Chapter 9, Problem T9.11SPT

(a)

To determine

To Explain: the appropriate test at the $\alpha =0.05$ significance level.

Answer to Problem T9.11SPT

No

Explanation of Solution

Given:

  $\begin{array}{l}x=16\\ n=60\end{array}$

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 hypotheses:

  $\begin{array}{l}{H}_0;p=20\%=0.20\\ H_1:p>0.20\end{array}$

The sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{16}{60}=0.2667$

The test-statistic is

  $\begin{array}{c}z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}\\ =\frac{0.2667-0.2}{\sqrt{\frac{0.2\left(1-0.2\right)}{60}}}\\ =1.29\end{array}$

The P-value is the probability of getting the value of the test statistic

  $\begin{array}{c}P=P\left(Z>1.29\right)\\ =P\left(Z<-1.29\right)\\ =0.0985\end{array}$

If the P-value is lesser than the significance level, reject the null hypothesis:

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

There is not sufficient proof to help the claim.

(b)

To determine

To find: the more serious mistake in this setting a Type I error or Type II error, justify the answer.

Answer to Problem T9.11SPT

Type I error

Explanation of Solution

Given:

  $\begin{array}{c}{H}_0:p=20\%=0.20\\ H_1:p>0.20\end{array}$

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

Consequence: few people are willing to pay $100 than it seems from the results of the test.

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

Consequence: Many people are willing to pay $100 than it seems from the results of the test.

Type I error is worse, the reason is that then the upgrade is less likely to be profitable.

(c)

To determine

To Explain: the power of the test to detect this fact is 0.60, interpret this value.

Answer to Problem T9.11SPT

If 30% of the company’s customers would be willing to pay $100 for the upgrade, then have a 60% probability that find convincing proof to help the alternative hypothesis $H_1:p>0.20$

Explanation of Solution

Given:

  $\begin{array}{c}{P}_A=30\%\\ n=60\\ \text{Power=0}\text{.60=60\%}\end{array}$

Claim is more than 20%

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.

  $\begin{array}{c}{H}_0:p=20\%=0.20\\ H_1:p>0.20\end{array}$

The power is that the chance of rejecting the null hypothesis once the choice the alternative hypothesis is true.

If 30% of the company’s customers would be willing to pay $100 for the upgrade, then have a 60% probability that find convincing proof to help the alternative hypothesis $H_1:p>0.20$ .