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In November 2005 , an international study to assess public opinion on the treatment of suspected terrorists was conducted ("Most in U.S., Britain, S. Korea and France Say Torture Is OK in at Least Rare Instances," Associated Press, December 7,2005 ). Each individual in random samples of 1000 adults from each of nine different countries was asked the following question: "Do you feel the use of torture against suspected terrorists to obtain information about terrorism activities is justified?" Responses consistent with percentages given in the article for the samples from Italy, Spain, France, the United States, and South Korea are summarized in the table at the top of the next page. Based on these data, is it reasonable to conclude that the response proportions are not the same for all five countries? Use a .01 significance level to test the appropriate hypotheses. $$ \begin{array}{l|ccccc} & & & \text { Response } \\ \hline \text { Country } & \text { Never } & \text { Rarely } & \begin{array}{c} \text { Some- } \\ \text { times } \end{array} & \text { Often } & \begin{array}{c} \text { Not } \\ \text { Sure } \end{array} \\ \hline \text { Italy } & 600 & 140 & 140 & 90 & 30 \\ \text { Spain } & 540 & 160 & 140 & 70 & 90 \\ \text { France } & 400 & 250 & 200 & 120 & 30 \\ \text { United } & 360 & 230 & 270 & 110 & 30 \\ \begin{array}{c} \text { States } \\ \text { South } \\ \text { Korea } \end{array} & 100 & 330 & 470 & 60 & 40 \\ \hline \end{array} $$

Step by step solution

01

State the Hypotheses

The Null Hypothesis, \(H_0\), will be that there's no difference in the response proportions among all five countries i.e., the observed frequencies are similar to expected frequencies. The Alternative Hypothesis, \(H_a\), states that at least one country has a different response proportion.

02

Perform a Chi-Square Test

Firstly, calculate the expected frequencies for each country and response category. The expected value for each cell of the table is computed as \((\text{sum of row values} * \text{sum of column values})/\text{total overall sum}\). Then, compute the Chi-Square statistic which is given by \(\sum_{i=1}^{n}\frac{(O_i - E_i)^2}{E_i}\) where \(O_i\) and \(E_i\) are the observed and expected frequencies respectively.

03

Compute Degrees of Freedom and the P-value

Degrees of freedom is given by \((\text{number of rows} - 1)*(\text{number of columns} - 1)\). Use the Chi-Square statistic and degrees of freedom to find the P-value from a Chi-Square distribution table. If the computed p-value is less than the specified significance level of 0.01, then reject the Null Hypothesis.

04

Conclusion

Based on the result of the P-value in comparison with the significance level, make a conclusion. If the P-value is less than 0.01, it would indicate that the response proportions across the five countries are not the same while a P-value ≥ 0.01 would fail to reject the Null Hypothesis and imply that there might be no difference in the response proportions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a critical component in statistics used to determine whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. In our exercise, we want to find out if different countries have different opinions on the use of torture, which is where hypothesis testing comes into play.

With hypothesis testing, we start by setting up two opposing statements: the Null Hypothesis (, often denoted as \(H_0\)) and the Alternative Hypothesis (\(H_a\)). In our context:

• **Null Hypothesis (\(H_0\))**: There is no difference in response proportions among the five countries. In simple terms, people in all countries feel the same about the use of torture.
• **Alternative Hypothesis (\(H_a\))**: At least one country has a different opinion, indicating variations in the response proportions.

Once these hypotheses are set, we use statistical tests, like the Chi-Square Test, to analyze our data and make a decision regarding these hypotheses.

Significance Level

The significance level is a threshold that determines how strong your evidence must be before you reject the Null Hypothesis. This threshold is predefined and denotes the probability of rejecting a true Null Hypothesis, known as a Type I error.

In the given exercise, we use a significance level of 0.01, which means we are willing to accept just a 1% chance of incorrectly rejecting the Null Hypothesis.

So, if the calculations from our Chi-Square Test yield a P-value less than our significance level of 0.01, it suggests strong evidence against the Null Hypothesis, implying likely real differences in opinions among the countries regarding the issue at hand.

• A low significance level means we need strong evidence to reject the Null Hypothesis.
• Choosing a significance level is a balance between the risk of making mistakes and the reliability of the test.

Degrees of Freedom

Degrees of freedom is a concept that refers to the number of values in a calculation that are free to vary. In the Chi-Square Test for independence, degrees of freedom help determine the critical value against which the test statistic is compared.

In our exercise involving a contingency table, we calculate the degrees of freedom using the formula:\[( ext{number of rows} - 1) imes ( ext{number of columns} - 1)\]For this exercise, assuming each country is a row and each response category is a column, understanding degrees of freedom helps ensure we're comparing our test statistic to the correct value in statistical tables.

• Calculating the degrees of freedom ensures the accuracy of the conclusions drawn.
• Correct degree of freedom calculations contribute to the validity of the hypothesis test.

P-Value

A P-value is a measure used in hypothesis testing to help you decide whether to reject the Null Hypothesis. It quantifies the strength of evidence against the Null Hypothesis.

The P-value tells you the probability that the differences observed in your data could have occurred by random chance under the Null Hypothesis. In our exercise, once we conduct the Chi-Square Test, we calculate a P-value. If this value is less than our significance level of 0.01, it suggests that such a difference in opinions on torture among countries is statistically significant.

• A lower P-value indicates stronger evidence against the Null Hypothesis.
• If the P-value is less than the significance level, we reject the Null Hypothesis.
• This means we conclude that differences in opinions are likely real, not due to random variation.

Expected Frequency

Expected frequency is a key component in conducting a Chi-Square Test. It represents the frequency that one would "expect" per category if there were no real differences among the groups in question.

To calculate expected frequencies in our exercise, for each cell in the table, divide the product of the sum of the row totals and the sum of the column totals by the overall total sum. This calculation assumes that there is no association between the countries and the opinion categories being examined.

• Expected frequencies provide a baseline for comparison with observed frequencies.
• They help determine if there is a significant difference between what was observed and what was expected under the Null Hypothesis.
• The comparison between observed and expected frequencies forms the basis for calculating the Chi-Square statistic.