Chapter PVI, Problem 10RE

(a)

To determine

To find out what percent of the sample signatures were valid.

Answer to Problem 10RE

  $88.6\%$ of the sample signatures were valid.

Explanation of Solution

It is given in the question that to get a voter initiative on a state ballot petitions that contain at least $250000$ valid voter signatures must be filed with the Elections Commission. Now, let us assume that:

  $\alpha =0.05$

And as we know that $1772\text{ of the 2000}$ signatures in the simple random sample were valid. The sample proportion is the number of successes divided by the sample size:

  $\begin{array}{l}\widehat{p}=\frac{x}{n}\\ =\frac{1772}{2000}\\ =0.886\\ =88.6\%\end{array}$

Thus, $88.6\%$ of the sample signatures were valid.

(b)

To determine

To find out what percent of the petitions signatures submitted must be valid in order to have the initiative certified by the Elections Commission.

Answer to Problem 10RE

  $82.18\%$ of the petitions signatures submitted must be valid in order to have the initiative certified by the Elections Commission.

Explanation of Solution

It is given in the question that to get a voter initiative on a state ballot petitions that contain at least $250000$ valid voter signatures must be filed with the Elections Commission. Now, as we know that $250000\text{ of the 304226}$ signatures should be valid to have the initiative certified, then the claim is either the null 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.

Thus, the population proportion is the number of successes divided by the sample size as:

  $\begin{array}{l}\widehat{p}=\frac{x}{n}\\ =\frac{250000}{304226}\\ =0.8218\\ =82.18\%\end{array}$

Thus, $82.18\%$ of the petitions signatures submitted must be valid in order to have the initiative certified by the Elections Commission.

(c)

To determine

To explain what will happen if the Elections Commission commits a Type I error.

Explanation of Solution

It is given in the question that to get a voter initiative on a state ballot petitions that contain at least $250000$ valid voter signatures must be filed with the Elections Commission.

Given claim is more than $82.18\%$ of the votes in the sample are valid. The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states 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 as:

  $\begin{array}{l}{H}_0:p=0.8218\\ H_a:p>0.8218\end{array}$

Type I error is defined when we reject the null hypothesis when the null hypothesis is true. Thus, this means the Election Commission certifies the initiative while it should not have certified it.

(d)

To determine

To explain what will happen if the Elections Commission commits a Type II error.

Explanation of Solution

It is given in the question that to get a voter initiative on a state ballot petitions that contain at least $250000$ valid voter signatures must be filed with the Elections Commission.

Given claim is more than $82.18\%$ of the votes in the sample are valid. The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states 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 as:

  $\begin{array}{l}{H}_0:p=0.8218\\ H_a:p>0.8218\end{array}$

Type II error is defined when we fail to reject the null hypothesis when the null hypothesis is false. Thus, this means the Election Commission does not certify the initiative while it should have certified it.

(e)

To determine

To explain does the sample provide evidence in support of certifications.

Answer to Problem 10RE

There is sufficient evidence to support the certification.

Explanation of Solution

It is given in the question that to get a voter initiative on a state ballot petitions that contain at least $250000$ valid voter signatures must be filed with the Elections Commission.

Given claim is more than $82.18\%$ of the votes in the sample are valid. The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states 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 as:

  $\begin{array}{l}{H}_0:p=0.8218\\ H_a:p>0.8218\end{array}$

Thus, to find out any test for proportion we will use the calculator $\text{TI}-89$ . Then select $5:1-$ PropZTest from the STAT TESTSmenu. Specify the hypothesized proportion. Enter the observed value of x . Specify the sample size. Indicate what kind of test you want: one-tail lower tail, two-tail, or one-tail upper tail. Specify whether to calculate the result or draw the result.

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}z=7.50\\ P=0\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

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

Thus, we conclude that there is sufficient evidence to support the certification.

(f)

To determine

To explain what could the Elections Commission do to increase the power of the test.

Answer to Problem 10RE

By increasing the sample size and increasing the significance level.

Explanation of Solution

It is given in the question that to get a voter initiative on a state ballot petitions that contain at least $250000$ valid voter signatures must be filed with the Elections Commission. Thus, to increase the power of the test, the Elections Commission should,

Firstly, increasing the sample size because having more information about the population will allow us to make better estimations.

Secondly, increasing the significance level because this increases the probability of making a Type I error and decrease the probability of making Type II error