Chapter 9.3, Problem 92E

(a)

To determine

To find: the power of the test to detect that p=0.45 based on a random sample of 200 students and a significance level of $\alpha =0.05$ is 0.75 and interpret this value.

Answer to Problem 92E

If the proportion of the students at school who are satisfied with the parking situation after the change is 0.45, then have a 75% probability that find convincing proof to help the alternative hypothesis $H_1:p>0.37$ .

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:p=0.37\\ H_1:p>0.37\\ {\mu }_A=0.45\\ n=200\\ \alpha =0.05\\ \text{Power = 0}\text{.75=75\%}\end{array}$

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true.

If the proportion of the students at school who are satisfied with the parking situation after the change is 0.45, then have a 75% probability that find convincing proof to help the alternative hypothesis $H_1:p>0.37$ .

(b)

To determine

To Calculate: the probability of a Type-I error and the probability of a Type-II error for the test in part (a)

Answer to Problem 92E

  $\begin{array}{l}P\left(\text{Type I error}\right)=5\%\\ P\left(\text{Type II error}\right)\text{=25\%}\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:p=0.37\\ H_1:p>0.37\\ {\mu }_A=0.45\\ n=200\\ \alpha =0.05\\ \text{Power = 0}\text{.75=75\%}\end{array}$

The significance level $\alpha$ shows the probability of a type I error.

  $P\left(\text{Type I error}\right)=\alpha =0.05=5\%$

The probability of type II error is

  $P\left(\text{Type II error}\right)=1-\text{Power =1-0}\text{.75=0}\text{.25=25\%}$

(c)

To determine

To Explain: two ways to increase the power of the test in part (a).

Answer to Problem 92E

Increase significance level

Making the alternative proportion more extreme

Increase sample size

Explanation of Solution

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true.

You can increase the power by:

Increasing the sample size because the having more information about the population will allow making the best estimations.

Increase the significance level (because this increases the probability of making a Type I error and decreases the probability of making a Type II error.

Making the alternative proportion $p$ more extreme, therefore bigger