Chapter 9, Problem 3CRE

(a)

To determine

To explain the null hypothesis and the alternative hypothesis and write their symbols

Explanation of Solution

Given information:

A company manufactures classroom chairs for the high school students. The mean breaking strength of the chairs is $300$ pounds. One chair with strength $220$ pounds collapsed.

Let $\mu$ be the strength of the chair.

The null hypothesis is the hypothesis that is considered to be true.

It is denoted as $H_0$ .

Define the null hypothesis $H_0:\mu =300$

One of the chairs with weight less than $300$ pounds collapsed.

Define the alternate hypothesis $H_1:\mu <300$

(b)

To determine

To explain: What is Type I and Type II error and what are the consequences of each.

Explanation of Solution

Type I error is defined as the error when $H_0$ is rejected when $H_0$ is true.

The null hypothesis $H_0:\mu =300$ .

The mean breaking strength of the chair is $300$ pounds.

Type II error is defined as the error when $H_0$ is accepted when $H_0$ is not true.

The alternate hypothesis is $H_1:\mu <300$ .

This indicates that the mean breaking strength of the chair is less than $300$ pounds.

In the given problem, Type II error indicate that the chances of collapsing a chair are less.

Type II error indicate that the chance that chair will collapse are more.

(c)

To determine

To find out: Out of the significance levels given below which is recommended.

1. 0.01
2. 0.05
3. 0.10

Explanation of Solution

Given Information : The three level of significance $0.01$ , $0.05$ and $0.10$ .

Type I error is defined as the error when $H_0$ is rejected when $H_0$ is true.

The null hypothesis $H_0:\mu =300$

Type II error is defined as the error when $H_0$ is accepted when $H_0$ is not true.

The alternate hypothesis is $H_1:\mu <300$ .

In response to the given problem, Type I error indicate that chair will collapse less.

Type II error indicate that the chair will collapse more often.

The main objective is to keep the Type II error as less as possible as its more serious.

This will happen when Type I error $\alpha$ is more.

The value $\alpha$ also signifies the level of significance.

Out of the given values $0.01$ , $0.05$ and $0.10$ .

The greatest value is $0.10$ .

Therefore, choose significance level $0.10$

(d)

To determine

To explain: the meaning of Power of Test.

Explanation of Solution

Given Information: The power of the test to detect $\mu =294$ is given as $0.71$

The null hypothesis $H_0:\mu =300$ .

The alternate hypothesis is $H_1:\mu <300$ .

The power of the test to detect $\mu =294$ is given as $0.71$

The power of the test is defined as the probability of rejecting the null hypothesis if the alternative hypothesis is true.

The power of the test is $0.71$ , this implies that when $\mu =294$ probability of rejecting the null hypothesis is $0.71$ .

(e)

To determine

To find out: the ways the increase the power of the test

Explanation of Solution

The following are the ways the increase the power of the test.

1. Increase the sample size: Increasing the sample size will give better estimation.
2. Increase the level of significance: This will increase the probability of committing a Type I error which will decrease the Type II error. Therefore, by definition it will increase the power of test.