Chapter 9.1, Problem 27E

(a)

To determine

To Explain: the appropriate hypotheses for performing significance test.

Answer to Problem 27E

  $H_0:p=10\%=0.10$

  $H_1:p>0.01$

Explanation of Solution

Given:

Claim is proportion is more than 10%

The null hypothesis statement is that the population value is equal to the value given in the claim:

  $H_0:p=10\%=0.10$

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.

  $H_1:p>0.01$

  $p$ is the population proportion of all students at the school who are left-handed.

(b)

To determine

To Explain: the P-value using the simulation results.

Answer to Problem 27E

There is a 12% possibility of getting at least 8 left-handed students in a sample of 50 students, when the proportion of left-handed students is 0.10

Explanation of Solution

Given:

  $\widehat{p}=8/50=0.16$

Claim is proportion is more than 10%

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.

  $\begin{array}{l}{H}_0:p=10\%=0.10\\ H_1:p>0.10\end{array}$

The P-value is the probability of getting the sample results or more extreme, when the null hypothesis is true.

In the dot plot, it is observed that 24 of the 200 dots are at 0.16 or to the right of 0.16.

  $\text{P-value =}\frac{24}{200}=\frac{3}{25}=0.12=12\%$

There is a 12% possibility of getting at least 8 left-handed students in a sample of 50 students, when the proportion of left-handed students is 0.10

(c)

To determine

To Explain: the conclusion would make.

Answer to Problem 27E

There is no sufficient convincing evidence that more than 10% of the students at the school are left-handed.

Explanation of Solution

Given:

  $\widehat{p}=8/50=0.16$

Claim is proportion is more than 10%

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.

  $\begin{array}{l}{H}_0:p=10\%=0.10\\ H_1:p>0.10\end{array}$

The P-value is the probability of getting the sample results or more extreme, once the null hypothesis is true.

In the dot plot, it is observed that 24 of the 200 dots are at 0.16 or to the right of 0.16.

  $\text{P-value =}\frac{24}{200}=\frac{3}{25}=0.12=12\%$

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

The common significance levels are $\alpha =0.01,\alpha =0.05$ and $\alpha =0.10$ . It is observed that the P-value is larger than all three common significance levels and therefore it is fails to reject the null hypothesis in each case.

There is no sufficient convincing evidence that more than 10% of the students at the school are left-handed.