Construct a 90% confidence interval for the sample mean based on the following
data: x=145, n=90, and σ = 3.8.
(1 point)
O (144.068, 145.932)
(144.215, 145.785)
(144.341, 145.659)
O (144.599, 145.401)



Answer :

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.