Chapter 23, Problem 28E

(a)

To determine

To explain will the probability of a Type II error increase, decrease or remain the same.

Answer to Problem 28E

The probability of a Type II error increases.

Explanation of Solution

In the previous question, during an angiogram, heart problems can be examined via a small tube threaded into the heart from a vein in the patient’s leg. It is important that the company that manufacturers the catheter maintain a diameter of $2$ mm. It is also given that,

  $\begin{array}{l}{H}_0:\mu =2\\ H_1:\mu \ne 2\end{array}$

  $\alpha =0.05$

Now, suppose that the significance level is changed to $\alpha =0.01$ . This means that the significance level is decreased. Thus, if the significance level $\alpha$ decreases then the probability of making Type I error decreases and the probability of making a Type II error increases.

(b)

To determine

To explain what is meant by the power of test the company conducts.

Explanation of Solution

In the previous question, during an angiogram, heart problems can be examined via a small tube threaded into the heart from a vein in the patient’s leg. It is important that the company that manufacturers the catheter maintain a diameter of $2$ mm. It is also given that,

  $\begin{array}{l}{H}_0:\mu =2\\ H_1:\mu \ne 2\end{array}$

The power of test is the probability to reject the null hypothesis when it is false. Thus, the power of test interprets that the probability of deciding that the quality is bad when the quality is actually bad.

(c)

To determine

To explain will the power of the quality control test increase, decrease or remain the same.

Answer to Problem 28E

The power of the quality control test will increase.

Explanation of Solution

In the previous question, during an angiogram, heart problems can be examined via a small tube threaded into the heart from a vein in the patient’s leg. It is important that the company that manufacturers the catheter maintain a diameter of $2$ mm. It is also given that,

  $\begin{array}{l}{H}_0:\mu =2\\ H_1:\mu \ne 2\end{array}$

Now, if the desired mean is farther and farther above the desired $2$ mm then it will become easier to reject the null hypothesis and thus the power of the quality control test will increase.

(d)

To determine

To explain what could they do to improve the power of the test.

Explanation of Solution

In the previous question, during an angiogram, heart problems can be examined via a small tube threaded into the heart from a vein in the patient’s leg. It is important that the company that manufacturers the catheter maintain a diameter of $2$ mm. It is also given that,

  $\begin{array}{l}{H}_0:\mu =2\\ H_1:\mu \ne 2\end{array}$

Now, they could do to improve the power of the test that they could increase the sample size, decrease the standard deviation and increase the significance level. If we increase the sample size then the advantage of this is the increase in the power and the disadvantage of this is that it is more costly and very time consuming. If we decrease the standard deviation then the advantage is that it increases the power and the disadvantage is that it becomes more costly. Now, if we increase $\alpha$ then the advantage is that it will increase the power and the disadvantage of this will be that it will be of more Type I error.