Lead exposure can cause many neurological problems in young children. Lead exposure is measured as the lead blood level in micrograms per 100 mL. A study in El Paso Texas wanted to determine if lead blood level varied by distance to a lead-emitting smelter, which was defined as 0-1 miles from the smelter, 1-2.5 miles from the smelter, and 2.5-4 miles from the smelter. Here are the ANOVA results with a 0.05 level of significance. Which of the following statements is true?

Df Sum Sq Mean Sq F value Pr(F)
Distance 2 601.518 300.759 3.4974 0.033
Residuals 121 10405.516 85.996

Question

Lead exposure can cause many neurological problems in young children. Lead exposure is measured as the lead blood level in micrograms per 100 mL. A study in El Paso Texas wanted to determine if lead blood level varied by distance to a lead-emitting smelter, which was defined as 0-1 miles from the smelter, 1-2.5 miles from the smelter, and 2.5-4 miles from the smelter. Here are the ANOVA results with a 0.05 level of significance. Which of the following statements is true?

Df	Sum Sq	Mean Sq	F value	Pr(>F)
Distance	2	601.518	300.759	3.4974	0.033
Residuals	121	10405.516	85.996

BlackTom AI · Accepted Answer

The question presents an ANOVA result comparing lead blood levels across distance categories from a smelter, with a significance level of 0.05.

Option 1: 'The null hypothesis is rejected, but post tests are required to determine which group mean significantly differs from which.' This aligns with standard ANOVA procedure: a significant F statistic (p = 0.033) indicates at least one group mean differs, so we reject H0 overall. To identify exactly which distance groups differ from each other, post hoc pairwise comparisons (such as Tukey's HSD) are typically performed.

Option 2: 'The average variability in lead blood level exposure between the distance categories is smaller than within the distance categories.' This would imply a smaller between-group sum of squares relative to within-group sum of squares, contradicting the observed F statistic and the small p-value. Since F = 3.497 with p = 0.033, there is evidence that between-group variability is not negligible compared to within-group variability; in fact, the test result supports differences among group means.

Option 3: 'We can make causal conclusions about the relationship between distance to the smelter and lead blood level based on these data.' This is not appropriate because ANOVA on observational data supports associations but does not establish causation. Without random assignment or other causal inference methods, we should be cautious about claiming causality.

Option 4: 'Distance from the smelter is not significantly associated with lead blood level at 0.05 level of significance.' This directly contradicts the reported p-value of 0.033, which is below 0.05, indicating a statistically significant association in the ANOVA.

In summary, option 1 correctly describes the standard interpretation of the results and the next step (post hoc tests) after a significant ANOVA. The other options misinterpret the variance relationship, overstate causal inferences from observational data, or overlook the reported significance.

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