Question

Awful accidents 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 life-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 times. She then performs a test at the $\alpha =0.05$level of $H_0$: $p=0.22$versus $H_a:p<0.22$, 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 I error and a Type II error in this setting.

b. Which type of error is more serious in this case? Justify your answer.

c. Based on your answer to part (b), do you agree with the manager’s choice of $\alpha =0.05$? Why or why not?

Short answer

a. Type I error is rejecting null hypothesis $H_0$, once $H_0$is true and Type II error fail to reject null hypothesis when $H_0$is false.

b. In this case, type II error is more serious.

c. It is better to use$\alpha =0.05$.

Step-by-step solution

Given Information

It is given that $H_0:p=0.22$

$H_1:p<0.22$

Explaining type I and type II error

Type I error:Enough convincing evidence is present that correct proportion of calls involves life threatening injuries in case personnel takes more than $8$minutes to arrive is $<0.22$when correct proportion of calls involves life threatening injuries in case personnel takes more than $8$minutes to arrive is actually $0.22$

Type II error: No convincing evidence is present that correct proportion of calls involves life threatening injuries in case personnel takes more than $8$minutes to arrive is less than $0.22$when correct proportion of calls involves life threatening injuries in case personnel takes more than $8$minutes to arrive is actually over$0.22$

More serious type of error

Comparison given type I error is worse as it leads to underestimating the time it takes for personal to come and it results in death of people.

Use of significance level

Since type I error is worse,.

Hence, it is better to use $\alpha =0.01$instead of$0.05$