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Difficulty: HardResolving Statistical and Proportional Paradoxes

A public health organization recently analyzed the incidence of a specific respiratory ailment across the adult populations of two neighboring regions, Westvale and Eastboro. The data revealed that the incidence rate of the ailment among adults over 65 is significantly higher in Westvale than in Eastboro. Similarly, the incidence rate of the ailment among adults 65 and under is also higher in Westvale than in Eastboro. Despite these findings, the organization concluded that the overall incidence rate of the respiratory ailment for the entire adult population is substantially higher in Eastboro than in Westvale.

Which one of the following, if true, most helps to resolve the apparent discrepancy described above?

  1. Adults over 65 are vastly more susceptible to the respiratory ailment than younger adults, and they constitute a much larger proportion of the adult population in Eastboro than in Westvale.Answer
  2. B
    The environmental pollutants that are the primary cause of the respiratory ailment are found in significantly higher concentrations in Westvale than in Eastboro.
  3. C
    The absolute number of adults residing in Eastboro is substantially larger than the total number of adults residing in Westvale.
  4. D
    Public health officials in Eastboro recently implemented a successful preventative health program that significantly lowered the incidence of the ailment among adults 65 and under.
  5. E
    The respiratory ailment typically presents with much more severe symptoms in older adults, requiring longer periods of hospitalization.

Answer

The statement explaining that adults over 65 are highly susceptible to the ailment and make up a much larger proportion of Eastboro's population than Westvale's.
The apparent discrepancy is a classic example of Simpson's Paradox, where trends observed in different groups disappear or reverse when those groups are combined. The overall incidence rate of a region is a weighted average of the rates of its demographic subgroups. Even if Westvale has higher disease rates within each specific age group, Eastboro can have a higher overall rate if its population is heavily concentrated in the age group that is most susceptible to the ailment. The correct answer provides exactly this mechanism: it establishes that adults over 65 have a much higher baseline risk, and that Eastboro has a much larger proportion of these high-risk adults. Therefore, Eastboro's overall average is heavily weighted by this high-risk group, driving its total rate higher than Westvale's, whose overall average is pulled down by a predominantly younger, low-risk population.

Step-by-Step Solution

1
Analyze the conflicting premises in the stimulus.
Westvale has a higher incidence rate in BOTH demographic subgroups (over 65, and 65 and under). However, Eastboro has a higher OVERALL incidence rate.
Understanding the exact nature of the statistical contradiction is required before evaluating the answer choices.
2
Identify the mathematical mechanism that governs overall rates.
An overall rate is a weighted average of its subgroup rates. It depends not only on the rates within each subgroup but also on the relative size (weight) of each subgroup within the total population.
This establishes the logical framework needed to resolve the paradox, often referred to as Simpson's Paradox.
3
Evaluate the options to find the one that utilizes this mathematical mechanism.
The correct option reveals that Eastboro's population is heavily weighted toward a demographic (over 65) that has a naturally high risk for the ailment, pulling their overall average upward.
This is the only way to mathematically reconcile the seemingly contradictory statistics provided in the stimulus.

Key Concept

Simpson's Paradox (Resolving proportional discrepancies through weighted averages and shifting population baselines).
Estimated Time:2m 0s
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Topics

Resolving Statistical and Proportional Paradoxes

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