Chapter 9.1, Problem 26E

(a)

To determine

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

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:\mu =1.3\\ H_1:\mu >1.3\end{array}$

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

There is enough convincing proof that the true mean copper content of the water from the new source is over 1.3 mg/litre, when the true mean copper content of the water from the new source is 1.3 mg/litre.

Type II error is fails to reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is false

There is no enough convincing evidence that the true mean copper content of the water from the new source is 1.3 mg/litre, when the true mean copper content of the water from the new source is over 1.3 mg/litre.

(b)

To determine

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

Answer to Problem 26E

Type-II error

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:\mu =1.3\\ H_1:\mu >1.3\end{array}$

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

There is convincing evidence that the true mean copper content of the water from the new source is over 1.3 mg/litre, when the true mean copper content of the water from the new source is 1.3 mg/litre.

Type II error is fails to reject the null hypothesis $H_0$ , when the null hypothesis $H_0$ is false

There is no enough convincing evidence that the true mean copper content of the water from the new source is 1.3 mg/litre, when the true mean copper content of the water from the new source is over 1.3 mg/litre.

Comparison

A type II error would be worse, because people might drink the water when the water is not safe drinking water.

(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 26E

No, a significance level of $\alpha =0.10$ would be better than the significance level of $\alpha =0.05$

Explanation of Solution

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

Type II error: Fail to reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is false

The probability of a type II error decreases as the probability of a type I error increases and therefore a larger significance level $\alpha$ would be better (as the significance level represents the probability of a type I error). This than means that a significance level of $\alpha =0.10$ would be better than the significance level of $\alpha =0.05$