Chapter 9.1, Problem 21E

(a)

To determine

The hypotheses for the significance test based on the mean income to check whether to open a restaurant or not.

Answer to Problem 21E

For null hypothesis $H_0$ mean income is equal to $85,000, for alternative hypothesis $H_a$ mean income is more than $85,000.

Explanation of Solution

Given Information:

  $\text{Estimated mean income}\left(\mu \right)=\$85,000$.

By the null hypothesis, mean incomeis equal to $85,000.

Thus, $H_0:\mu =\$85,000$

By the alternative hypothesis, mean income is more than $85,000.

Thus, $H_a:\mu >\$85,000$

Hence, for null hypothesis $H_0$ mean income is equal to $85,000, for alternative hypothesis $H_a$ mean income is more than $85,000.

(b)

To determine

Explain Type I error and Type II error and their consequences.

Answer to Problem 21E

In Type I error the mean income is equal to $85,000 but the test indicates that the mean income is more.

In Type II error mean income is more than $85,000 while the test indicates that the mean income is equal to $85,000.

Explanation of Solution

Given Information:

  $\text{Estimated mean income}\left(\mu \right)=\$85,000$.

In Type I error, when the null hypotheses is true than $H_o$ is rejected.

The mean income is equal to $85,000but the test indicates that the mean income is more.

Hence, the consequence is that the restaurant will open in that location since the mean income more than the required income, but in reality it is not a good location for the restaurant.

In Type II error,when the null hypotheses is true than rejection of $H_o$ also failed.

The mean income is more than $85,000 while the test indicates that the mean income is equal to $85,000.

Hence, the consequence is that the restaurant will not open in that location since the mean income smaller than the required income, but in reality it is a good location for the restaurant.

(c)

To determine

Explain which of the significance level of $\alpha$ will be better for the significance test.

Answer to Problem 21E

Significance level $\alpha =0.01$ will be better to minimize the probability of making the wrong decision.

Explanation of Solution

Given Information:

  $\text{Estimated mean income}\left(\mu \right)=\$85,000$.

  $\text{Significance level}\left(\alpha \right)=0.01or0.05$ .

To open a restaurant,requires a large investment. Thus, there is a chance of making wrong decision which can cost a lot of money.

Hence, smaller significance level $\alpha =0.01$ will be better to minimize the probability of making the wrong decision.