Bivariate Analysis: Chi-Square and T-Test Applications
This paper presents a series of worked bivariate analysis problems demonstrating the application of chi-square tests and independent-samples t-tests. Problems address selecting appropriate tests for different data types, computing chi-square statistics from observed and expected frequency tables, collapsing response categories to meet chi-square assumptions, and interpreting results relative to critical values. T-test calculations are demonstrated for comparing loan repayment rates across institution types and manager performance ratings across geographic regions. Throughout, the paper emphasizes null hypothesis logic, degrees of freedom, significance thresholds, and the conditions under which standard tests may be inappropriate.
- Selecting the Appropriate Test of Differences: Matching chi-square and t-tests to scenarios
- Chi-Square Tests: Workplace Regulation, Home Ownership, and Shopper Age: Three worked chi-square problems with tables
- Collapsing Categories and Testing Home Ownership by Education: Collapsing cells to meet chi-square assumptions
- Goodness-of-Fit and Commuting Pattern Chi-Square Tests: Sample distribution and commuting pattern tests
- T-Test: Loan Repayment Rates Across Institution Types: T-test comparing Savings and Loan vs. other lenders
- T-Test: Manager Ratings in Eastern vs. Western Regions: Regional manager performance comparison by t-test
- Interpreting Borderline Results and Test Assumptions: One-tailed tests, borderline significance, assumption violations
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What makes this paper effective
- Each problem is fully worked through — expected value tables are constructed, cell-level calculations are shown, and chi-square or t-statistics are derived step by step, making the logic transparent and reproducible.
- Interpretive statements consistently tie the computed statistic back to the critical value and decision rule, reinforcing the null hypothesis framework rather than just reporting a number.
- The final problem demonstrates mature statistical reasoning by flagging a potential assumption violation (non-independence of variables) and suggesting an alternative analytical approach.
Key academic technique demonstrated
The paper consistently applies the hypothetico-deductive structure of null hypothesis significance testing: state the null, derive expected values from it, compute the test statistic, compare to a critical value, and render a decision. This explicit scaffolding — repeated across seven distinct problems — shows how a single inferential framework adapts to different data types and research questions.
Structure breakdown
The paper is organized as a problem-set response, with each numbered question forming its own analytical unit. Questions 1–5 focus on chi-square logic (test selection, computation, category collapsing, goodness-of-fit, and pattern detection), while Questions 6–8 shift to t-tests of means. The progression moves from categorical to continuous data and from straightforward significant/non-significant decisions to a nuanced discussion of one-tailed tests and violated assumptions.
Selecting the Appropriate Test of Differences
The following test types are appropriate for each situation described:
a) Chi-square, with the base hypothesis that all political groups contribute equal amounts.
b) Chi-square, with a base hypothesis appropriate to the attitude question being asked.
c) T-test.
d) Chi-square, with the base hypothesis of equal average salaries between regions.
The choice between a chi-square test and a t-test depends primarily on the level of measurement involved: chi-square tests are used for categorical (nominal or ordinal) data, while t-tests are used when comparing means of continuous variables.
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