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 after 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.

(a) State hypotheses for a significance test to determine whether the average response time has decreased. 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.

Short answer

a. Mean response time is $6.7$minutes for null hypothesis and less than $6.7$for alternative hypothesis

b. Type I error is training needs improvement and type II error is training needs no improvement

c. Type I error is more serious in this setting

Step-by-step solution

introduction

Given,

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.

Mean population is $125.7$

The sample size is $45$

explanation part (a)

Calculating the null and alternative hypotheses,

$H_0:\mu =6.7$

The mean response time is equal to $6.7$minutes as per the null hypothesis.

$H_0:\mu <6.7$

The mean response time is less $6.7$minutes as per the alternative hypothesis.

explanation part (b)

In Type l error, when the null hypothesis is valid then it is dismissed.

Accordingly, the mean reaction time is equal to $6.7$minutes, however, the test shows that the mean reaction time is less. Consequently, the reaction time has improved and the preparation doesn't need to be improved, however actually the preparation needs to move along.

In Type Il error, when the null hypothesis is valid then its dismissal also failed.

Along these lines, the mean reaction time is less than$6.7$ minutes, while the test demonstrates that it isn't less. Thus, rather than the reaction time, preparation should be improved, yet in reality, the preparation should not be gotten to the next level.

explanation part (c)

Type I error is more serious in this setting as the preparation has been worked on in Type II error however in Type I error, the preparation isn't worked on yet in reality preparing should be worked on in Type I error.