Question

a local courthouse has a metal detector to stop people from bringing weapons into the courthouse. as a person walks through the metal detector the alarm will sound if the machine detects any metal. let \\( \alpha = 0.05 \\). the hypotheses are: \\( h _ { 0 } \\): a weapon is detected. \\( h _ { 1 } \\): a weapon is not detected. what is a type ii error in this situation? the alarm sounds but the person really did not have a weapon on them and they were searched needlessly. the alarm does not sound when a person has a weapon on them and the armed person enters the courthouse. there is a 95% chance that the alarm will sound given that a person has a weapon. there is a 5% chance that the alarm will sound given that a person has a weapon.

Explanation:

A Type II Error is failing to reject the null hypothesis (\(H_0\)) when the alternative hypothesis (\(H_1\)) is true. In this context, \(H_0\) is "A weapon is detected" and \(H_1\) is "A weapon is not detected". So, a Type II Error would be when we do not reject \(H_0\) (alarm does not sound, meaning we assume no weapon is detected) but \(H_1\) is actually true (a weapon is present).

Answer:

The alarm does not sound when a person has a weapon on them and the armed person enters the courthouse.