Chapter 9.1, Problem 20E

(a)

To determine

The hypotheses for the significance test to check whether the average response time has decreased or not.

Answer to Problem 20E

For null hypothesis $H_0$ proportion is equal to 0.78, for alternative hypothesis $H_a$ proportion is more than 0.78.

Explanation of Solution

Given Information:

Population proportion $p=78\%$

Population proportion is 78%.

Convert 78% into decimal.

  $\frac{78}{100}=0.78$

By the null hypothesis,proportionis equal to 0.78.

Thus, $H_0:p=0.78$ .

By the alternative hypothesis proportion is more than 0.78.

Thus, $H_a:p>0.78$ .

Hence, for null hypothesis $H_0$ proportion is equal to 0.78, for alternative hypothesis $H_a$ proportion is more than 0.78.

(b)

To determine

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

Answer to Problem 20E

In Type I error the response time has improved and the training does not have to be improved, but actually the training needs to improve.

In Type II error instead of the response time, training should be improving, but in reality the training should not be improved.

Explanation of Solution

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

The proportion is equal to 0.78 that the first responder arrived in 8 minutes after the accident, but it indicates that it is more.

Hence, the consequence is that the response time has improved and the training does not have to be improved, but actually the training needs to improve.

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

The proportion is more than 0.78 for the responder that arrived 8 minutes after the accident.

Hence, instead of the response time,training should beimproving, but in reality the training should not be improved.

(c)

To determine

Explain which type of error is more serious.

Answer to Problem 20E

Type I error is more serious.

Explanation of Solution

The training has improved in Type II error but in Type I error, the training is not improved but in reality training needs to be improved in Type I error.

Hence,Type I error is more serious.

(d)

To determine

Explain in which of the significance test of $H_o$ will be more interested for quick medical treatment.

Answer to Problem 20E

Population proportion of 0.78 will be better for quick medical treatment.

Explanation of Solution

From part (a) significance test, the population proportion of first responders who arrive within 8 minutes of an accident is more than 0.78.

Since the significance test population mean is less than 6.7 minutes.

Hence, more interested in higher population proportion.