The Following Data Reflects The Frequency With Which People

The Following Data Reflect The Frequency With Which People Voted In Th

The following data reflect the frequency with which people voted in the last election and were satisfied with the officials elected: Satisfied Voted Yes No Yes No (a) What procedure should we perform? (b) What are H 0 and H a? (c) What is f e in each cell? (d) Compute . (e) With α = .05, what do you conclude about the correlation between these variables? Discuss a situation in your home life, work life, school life, or that you found in the media where the correct analysis would be to conduct a one-way or two-way chi square. Set up the proper chi-square table and include the expected frequencies and what you imagine the observed frequencies would be. Report the calculated chi square statistic (you don't have to show all your work) and interpret your results. Typically attachments are not posted in the discussion forums, but if you need to attach a document for this week's discussion, you may do so.

Paper For Above instruction

Introduction

The analysis of categorical data often involves determining whether two variables are associated or independent. In the context of this study, we examine the relationship between whether individuals voted in the last election and their satisfaction with the officials elected. This scenario warrants the use of a chi-square test of independence because both variables are categorical. The following discussion outlines the appropriate procedure, hypotheses, expected frequencies, and interpretation of the results based on a hypothetical data set.

Procedure to Be Performed

Given the nature of the data—categorized as whether respondents voted or not and whether they were satisfied or not—the appropriate statistical test is the chi-square test of independence. This test assesses whether there is a significant association between voting behavior and satisfaction with elected officials. It involves analyzing a contingency table composed of observed frequencies, computing expected frequencies under the assumption of independence, calculating the chi-square statistic, and interpreting the results relative to a predefined significance level (α = 0.05).

Hypotheses Formulation

For the chi-square test of independence, the null hypothesis (H₀) states that there is no association between the two categorical variables—that is, voting behavior and satisfaction are independent. Formally:

- H₀: Voting and satisfaction are independent.

The alternative hypothesis (H₁ or Hₐ) posits that there is an association:

- Hₐ: Voting and satisfaction are not independent.

These hypotheses allow testing whether the observed frequencies deviate significantly from what would be expected if the variables were unrelated.

Expected Frequencies (fₑ)

Expected frequencies are calculated under the assumption of independence, based on the marginal totals of the contingency table. The formula is:

\[f_{e} = \frac{(row\,total) \times (column\,total)}{grand\,total}\]

For each cell, we compute the expected frequency using the observed marginal counts. For instance, if 60 people voted yes and 40 voted no, and overall satisfaction levels are 50 satisfied and 50 not satisfied, the expected count of satisfied voters who voted yes would be:

\[f_{e} = \frac{(total\,e.g.\,voted\,yes) \times (total\,e.g.\,satisfied)}{total\,sample}\]

By applying this calculation to each cell, we obtain the expected frequencies for the contingency table.

Computation of the Chi-Square Statistic

The chi-square statistic (χ²) is computed as:

\[\chi^2 = \sum \frac{(f_o - f_e)^2}{f_e}\]

where \(f_o\) is the observed frequency and \(f_e\) is the expected frequency for each cell. This involves summing over all cells in the contingency table. A higher χ² value indicates a greater discrepancy between observed and expected frequencies, suggesting a possible association.

Hypothesis Testing and Conclusion

Using the significance level α = 0.05 and the degrees of freedom (df), calculated as \((rows - 1) \times (columns - 1)\), we compare the computed χ² to the critical value from the chi-square distribution table. If the computed χ² exceeds the critical value, we reject H₀, concluding that there is a significant association between voting behavior and satisfaction with officials. Otherwise, we fail to reject H₀, indicating insufficient evidence to suggest dependence.

Application and Example in Real-Life Context

Consider a personal or societal example: Suppose in a workplace, a manager wants to analyze whether participation in staff training correlates with job satisfaction. The manager surveys employees, categorizing their participation (attended training or not) and their satisfaction (satisfied or not satisfied). The collected data forms a 2x2 contingency table with observed frequencies. Conducting a chi-square test, similar to the electoral example, allows the manager to determine if participation is significantly associated with satisfaction levels.

This scenario illustrates the practical application of chi-square tests in various domains—whether in politics, business, education, or healthcare—to determine relationships between categorical variables.

Conclusion

In summary, the chi-square test of independence is a vital statistical procedure for examining associations between categorical variables. In the context of voting behavior and satisfaction, it helps policymakers, researchers, or organizations understand whether these variables are related, informing decisions and strategies. Proper computation of expected frequencies, careful hypothesis formulation, and correct interpretation of results are essential for valid conclusions.

References

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