Chapter 9.1, Problem 25E

(a)

To determine

To Explain: the Type-I error and Type-II error in this setting.

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:p=0.22\\ H_1:p<0.22\end{array}$

Type I error is rejecting the null hypothesis $H_0$ , once the null hypothesis $H_0$ is true.

There is enough convincing evidence that the true proportion of calls involving life-threatening injuries where the personnel took more than 8 minutes to arrive is less than 0.22, when the true proportion of calls involving life-threatening injuries where the personnel took more than 8 minutes to arrive is actually 0.22.

Type II error: fail to eject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is false

There is no enough convincing evidence that the true proportion of calls involving life- threatening injuries where the personnel took more than 8 minutes to arrive is less than 0.22, when the true proportion of calls involving life-threatening injuries where the personnel took more than to arrive is actually over 0.22.

(b)

To determine

To find: the type of error is more serious in this case, justify the answer.

Answer to Problem 25E

Type I error

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:p=0.22\\ H_1:p<0.22\end{array}$

Type I error: reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is true.

There is enough convincing evidence that the true proportion of calls involving life-threatening injuries where the personnel took more than 8 minutes to arrive is less than 0.22, when the true proportion of calls involving life-threatening injuries where the personnel took more than 8 minutes to arrive is actually 0.22.

Type II error: fail to eject the null hypothesis $H_0$ , when the null hypothesis $H_0$ is false

There is no convincing evidence that the true proportion of calls involving life- threatening injuries where the personnel took more than 8 minutes to arrive is less than 0.22, when the true proportion of calls involving life-threatening injuries where the personnel took more than to arrive is actually over 0.22.

Comparison

A type I error would be worse, the reason is that then it would be underestimating the time it takes for the personal to arrive and therefore people could then die because it takes too long for the personnel to arrive.

(c)

To determine

To Explain: the answer on the basis of part (b), agree with the manager’s choice of $\alpha =0.05$ yes or not.

Answer to Problem 25E

No, it would be better to use the significance level $\alpha =0.01$ instead of $\alpha =0.05$ .

Explanation of Solution

In the previous exercise, finding the type I error was worse and therefore it require to minimize the likelihood of a type I error.

Type I error: Reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is true.

Since the significance level $\alpha$ and therefore it would be better to use the significance level $\alpha =0.01$ instead of $\alpha =0.05$ .