Is there an association between eye color and meningitis-related deafness? Which test would be used?

Is there an association between eye color and meningitis-related deafness? Which test would be used?

• A. Chi-square goodness-of-fit
• B. Chi-square test of independence
• C. Fisher's exact test
• D. Z-test for proportions

Answer: (B)

When you want to know if two categorical variables are related, you test for association in a contingency table. Here, eye color is one categorical variable and meningitis-related deafness (yes or no) is the other. The chi-square test of independence asks whether the pattern of deafness across eye-color groups could occur if there were no relationship between the two variables. If eye color and deafness are independent, the proportion of cases with deafness should be similar across different eye colors. The test compares what you actually observe in the table to what you would expect under independence, using a chi-square statistic derived from the differences between observed and expected counts. A small p-value suggests the variables are linked rather than independent. This method works for tables with multiple eye-color categories and two outcomes. If some cells have very small expected counts (typically less than 5), Fisher’s exact test becomes a better option because it provides an exact probability rather than relying on the chi-square approximation. The other options fit different questions: a goodness-of-fit test checks whether a single categorical variable fits a specified distribution, not whether two variables are related; a Z-test for proportions compares proportions between groups, not the full association across a contingency table; and Fisher’s exact test is the exact alternative to the chi-square test of independence for small samples, especially 2x2, but the standard choice for larger samples is the chi-square test of independence.

When you want to know if two categorical variables are related, you test for association in a contingency table. Here, eye color is one categorical variable and meningitis-related deafness (yes or no) is the other. The chi-square test of independence asks whether the pattern of deafness across eye-color groups could occur if there were no relationship between the two variables.

If eye color and deafness are independent, the proportion of cases with deafness should be similar across different eye colors. The test compares what you actually observe in the table to what you would expect under independence, using a chi-square statistic derived from the differences between observed and expected counts. A small p-value suggests the variables are linked rather than independent.

This method works for tables with multiple eye-color categories and two outcomes. If some cells have very small expected counts (typically less than 5), Fisher’s exact test becomes a better option because it provides an exact probability rather than relying on the chi-square approximation.

The other options fit different questions: a goodness-of-fit test checks whether a single categorical variable fits a specified distribution, not whether two variables are related; a Z-test for proportions compares proportions between groups, not the full association across a contingency table; and Fisher’s exact test is the exact alternative to the chi-square test of independence for small samples, especially 2x2, but the standard choice for larger samples is the chi-square test of independence.