Question

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, the mean response time to all accidents involving life-threatening injuries last year was m = 6.7 minutes. Emergency personnel arrived within 8 minutes on 78% of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to do better. At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times. Awful accidents (a) State hypotheses for a significance test to determine whether first responders are arriving within 8 minutes of the call more often. Be sure to define the parameter of interest. (b) Describe a Type I error and a Type II error in this setting and explain the consequences of each. (c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

Answer 1

Answer:

a)

$H_0: \pi\geq0.78\\\\H_a: \pi$

b) The Type I error occurs when we reject a null hypothesis that is actually true. In this case, it means we conclude that the arrival time have improved, when it didn't.

The Type II error occurs when we accept a null hypothesis that is actually false. In this case, although the arrival times have really improved, the evidence from the sample was not enough to show that improvement.

c) In this case, the Type I error is more serious, because it gives the wrong impression of improvement and no further actions will be taken to reduce the times.

Step-by-step explanation:

a) If you want to determine if the responders are arriving within 8 minutes of the call more often, you have to evaluate the proportion of accidents in which the arrival time is less than 8 minutes and compare it with the known proportion of π=0.78.

The sample parameter "p: proportion of accidents with arrival time of 8 minutes or less" will be used to test the hypothesis.

The null and alternative hypothesis will be:

$H_0: \pi\geq0.78\\\\H_a: \pi$