Chapter 12, Problem 37CR

To determine

Check whether it is reasonable to conclude that the response proportions are not the same for all five countries at 0.01 significance level.

Answer to Problem 37CR

There is convincing evidence that the response proportions are not all the same for all five countries.

Explanation of Solution

Calculation:

The given data represent classifications of different countries for the response to the question of torture against suspected terrorists to obtain information about the terrorism activities.

The steps in chi-square test for homogeneity are as follows:

Null and alternative Hypotheses:

Null hypothesis:

$H_0:$ The proportions falling into the response categories for all five countries are all the same.

Alternative hypothesis:

$H_a:$ The proportions falling into the response categories for all five countries are not the same.

Level of significance:

$\alpha =0.01$

Assumptions:

• It is given that the samples are random samples.
• The expected cell counts are calculated as shown below:

Country	Response					Total
	Never	Rarely	Sometimes	Often	Not Sure
Country I	$\begin{array}{r}\frac{1,000\times 2,000}{5,000}\\ =400\end{array}$	$\begin{array}{r}\frac{1,000\times 1,100}{5,000}\\ =222\end{array}$	$\begin{array}{r}\frac{1,000\times 1,220}{5,000}\\ =244\end{array}$	$\begin{array}{r}\frac{1,000\times 450}{5,000}\\ =90\end{array}$	$\begin{array}{r}\frac{1,000\times 220}{5,000}\\ =44\end{array}$	1,000
Country S	$\begin{array}{r}\frac{1,000\times 2,000}{5,000}\\ =400\end{array}$	$\begin{array}{r}\frac{1,000\times 1,100}{5,000}\\ =222\end{array}$	$\begin{array}{r}\frac{1,000\times 1,220}{5,000}\\ =244\end{array}$	$\begin{array}{r}\frac{1,000\times 450}{5,000}\\ =90\end{array}$	$\begin{array}{r}\frac{1,000\times 220}{5,000}\\ =44\end{array}$	1,000
Country F	$\begin{array}{r}\frac{1,000\times 2,000}{5,000}\\ =400\end{array}$	$\begin{array}{r}\frac{1,000\times 1,100}{5,000}\\ =222\end{array}$	$\begin{array}{r}\frac{1,000\times 1,220}{5,000}\\ =244\end{array}$	$\begin{array}{r}\frac{1,000\times 450}{5,000}\\ =90\end{array}$	$\begin{array}{r}\frac{1,000\times 220}{5,000}\\ =44\end{array}$	1,000
Country U	$\begin{array}{r}\frac{1,000\times 2,000}{5,000}\\ =400\end{array}$	$\begin{array}{r}\frac{1,000\times 1,100}{5,000}\\ =222\end{array}$	$\begin{array}{r}\frac{1,000\times 1,220}{5,000}\\ =244\end{array}$	$\begin{array}{r}\frac{1,000\times 450}{5,000}\\ =90\end{array}$	$\begin{array}{r}\frac{1,000\times 220}{5,000}\\ =44\end{array}$	1,000
Country SK	$\begin{array}{r}\frac{1,000\times 2,000}{5,000}\\ =400\end{array}$	$\begin{array}{r}\frac{1,000\times 1,100}{5,000}\\ =222\end{array}$	$\begin{array}{r}\frac{1,000\times 1,220}{5,000}\\ =244\end{array}$	$\begin{array}{r}\frac{1,000\times 450}{5,000}\\ =90\end{array}$	$\begin{array}{r}\frac{1,000\times 220}{5,000}\\ =44\end{array}$	1,000
Total	2,000	1,110	1,220	450	220	5,000

From the expected cell counts table, it is observed that all the expected counts are greater than 5.

Test Statistic:

${\chi }^2={{\displaystyle \sum \frac{(\text{observed count}-\text{expected count})}{\text{expected count}}}}^2$

Software procedure:

Step-by-step procedure to obtain the test statistic and P-value using the MINITAB software:

• Choose Stat > Tables > Cross Tabulation and Chi-Square.
• Select Summarized data in two-way table.
• In Columns containing the table, enter the columns of Never, Rarely, Sometimes, Often, and Not Sure.
• Check Chi-square test under Chi-square.
• Click OK.

Output using the MINITAB software is given below:

From the output, ${\chi }^2=881.360$ and $df=16$.

P-value:

From the MINITAB output, P-value is 0.000.

Decision rule:

• If P-value is less than or equal to the level of significance, reject the null hypothesis.
• Otherwise, fail to reject the null hypothesis.

Conclusion:

Here the level of significance is 0.01.

Here, P-value is less than the level of significance.

That is, $0.000<0.01$.

Hence, reject the null hypothesis.

Therefore, there is convincing evidence that the response proportions are not the same for all the five countries at 0.01 significance level.