Chapter 10, Problem 86CR

The report “A Crisis in Civic Education” (American Council of Trustees and Alumni, January 2016, goacta.org/images/download/A_Crisis_in_Civic_Education.pdf, retrieved November 30, 2016) summarizes data from a survey of a representative sample of 1000 adult Americans regarding their understanding of U.S. government. Only 459 of the adults in the sample were able to give a correct response to a question asking them to choose a correct definition of the Bill of Rights from a list of five possible answers. Using a significance level of 0.01, determine if there is convincing evidence that less than half of adult Americans could identify the correct definition of the Bill of Rights.

To determine

Check whether the sample data provide convincing evidence that less than 50% of adults of Country A identified the correct definition of the Bill of Rights at a significance level, $\alpha =0.01$.

Answer to Problem 86CR

The data provide the convincing evidence that less than 50% of adults of Country A identified the correct definition of the Bill of Rights at a significance level, $\underset{\_}{\alpha =0.01}$.

Explanation of Solution

Calculation:

In a sample of 1,000 adults of Country A, who responded to the given survey regarding their understanding of the U.S government, 459 were found to know the correct definition of the Bill of Rights. It is assumed that the sample of adults of Country A is representative of all the adults of Country A in the U.S.

Step 1:

Population characteristic of interest:

Let p be the population characteristic of interest.

In this context, p denotes population proportion of adults of Country A who identified the correct definition of the Bill of Rights.

Step 2:

Null hypothesis: $H_0:p=0.5$

That is, the proportion of adults of Country A, who identified the correct definition of the Bill of Rights, is 0.5.

Step 3:

Alternative hypothesis: $H_a:p<0.5$

That is, the proportion of adults of Country A, who identified the correct definition of the Bill of Rights, is less than 0.5.

Step 4:

Significance level, $\alpha$:

It is given that the significance level, $\alpha =0.01$.

Step 5:

Test statistic, z:

$z=\frac{\widehat{p}-p}{\sqrt{\frac{p\left(1-p\right)}{n}}}$,

Where, $\widehat{p}$ is the sample proportion, p is the hypothesized proportion, and n be the sample size.

By substituting the hypothesized proportion, $p=0.5$ in the test statistic, the following can be obtained:

$z=\frac{\widehat{p}-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{n}}}$

Here, the sample proportion $\widehat{p}$ is not known.

Step 6:

Assumptions:

• Let $\widehat{p}$ be the sample proportion from a random sample.
• The large sample z test can be used if the sample size n satisfies the conditions: $n\left(\text{hypothesized value}\right)\ge 10$ and $n\left(1-\text{hypothesized value}\right)\ge 10$.
• The sample size should not be greater than 10% of the population size.

Requirement check:

• It is assumed that the sample is from a random sample.
• Check the conditions: $n\left(\text{hypothesized value}\right)\ge 10$ and $n\left(1-\text{hypothesized value}\right)\ge 10$.

$\begin{array}{c}n\left(\text{hypothesized value}\right)=np\\ =1,000\left(0.5\right)\\ =500\\ >10\end{array}$

$\begin{array}{c}n\left(1-\text{hypothesized value}\right)=n\left(1-p\right)\\ =1,000\left(1-0.5\right)\\ =1,000\left(0.5\right)\\ =500\\ >10\end{array}$

Since $n\left(\text{hypothesized value}\right)$ and $n\left(1-\text{hypothesized value}\right)$ are greater than 10, the sample size requirement is met.

Therefore, the large sample z test is appropriate.

• Although the population size is not known, it is reasonable to assume that the sample size of 1,000 acts as the representative for all the adults in the U.S. It is also definite that the sample size is less than 10% of the population of the adults in the U.S.

Step 7:

The value of the test statistic is obtained as follows:

$z=\frac{\widehat{p}-0.1}{\sqrt{\frac{0.1\left(1-0.1\right)}{n}}}$

The sample proportion, $\widehat{p}$ is obtained as follows:

$\widehat{p}=\frac{x}{n}$

Where, x is the number of observations of interest, and n be the sample size.

Substitute the corresponding values to get the sample proportion:

$\begin{array}{c}\widehat{p}=\frac{459}{1,000}\\ =0.459\end{array}$

By substituting the value of sample proportion, the test statistic is obtained as follows:

$\begin{array}{c}z=\frac{0.459-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{1,000}}}\\ =\frac{-0.041}{0.0158}\\ \approx -2.59\end{array}$

Thus, the value of test statistic is –2.59.

Step 8:

P-value:

In this context, the alternative hypothesis denotes that the test carried out is a left-tailed test. Therefore, the P-value is the area under the z curve and to the left of the calculated z value.

The P-value for the test statistic value of –2.59 is obtained as follows:

$\begin{array}{c}P\text{-value}=\text{Area to the left of 2}\text{.59}\\ =P\left(z<-2.59\right)\end{array}$

Use Standard Normal Probabilities (Cumulative z curve areas) table to find the z-value

Procedure:

For z at –2.59,

• Locate –2.5 in the left column of the table.
• Obtain the value in the corresponding row below .09.

That is, $P\left(z<-2.59\right)=0.0048$

Thus, the P-value for the test statistic of –2.59 is 0.0048.

Step 9:

Decision rule:

If $P\text{-value}\le \text{significance level}$, then reject the null hypothesis $H_0$.

If $P\text{-value}>\text{significance level}$, then fail to reject the null hypothesis $H_0$.

Here, the $P\text{-value}$ of 0.0048 is less than the significance level 0.01.

That is, $P\text{-value}\left(=0.0048\right)<\text{significance level}\left(=0.01\right)$.

The decision is that the null hypothesis is rejected.

Conclusion:

Hence, the sample data provide the evidence that the proportion of adults of Country A, who identified the correct definition of the Bill of Rights, is less than 0.5 at a significance level of 0.01.