, \( Z_{\alpha/2} \), which corresponds to the confidence level. For a 90% confidence interval, the critical value is approximately 1.645.
Substitute the values into the formula:
\[ \bar{x} \pm 1.645 \times \left( \frac{3.8}{\sqrt{90}} \right) \]
Calculate the margin of error:
\[ 1.645 \times \left( \frac{3.8}{\sqrt{90}} \right) \approx 0.732 \]
Now, we can calculate the confidence interval:
\[ 145 \pm 0.732 \]
Therefore, the 90% confidence interval for the sample mean is approximately (144.268, 145.732).
Out of the given options, the closest interval is (144.215, 145.785), which is the best choice for the 90% confidence interval based on the provided data.
