The Problem:
Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. IN the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, emergency personnel took more than 8 minutes to arrive on 22% of all calls involving threatening injuries last year. The city manager shares this information and encourages these first responders to "do better". After 6 months, the city manager selects an SRS of 400 calls involving life threatening injuries and examines the response time. She then performs a test at the alpha level= 0.05 level of null hypothesis: p=0.22 vs alternative hypothesis: p<0.22, where p is the true proportion of calls involving life threatening injuries during this 6 month period for which emergency personnel took more than 8 minutes to arrive.
A) Describe a Type 1 error and a Type 2 error in this setting.
My answer:
Type 1: The emergency personnel took more than 8 minutes to arrive on 0.22 of the calls, but they didn't do better.
Type 2: The emergency personnel took less than 8 minutes to arrive on 0.22 of the calls, but they tried to do better.
Is the way I described it wrong?
Your description of Type 1 and Type 2 errors is not entirely accurate. Let me explain the correct definitions:
Type 1 error: In this context, a Type 1 error would occur if the city manager rejects the null hypothesis (p = 0.22) and concludes that the emergency personnel did not do better, even though the true proportion of calls where they took more than 8 minutes to arrive is actually less than 0.22. In simpler terms, it means falsely concluding that there is a problem with the emergency response times when there isn't.
Type 2 error: A Type 2 error would occur if the city manager fails to reject the null hypothesis (p = 0.22) and concludes that the emergency personnel did better, even though the true proportion of calls where they took more than 8 minutes to arrive is actually greater than 0.22. In other words, it means failing to detect a problem with the emergency response times when there actually is one.
So to summarize:
Type 1 error: Falsely concluding there is a problem when there isn't.
Type 2 error: Failing to detect a problem when there actually is one.
I hope this clears up any confusion! Let me know if you have any further questions.