Chapter 9.1, Problem 22E

(a)

To determine

To Explain: the appropriate hypothesis.

Answer to Problem 22E

  $H_0:p=0.12$

  $H_1:p>0.12$

Explanation of Solution

Given:

Claim is proportion is higher than 0.12

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

  $H_0:p=0.12$

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.

  $H_1:p>0.12$

  $p$ is the proportion of Phillies fans who drive a red car.

(b)

To determine

To find: the reason there is some evidence for the alternative hypothesis.

Answer to Problem 22E

There is some proof for the alternative hypothesis, the reason is that the sample proportion of 0.1667 is greater than 0.12 and this corresponds with the claim of the alternative hypothesis that the proportion is greater than 0.12

Explanation of Solution

Given:

  $\begin{array}{l}x=\text{Number of succes = 35}\\ n\text{ = Sample size =210}\end{array}$

Claim is Proportion is greater than 0.12.

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 mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis states the opposite of the null hypothesis.

  $\begin{array}{l}{H}_0:p=0.12\\ H_1:p>0.12\end{array}$

The sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{35}{210}=\frac{1}{6}=0.1667$

There is some proof for the alternative hypothesis, the reason is that the sample proportion of 0.1667 is greater than 0.12 and this corresponds with the claim of the alternative hypothesis that the proportion is greater than 0.12

(c)

To determine

To Explain: the P-value.

Answer to Problem 22E

There is a 1.87% possibility that 35 or more out of 210 randomly selected Phillies fans own a red car, when the proportion of red cars among all Phillies fans is actually 0.12.

Explanation of Solution

Given:

  $\begin{array}{l}x=\text{Number of succes = 35}\\ n\text{ = Sample size =210}\\ P\text{=0}\text{.0187=1}\text{.87\%}\end{array}$

Claim is proportion is greater than 0.12

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.

  $\begin{array}{l}{H}_0:p=0.12\\ H_1:p>0.12\end{array}$

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.

There is a 1.87% possibility that 35 or more out of 210 randomly selected Phillies fans own a red car, when the proportion of red cars among all Phillies fans is actually 0.12.

(d)

To determine

To find: the conclusion would make at the $\alpha =0.05$

Answer to Problem 22E

There is enough convincing proof that the proportion of red cars among Phillies fans is greater than 0.12.

Explanation of Solution

Given:

  $\begin{array}{l}x=\text{ 35}\\ n\text{ = 210}\\ P\text{=0}\text{.0187=1}\text{.87\%}\\ \alpha \text{=0}\text{.05}\end{array}$

Claim is proportion is greater than 0.12.

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.

  $\begin{array}{l}{H}_0:p=0.12\\ H_1:p>0.12\end{array}$

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.

  $0.0187<0.05\Rightarrow \text{Reject }{H}_0$

There is enough convincing proof that the proportion of red cars among Phillies fans is greater than 0.12.