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. fo 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 the Variables

Let's identify the response variable and the explanatory variable. - **Response Variable**: Belief about whether the government can eventually prevent all major terrorist attacks. - **Categories**: Believes government can prevent all attacks, Does not believe government can prevent all attacks. - **Explanatory Variable**: How effective the government is perceived in preventing terrorism. - **Categories**: Very effective, Somewhat effective, Not too effective, Not at all effective.

02

Calculate Number of People Believing in Complete Prevention

We know - 510 adults consider the government to be very effective.- 37.06% of these 510 adults believe that the government can prevent all major attacks.Calculate: \[0.3706 \times 510 = 189\]So, 189 adults out of 510 believe that the government can prevent all major attacks.

03

Calculate for Somewhat, Not Too, or Not At All Effective

Now consider the next group.- Total surveyed = 1515- 510 considered very effective, so the remaining: \[1515 - 510 = 1005\]- Of these 1005, 28.36% believe government can prevent all attacks:\[0.2836 \times 1005 = 285\]So, 285 adults believe in prevention among those who consider the government to be somewhat, not too, or not at all effective.

04

Construct the Contingency Table

Using the counts calculated: - **Response: Believes Government Can Prevent All** - Very Effective: 189 - Somewhat/Not Too/Not At All Effective: 285 - **Response: Does Not Believe Government Can Prevent All** - Very Effective: 510 - 189 = 321 - Somewhat/Not Too/Not At All Effective: 1005 - 285 = 720

05

Compute and Display Percentages

Calculate the percentage regarding each category of the response variable:For 'Very Effective' category:- Prevent All Attacks: \[\frac{189}{510} \times 100 = 37.06\%\]- Cannot Prevent All Attacks: \[\frac{321}{510} \times 100 = 62.94\%\]For 'Somewhat/Not Too/Not At All Effective' category:- Prevent All Attacks: \[\frac{285}{1005} \times 100 = 28.36\%\]- Cannot Prevent All Attacks: \[\frac{720}{1005} \times 100 = 71.64\%\]

06

Conditional Percentages Analysis

The percentages in Step 5 are conditional because they describe the response variable distribution within each category of the explanatory variable.

07

Graphical Representation

Draw a bar graph showing the percentage of each response for the two categories of the explanatory variable: - X-axis: Effectiveness Categories (Very Effective, Somewhat/Not Too/Not At All Effective) - Y-axis: Percentage - Two bars for each category: 'Believes Can Prevent' and 'Does Not Believe'. Use one color for each bar to compare within each category.

08

Difference and Ratio of Proportions

Calculate the difference and ratio:- Difference = 37.06% - 28.36% = 8.7%- Ratio = \( \frac{37.06}{28.36} \approx 1.306 \)This implies a higher proportion of belief in government effectiveness when categorized as 'Very Effective.'

09

No Evidence of Association

If the proportions of those who believe and do not believe in government effectiveness were the same regardless of perceived effectiveness, it would indicate no association. The belief percentages would be equal (e.g., both are 37.06% and 28.36% in all categories).

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Key Concepts

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

Response Variable

When examining survey data, like the one conducted in March 2013 on the perception of government effectiveness in preventing terrorism, it is crucial to identify the response variable. The response variable is the outcome of interest, the particular metric we are looking to analyze or explain. In this survey, the response variable is the belief about whether the government can eventually prevent all major terrorist attacks. This belief is what researchers want to understand in relation to other factors.
Any survey needs to clarify these types of variables so researchers can track how other influences might cause changes in the response variable. By isolating this variable, differences in belief can be assessed against other data points, such as how effective the government is perceived to be in this context.

Explanatory Variable

An explanatory variable, in simple terms, helps us to explain changes in our response variable. It provides categories or groups that potentially influence the response variable. In the surveyed data on terrorism prevention, the explanatory variable is the perceived effectiveness of government actions in preventing terrorism.
This variable categorizes how individuals rate the government’s effectiveness with options like very effective, somewhat effective, not too effective, and not at all effective. By categorizing perceptions, researchers aim to find patterns or insights into how these perceptions relate to the belief in the government's ability to prevent all terrorist attacks. Understanding this relationship is key to identifying any potential associations or dependencies between public perceptions and beliefs in effectiveness.

Conditional Percentages

Conditional percentages are calculated within specific categories of another variable. This approach breaks down general percentages into smaller, more specific insights, providing a clearer picture of the relationship between variables. In this survey, conditional percentages describe the belief in government effectiveness within each effectiveness category.
For example, among those who view the government as very effective, 37.06% believe it can prevent all attacks. Comparatively, 28.36% of those who see the government as less effective share this belief. These percentages are 'conditional' because they rely on the explanatory variable's categories rather than generalizing across all survey participants.
By examining conditional percentages, we can identify disparities and variations within groups that might be overlooked if only overall percentages were considered.

Graphical Representation

Representing data graphically is a powerful tool to quickly convey relationships and patterns. A bar graph is particularly effective for showing categorical data, such as the survey on government effectiveness and belief in preventing attacks.
In this case, a bar graph can display the percentages of belief across different effectiveness categories on the x-axis, with the percentage values on the y-axis.

• Each category, like 'Very Effective' or 'Somewhat Effective,' will have two bars.
• One bar shows the percentage who believe major attacks can be prevented.
• The second bar shows the percentage of those who do not hold this belief.

By using different colors for belief and non-belief bars within each category, this visualization helps quickly spot differences in responses across these categories, highlighting underlying trends effortlessly. Graphical representations make complex data more approachable and understandable, which aids both analysis and communication.