91Ó°ÊÓ

Multiple response variables Each subject in a sample of 100 men and 100 women is asked to indicate which of the following factors (one or more) are responsible for increases in crime committed by teenagers: \(\mathrm{A}-\) the increasing gap in income between the rich and poor, \(\mathrm{B}-\) the increase in the percentage of single-parent families, \(\mathrm{C}\) - insufficient time that parents spend with their children. To analyze whether responses differ by gender of respondent, we cross-classify the responses by gender, as the table shows. a. Is it valid to apply the chi-squared test of independence to these data? Explain. b. Explain how this table actually provides information needed to cross- classify gender with each of three variables. Construct the contingency table relating gender to opinion about whether factor \(A\) is responsible for increases in teenage crime. \begin{tabular}{lccc} \hline \multicolumn{3}{l} { Three Factors for Explaining Teenage Crime } \\ \hline Gender & A & B & C \\ \hline Men & 60 & 81 & 75 \\ Women & 75 & 87 & 86 \\ \hline \end{tabular}

Step by step solution

01

Understanding the Exercise Context

The task involves analyzing survey data collected from 100 men and 100 women about factors they believe are responsible for teenage crime. The factors are identified as A, B, and C. We need to determine if gender influences the perception of these factors and consider the chi-squared test applicability. Additionally, we'll construct a contingency table for factor A.

02

Analyzing the Validity of Chi-Squared Test

The chi-squared test requires categorical data with a sufficient sample size in each category. Given the sample size (100 men and 100 women) and the binary nature of response (yes/no for each factor), the data seem appropriate for chi-squared analysis. However, the test assumes independence, which may not hold here due to subjects responding to multiple factors simultaneously.

03

Creating Contingency Table for Factor A

To cross-classify gender with responses to factor A, we take data points from the table. For men, 60 out of 100 think factor A contributes to teenage crime, so 40 do not. For women, 75 out of 100 think factor A is a factor, leaving 25 who do not. Construct the table: \[ \begin{array}{|c|c|c|} \hline & \text{Yes (Factor A)} & \text{No (Factor A)} \ \hline \text{Men} & 60 & 40 \ \text{Women} & 75 & 25 \ \hline \end{array} \]

04

Summary of Insights

The constructed table indicates that a higher percentage of women than men attribute teenage crime to factor A. The chi-squared test could examine if the differences between men’s and women’s perceptions are statistically significant. However, results may be limited due to potential dependency between responses for different factors.

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Key Concepts

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

Multiple response variables

In surveys like the one discussed, respondents have the option to select multiple answers from a list of provided factors. This type of data collection is referred to as having "multiple response variables." Each factor represents a potential cause of teenage crime, and respondents can choose as many or as few as they believe are significant. This flexibility in response captures the complexity and nuances of individual perceptions.

However, handling such data can be challenging. With multiple response variables, each factor is treated as a separate binary variable. For example, for Factor A, a respondent either thinks it contributes to crime or doesn't. This creates multiple layers of data for each person in the survey. The complexity arises because each individual can respond "yes" or "no" to each question independently, potentially correlating responses as they might perceive certain factors as related.

Contingency table

To analyze relationships between two or more categorical variables, such as gender and opinion on crime factors, we use a contingency table. This is a type of data summary that cross-tabulates variables to showcase potential relationships. In simple terms, it allows us to see how the presence of one variable affects another.

For instance, regarding gender and Factor A in the crime survey, a contingency table can display how many men and women believe Factor A contributes to teenage crime. The columns could mark the 'Yes' and 'No' responses, while the rows represent the two genders. This table allows for easy visualization of differences and similarities, making it a valuable tool for statistical analysis.

• Provides a visual summary of data.
• Helps assess potential relationships between variables.
• Essential for chi-squared testing.

Gender differences in perception

When people look at causes for complex issues like teenage crime, their perspectives can differ, especially between groups such as genders. In this study, gender differences in perception are scrutinized. These differences refer to how men and women may view certain factors affecting teenage crime differently. Such variances can stem from societal influences, personal experiences, or inherent biases.

The contingency table constructed for Factor A from the survey data reveals gender differences. In our case, 75 women compared to 60 men attribute teenage crime to income disparity. Understanding these perceptions is crucial in designing policies or interventions aimed at crime reduction.

• Shows varying priorities or concerns across genders.
• Can be influenced by external societal factors.
• Important for creating balanced, effective social policies.