Question

e) in general, all other aspects of the situation remain the same; will smaller margins of error produce greater or less confidence in the interval?

Explanation:

Step1: Understand the relationship between margin of error and confidence level

The formula for the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\) (for population mean when \(\sigma\) is known). The confidence level \(C = 1-\alpha\). A smaller margin of error \(E\) implies a smaller \(z_{\alpha/2}\) value.

Step2: Analyze the impact on confidence level

Since \(z_{\alpha/2}\) is related to the confidence level. As \(z_{\alpha/2}\) decreases (to get a smaller \(E\)), \(\alpha\) increases. And since \(C = 1 - \alpha\), when \(\alpha\) increases, the confidence level \(C\) decreases.

Answer:

Smaller margins of error produce less confidence in the interval.