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Difficulty: Very hardFlaw in the Reasoning

Public Health Official: Critics of our recent influenza vaccine campaign argue that the vaccine is ineffective. They point out that this year, the absolute number of vaccinated individuals who contracted influenza was 15 percent greater than the number of unvaccinated individuals who contracted the disease. However, these critics fail to recognize that due to our aggressive outreach efforts over the past year, the total number of vaccinated individuals in our district has tripled. Therefore, contrary to the critics' claims, the vaccine has indeed effectively reduced an individual's likelihood of contracting influenza compared to remaining unvaccinated.

Which of the following most accurately describes a flaw in the public health official's reasoning?

  1. A
    challenges the critics' conclusion by implicitly rejecting the statistical accuracy of the infection data they provide
  2. B
    illicitly shifts the meaning of the term "ineffective" from failing to prevent infection entirely to merely failing to reduce the overall statistical probability of infection
  3. C
    infers a causal relationship between the aggressive outreach efforts and the tripling of vaccinated individuals without ruling out alternative explanations
  4. bases a conclusion about comparative probabilities on evidence concerning the proportional growth of a single group, without establishing the actual sizes of the groups being comparedAnswer
  5. E
    confuses a condition that is sufficient to lower the incidence of influenza with a condition that is necessary to achieve that same outcome

Answer

The correct answer is the option that identifies the flaw of relying on the proportional growth of a single group to draw a conclusion about comparative rates without establishing the actual base sizes of the groups being compared.
The correct answer accurately identifies the mathematical flaw in the official's reasoning. The official attempts to prove that the vaccine reduces the likelihood of infection (a probability or rate) by comparing the absolute number of sick individuals. While the official correctly notes that a tripled vaccinated population could explain the higher absolute number of sick vaccinated people, this is only logically sound if the vaccinated population is now significantly larger than the unvaccinated population. Because the official never establishes the actual baseline sizes of these two groups, knowing that one group tripled is strictly insufficient to determine their comparative infection rates.

Step-by-Step Solution

1
Identify the conclusion of the argument.
The official concludes that the vaccine has effectively reduced an individual's likelihood of contracting influenza compared to remaining unvaccinated.
Understanding the main point is required to evaluate the argument's validity.
2
Identify the premises supporting the conclusion.
Premise 1: The number of sick vaccinated individuals is 15% greater than sick unvaccinated individuals. Premise 2: The total number of vaccinated individuals has tripled recently due to outreach.
We must isolate the evidence used to justify the conclusion.
3
Evaluate the logical connection between the premises and conclusion.
The official attempts to prove a lower infection rate (likelihood) by pointing out that the base population of vaccinated people grew (tripled).
This reveals the structural relationship of the argument.
4
Identify the missing information or logical leap.
Knowing that the vaccinated group tripled does not reveal its size relative to the unvaccinated group. If the vaccinated group is still much smaller than the unvaccinated group, having more sick people would mean the infection rate is actually much higher.
This pinpoints the exact flaw: confusing a rate of growth with comparative absolute sizes required to determine probabilities.

Key Concept

Numbers and Percentages Flaw
Estimated Time:2m 0s
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Topics

Flaw in the Reasoning

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