Question

suppose you compute a 95% confidence interval. what will happen to the confidence interval if you change the confidence level to 99%? the confidence interval will become select an answer if the confidence level is changed from 95% to 99%. question help: message instructor submit question

Explanation:

Step1: Recall the formula for confidence interval

The formula for a confidence interval for a population mean (when population standard deviation \(\sigma\) is known) is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\), where \(z_{\alpha/2}\) is the critical value. The width of the confidence interval is \(2z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\).

Step2: Compare critical values for 95% and 99% confidence levels

For a 95% confidence level, \(\alpha = 1 - 0.95=0.05\), and \(\alpha/2 = 0.025\). The critical value \(z_{0.025}\) from the standard normal distribution table is \(z = 1.96\).
For a 99% confidence level, \(\alpha=1 - 0.99 = 0.01\), and \(\alpha/2=0.005\). The critical value \(z_{0.005}\) from the standard normal distribution table is \(z = 2.576\).

Step3: Analyze the effect on the width of the confidence interval

Since the width of the confidence interval is proportional to \(z_{\alpha/2}\) (because \(n\) and \(\sigma\) remain the same when only the confidence level is changed), and \(z_{0.005}(2.576)>z_{0.025}(1.96)\)

Answer:

The confidence interval will become wider.