Question

Better parking A local high school makes a change that should improve student satisfaction with the parking situation. Before the change, $37\%$ of the school’s students approved of the parking that was provided. After the change, the principal surveys an SRS of students at the school. She would like to perform a test of $$\begin{gathered} H_0:p=0.37 \\ {H}_a:p>0.37 \end{gathered}$$where p is the true proportion of students at school who are satisfied with the parking

situation after the change.

a. 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 $0.75$ Interpret this value.

b. Find the probability of a Type I error and the probability of a Type II error for the test in part (a).

c. Describe two ways to increase the power of the test in part (a).

Short answer

Part (a) If $0.45$of pupils at the school are satisfied with the parking situation following the modification then have a $75\%$probability that finds convincing proof to help the alternative hypothesis $H_1:p>0.37$

Part (b) $$\begin{gathered} P(TypeIerror)=5\% \\ P(TypeIIerror)=25\% \end{gathered}$$

Part (c) Increase significance level.

Making the alternative proportion more extreme.

Increase sample size.

Step-by-step solution

Part (a) Step 1: Given information

$$\begin{gathered} H_0:p=0.37 \\ {H}_1:p>0.37 \\ {\mu }_A=0.45 \\ n=200 \\ \alpha =0.05 \\ Power=0.75=75\% \end{gathered}$$

Part (a) Step 2: Explanation

When the alternative hypothesis is true, the power is the probability of rejecting the null hypothesis. If the proportion of kids at school who are content with the parking situation after the adjustment is $0.45$ there is a $75\%$ chance that the alternative hypothesis $H_1:p>0.37$ will be supported.

Part (b) Step 1: Explanation

The type I error likelihood is represented by the significance level.

$P(TypeIerror)=\alpha =0.05=5\%$

The probability of type II error is

$P(TypeIIerror)=1-Power=1-0.75=0.25=25\%$

Part (c) Step 1: Explanation

When the alternative hypothesis is true, the power is the probability of rejecting the null hypothesis. You can boost your power by Increasing the sample size because better estimations can be made with more information about the population. Increase the significance level (since doing so increases the likelihood of making a Type I error while lowering the likelihood of making a Type II error). Making the alternative percentage $p$larger by making it more severe.