Question

suppose you compute a 98% confidence interval. what will happen to the confidence interval if you decrease the confidence level to 94%?
the confidence interval will widen.
the confidence interval will narrow.
the width of the confidence interval will stay the same.

Explanation:

Step1: Understand confidence - level and interval relationship

The confidence level is related to the critical - value. A higher confidence level corresponds to a larger critical - value. For example, in a normal distribution, as the confidence level increases, the z - value (or t - value depending on the situation) increases.

Step2: Analyze the change in critical - value

When we decrease the confidence level from 98% to 94%, the critical - value (z or t) decreases. The formula for a confidence interval (for example, for a population mean with known population standard deviation in a normal distribution) is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. The margin of error $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. Since $z_{\alpha/2}$ decreases when the confidence level decreases, the margin of error decreases.

Step3: Determine the change in confidence interval

A decrease in the margin of error results in a narrower confidence interval.

Answer:

The confidence interval will narrow.