91Ó°ÊÓ

In a survey conducted in March 2013 by the National Consortium for the Study of Terrorism and Responses to Terrorism, 1515 adults were asked about the effectiveness of the government in preventing terrorism and whether they believe that it could eventually prevent all major terrorist attacks. \(37.06 \%\) of the 510 adults who consider the government to be very effective believed that it can eventually prevent all major attacks, while this proportion was \(28.36 \%\) among those who consider the government somewhat, not too, or not at all effective in preventing terrorism. The other people surveyed considered that terrorists will always find a way. a. Identify the response variable, the explanatory variable and their categories. b. Construct a contingency table that shows the counts for the different combinations of categories. c. Use a contingency table to display the percentages for the categories of the response variables, separately for each category of the explanatory variable. d. Are the percentages reported in part c conditional? Explain. e. Sketch a graph that compares the responses for each category of the explanatory variable. f. Compute the difference and the ratio of proportions. Interpret. g. Give an example of how the results would show that there is no evidence of association between these variables.

Step by step solution

01

Identify Variables and Categories

The **response variable** is whether the respondents believe the government can eventually prevent all major attacks (Yes, No). The **explanatory variable** is the effectiveness of the government in preventing terrorism (Very Effective, Somewhat/Not Too/Not At All Effective).

02

Totals for Each Group

Calculate the total number of respondents who consider the government to be 'very effective' and those who do not. - **Very Effective Group** = 510 adults - **Somewhat/Not Too/Not At All Effective Group** = 1515 - 510 = 1005 adults

03

Calculate Yes Counts

Calculate the number of respondents who believe the government can eventually prevent all major attacks in each effectiveness group. - **Very Effective Group: Yes** = 0.3706 * 510 = 189 - **Somewhat/Not Too/Not At All Effective Group: Yes** = 0.2836 * 1005 = 285

04

Construct Contingency Table (Counts)

Construct a table showing the counts for each category of the response and explanatory variables. | Government Effectiveness | Can Prevent All Attacks: Yes | Can Prevent All Attacks: No | |--------------------------------|-----------------------------|-----------------------------| | Very Effective | 189 | 321 (510-189) | | Somewhat/Not/Not at All Effective | 285 | 720 (1005-285) |

05

Contingency Table (Percentages)

Display the percentage of 'yes' within each effectiveness category.- **Very Effective: Yes Percentage** = \( \frac{189}{510} \times 100 = 37.06\% \)- **Somewhat/Not/Not at All Effective: Yes Percentage** = \( \frac{285}{1005} \times 100 = 28.36\% \)

06

Conditional Percentages Explanation

The percentages calculated in Step 5 are conditional because they represent the probability of a respondent answering 'yes' given their specific category of effectiveness assessment.

07

Graph Sketch Guidance

A bar graph can show the percentage of 'yes' responses for each effectiveness category. This helps visually compare how belief in the government's ability changes with perceived effectiveness.

08

Difference and Ratio of Proportions

Calculate and interpret the difference and ratio of proportions for the two groups.- **Difference of Proportions** = 37.06% - 28.36% = 8.7%- **Ratio of Proportions** = \( \frac{37.06}{28.36} = 1.31 \)**Interpretation**: Those who view the government as very effective are 8.7% more likely to believe it can prevent all major attacks, and about 31% more likely (times more likely) than those less confident in its effectiveness.

09

Lack of Association Example

If both groups had similar belief percentages regarding the prevention of major attacks (e.g., both around 37% or 28%), it would suggest no significant association between perceived government effectiveness and belief in preventing major attacks.

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Key Concepts

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

Response Variable

In statistical studies, the response variable is essentially the focus of the study. It is the outcome that researchers are interested in understanding or predicting.

In the context of this survey, the response variable is whether the respondents believe the government can eventually prevent all major terrorist attacks. Essentially, it is the outcome based on the level of confidence each respondent has in the government’s effectiveness.

Common categories for the response variable in this survey are:

• Yes - Respondents believe that eventually, all major attacks can be prevented by the government.
• No - Respondents believe that it is not possible for the government to prevent all major attacks.

It's important to categorize response variables to analyze data and understand trends clearly.

Explanatory Variable

An explanatory variable, often called an independent variable, is what researchers manipulate or observe to determine its relationship with the response variable.

In this particular study, the explanatory variable is how effective the respondents perceive the government to be in preventing terrorism. This perception is presumed to affect their belief about the ability to prevent all major attacks.

The categories for this explanatory variable often are:

• Very Effective - Respondents believe the government is highly capable of preventing terrorism.
• Somewhat Effective, Not Too Effective, or Not At All Effective - These categories reflect increasing skepticism about the government’s effectiveness.

Understanding this variable helps to see how various levels of perceived effectiveness impact beliefs about terrorism prevention.

Conditional Percentages

Conditional percentages provide insight into how a response differs across the levels of an explanatory variable.

These percentages are calculated within each category of the explanatory variable to show probabilities for the occurrence of certain outcomes.

In our survey, for example, the conditional percentage of respondents who believe that all major attacks can be prevented, given that they find the government "very effective," was 37.06%, while it was 28.36% for those who perceived the government as less effective.

In essence, conditional percentages are useful for understanding tailored probabilities within different sub-groups of data and allow us to perceive nuanced patterns and relationships.

Difference and Ratio of Proportions

The difference and ratio of proportions are statistical measures used to compare two rates or percentages from different groups.

In this case, the difference in proportions helps to quantify the absolute difference in belief about prevention effectiveness between the groups (37.06% - 28.36% = 8.7%). This indicates how much more likely one group is to hold a belief compared to another.

The ratio of proportions (e.g., 1.31) shows how many times more likely one group is to have a certain response compared to another group. A ratio of 1.31 implies that respondents who see the government as "very effective" are 31% more likely to believe in the prevention of all major attacks compared to those less convinced of its effectiveness.

These measures provide easy-to-understand comparisons and establish the strength of associations between variables.