Chapter 9.2, Problem 45E

(a)

To determine

To Explain: a Type-I error and a Type-II error in this setting, and give a possible consequence of each.

Explanation of Solution

Given:

Claim is proportion is higher than 37% (effective change)

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is that the population proportion is equal to the value given in the claim. If the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $\begin{array}{c}{H}_0:p=37\%=0.37\\ H_1:p>0.37\end{array}$

Type I error is rejecting the null hypothesis $H_0$ , once the null hypothesis $H_0$ is true.

There is enough convincing evidence that the proportion of students who approve of the new parking arrangement is bigger than 0.37, when the proportion of students who approve of the new parking arrangement is really 0.37.

A possible consequence is that it is concluded that the change was effective, where the change was actually not effective and therefore executing the change could lead to a large waste of money.

Type II error is fails to reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is false.

There is not convincing evidence that the proportion of students who approve of the new parking arrangement is really bigger than 0.37.

A possible consequence is that it is concluded that the change is not effective, when it actually was effective and therefore would then not improve the parking,

(b)

To determine

To Explain: that there is convincing evidence that the principal’s claim is true.

Answer to Problem 45E

No

There is no convincing evidence that the principal’s claim is true.

Explanation of Solution

Given:

  $\begin{array}{c}\alpha =0.05\\ n=200\\ x=83\end{array}$

Formula used:

  $\begin{array}{c}\widehat{p}=\frac{x}{n}\\ z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}\end{array}$

Calculation:

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportion is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis states the opposite of the null hypothesis.

  $\begin{array}{c}{H}_0:p=37\%=0.37\\ H_1:p>0.37\end{array}$

Normal: satisfied, because

  $\begin{array}{c}n{p}_0=200\left(0.37\right)\\ =74\\ \text{and }n\left(1-p_0\right)=200\left(1-0.37\right)\\ =200\left(0.63\right)\\ =126\end{array}$

are both at least 10.

Since all conditions are satisfied, it is appropriate to use a hypothesis test for the population proportion $p$ .

The sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{83}{200}=0.415$

The test-static is

  $\begin{array}{c}z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}\\ =\frac{0.415-0.37}{\sqrt{\frac{0.37\left(1-0.37\right)}{200}}}\\ =1.32\end{array}$

The P-value is the probability of getting the value of the test statistic, or a value more extreme, when the null hypothesis is true, find the P-value using the normal probability table

  $\begin{array}{c}P=P\left(z>1.32\right)\\ =1-P\left(Z<1.32\right)\\ =1-0.9066\\ =0.0934\end{array}$

If the P-value is lesser than the significance level $\alpha$ , then reject the null hypothesis:

  $P>0.05\Rightarrow \text{ Fail to reject }{H}_0$

There is no convincing evidence that the principal’s claim is true.