Among Americans, Type O 45%, Type A 40%, Type B 11%, Type AB 4%. A blood drive collected 132 pints with counts 51 O, 55 A, 17 B, 9 AB. Was this yield unusual in any way?

Among Americans, Type O 45%, Type A 40%, Type B 11%, Type AB 4%. A blood drive collected 132 pints with counts 51 O, 55 A, 17 B, 9 AB. Was this yield unusual in any way?

• A. Chi-square test of independence
• B. 2-proportion z-test
• C. Linear regression
• D. Chi-square goodness of fit with 3 degrees of freedom

Answer: (D)

We’re testing whether the observed distribution of blood types matches what would be expected from the known population proportions. This is a goodness-of-fit situation: you have one categorical variable with four categories (the blood types) and you want to see if the counts in your sample fit the expected counts derived from the population proportions. Compute the expected counts from the sample size (132) and the population proportions: O about 59.4, A about 52.8, B about 14.52, AB about 5.28. Compare each observed count to its expectation with (observed − expected)² / expected and sum these contributions. The calculation gives a chi-square statistic of roughly 4.3. The degrees of freedom for four categories is 4−1 = 3. The critical value at a 5% significance level is about 7.815. Since 4.3 is less than 7.815, there isn’t enough evidence to say the yield differs from the population distribution at that level. Other tests aren’t appropriate here because they’re designed for different scenarios: testing independence between two categorical variables, or comparing proportions between two groups, or modeling a continuous outcome with predictors. This situation specifically asks whether one categorical distribution matches a specified expected distribution, which is exactly what a chi-square goodness-of-fit test with three degrees of freedom is designed to do.

We’re testing whether the observed distribution of blood types matches what would be expected from the known population proportions. This is a goodness-of-fit situation: you have one categorical variable with four categories (the blood types) and you want to see if the counts in your sample fit the expected counts derived from the population proportions.

Compute the expected counts from the sample size (132) and the population proportions: O about 59.4, A about 52.8, B about 14.52, AB about 5.28. Compare each observed count to its expectation with (observed − expected)² / expected and sum these contributions. The calculation gives a chi-square statistic of roughly 4.3. The degrees of freedom for four categories is 4−1 = 3. The critical value at a 5% significance level is about 7.815. Since 4.3 is less than 7.815, there isn’t enough evidence to say the yield differs from the population distribution at that level.

Other tests aren’t appropriate here because they’re designed for different scenarios: testing independence between two categorical variables, or comparing proportions between two groups, or modeling a continuous outcome with predictors. This situation specifically asks whether one categorical distribution matches a specified expected distribution, which is exactly what a chi-square goodness-of-fit test with three degrees of freedom is designed to do.