Chi-Square Tests: Homogeneity, Independence, and Goodness-of-Fit

Chi-Square Tests in Business Statistics

Chi-square tests are a family of statistical tests used to analyze categorical data. They help determine whether observed frequencies differ from expected frequencies under certain hypotheses. The main types relevant to business statistics are the Test of Homogeneity, Test of Independence, and the Goodness-of-Fit Test.

Test of Homogeneity

The Test of Homogeneity is used to determine whether two or more populations are homogeneous, meaning they have the same proportion of a certain characteristic. This test is commonly applied when comparing proportions across different groups.

• Definition: Populations are homogeneous if they have the same proportion of a characteristic.

• Hypotheses:

  • Null hypothesis ($H_0$): All proportions are the same across populations.

  • Alternative hypothesis ($H_a$): At least one proportion is different.

• Example: Testing if the proportion of car ownership is the same for different age groups.

Comparison: Independence Test vs. Homogeneity Test

Both tests use similar calculations but differ in their hypotheses and study design:

• Independence Test: Examines if two variables are related within a single population.

• Homogeneity Test: Compares proportions of a characteristic across different populations.

Test Statistic and Calculation

• The test statistic for both tests is the chi-square statistic:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

• $O$ = observed frequency

• $E$ = expected frequency, calculated as:

$E = \frac{\text{row total} \cdot \text{column total}}{\text{Grand Total}}$

• Degrees of Freedom (df): For a contingency table, $df = (r-1)(c-1)$, where $r$ is the number of rows and $c$ is the number of columns.

Example calculation:

$\chi^2 = \frac{(18-26.4)^2}{26.4} + \frac{(37-28.6)^2}{28.6} + \frac{(30-21.6)^2}{21.6} + \frac{(15-23.4)^2}{23.4} = 11.42$

P-Value and Conclusion

• The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

• Decision rule: If $p$-value $< \alpha$, reject $H_0$; otherwise, fail to reject $H_0$.

• Conclusion: State whether there is enough evidence to claim a difference in proportions.

Application Example: Medical Study

Suppose a pharmaceutical company tests a new ADHD medication. At the 0.05 significance level, they test whether the distribution of symptom improvement differs between the placebo and treatment groups.

• Null hypothesis ($H_0$): The proportion of symptom improvement is the same for both groups.

• Alternative hypothesis ($H_a$): The proportions are different.

Symptoms	Placebo	Non-Placebo	Total
Improved	18	37	55
Not Improved	30	15	45
Total	48	52	100

Test of Independence

The Test of Independence determines whether two categorical variables are independent within a single population. It is often used to analyze survey or experimental data.

• Definition: Two variables are independent if the occurrence of one does not affect the probability of the other.

• Hypotheses:

  • Null hypothesis ($H_0$): Variables are independent.

  • Alternative hypothesis ($H_a$): Variables are dependent.

• Example: Testing if height and grade are independent among students.

Test Statistic and Calculation

• The test statistic is the same as for the homogeneity test:

$\chi^2 = \sum \frac{(O - E)^2}{E}$

• Degrees of freedom: $df = (r-1)(c-1)$

Calculator Application

Statistical calculators (e.g., TI-84) can be used to perform chi-square tests by entering observed data and selecting the appropriate test function.

Goodness-of-Fit Test (G.O.F.)

The Goodness-of-Fit Test compares observed frequencies to expected frequencies based on a claimed distribution. It is used to test if a sample matches a population distribution.

• Hypotheses:

  • Null hypothesis ($H_0$): Observed frequencies match the claimed distribution.

  • Alternative hypothesis ($H_a$): Observed frequencies do not match the claimed distribution.

• Test Statistic: Same chi-square formula as above.

• Degrees of Freedom: $df = k - 1$, where $k$ is the number of categories.

Assumptions and Criteria for Chi-Square Tests

• Random samples are used.

• Observed frequencies are recorded for each category.

• Expected frequency for each cell should be at least 5.