Chapter 23, Problem 25E

(a)

To determine

To find out is this an upper-tail or lower-tail test and explain why do you think this is important.

Answer to Problem 25E

It is an upper-tailed.

Explanation of Solution

In this question, the manufacturer of a metal stand for home TV sets must be sure that its product will not fail under the weight of the TV. Since some larger sets weigh nearly $300$ pounds, the company’s safety inspectors have a set a standard of ensuring that the stands can support an average of $500$ pounds. It is also given that,

  $\begin{array}{l}{H}_0:\mu =500\\ H_1:\mu >500\end{array}$

By this, we know that if the alternative hypothesis contains $<$ , then the test is lower-tailed.

And if the alternative hypothesis contains $>$ , then the test is upper-tailed.

Thus, according to the information, it is a upper −tailed test. This is important because we want to know that the stands can support $500$ pounds (or more) easily.

(b)

To determine

To explain what will happen if the inspectors commit a Type I error.

Explanation of Solution

In this question, the manufacturer of a metal stand for home TV sets must be sure that its product will not fail under the weight of the TV. Since some larger sets weigh nearly $300$ pounds, the company’s safety inspectors have a set a standard of ensuring that the stands can support an average of $500$ pounds. It is also given that,

  $\begin{array}{l}{H}_0:\mu =500\\ H_1:\mu >500\end{array}$

Now, if the inspectors commit a Type I error then Type I error means reject the null hypothesis, when it is true. Thus, it means that the inspector will decide that the stands are safe while the standard are not safe.

(c)

To determine

To explain what will happen if the inspectors commit a Type II error.

Explanation of Solution

In this question, the manufacturer of a metal stand for home TV sets must be sure that its product will not fail under the weight of the TV. Since some larger sets weigh nearly $300$ pounds, the company’s safety inspectors have a set a standard of ensuring that the stands can support an average of $500$ pounds. It is also given that,

  $\begin{array}{l}{H}_0:\mu =500\\ H_1:\mu >500\end{array}$

Now, if the inspectors commit a Type II error then Type II error means fail to reject null hypothesis when it is false. Thus, it means that the inspectors will decide that the stands are unsafe while the standards are safe.