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Difficulty: EasyPercentages vs. Numbers Flaws

A local news anchor reported: 'Last year, 20% of all crimes committed in our city were burglaries. This year, that figure has risen to 40%. Therefore, the absolute number of burglaries committed in our city must have increased this year.'

Which of the following represent statistical assumptions required to make the news anchor's conclusion valid? Select all that apply.

  1. The overall volume of crime in the city did not decrease by 50 percent or more from last year to this year.Answer
  2. The total number of all crimes committed this year is not significantly lower than the total number of crimes committed last year.Answer
  3. C
    The 20 percentage point increase in the share of burglaries guarantees a 20 percent increase in the absolute number of burglaries.
  4. D
    The rising proportion of burglaries directly caused a decrease in the absolute number of other types of crimes.
  5. E
    Because the percentage doubled, burglaries now account for the majority of all crimes committed in the city.

Answer

The correct statements are those establishing that the total volume of all crimes did not drop drastically (specifically, not by 50 percent or more).
The argument concludes that the absolute number of burglaries increased based solely on an increase in the percentage of burglaries (from 20% to 40%). For this deduction to be mathematically valid, the baseline (total crimes) must not have decreased drastically. Specifically, since the percentage rate doubled, the total number of crimes cannot have dropped by 50% or more (otherwise 40% of the new total would be less than or equal to 20% of the old total). The correct choices correctly articulate this necessary mathematical boundary.

Step-by-Step Solution

1
Identify the premise and the conclusion of the argument.
Premise: Burglaries went from 20% to 40% of all crimes. Conclusion: The absolute number of burglaries increased.
Separating the evidence from the claim clarifies the logical gap being tested.
2
Recognize the logical flaw being tested.
The argument confuses a proportion with an absolute number. A larger percentage of a total only means a larger absolute amount if the total itself did not shrink drastically.
Identifying the archetype of the flaw helps predict the necessary assumption.
3
Translate the numerical relationship into a mathematical inequality.
Let last year's total crimes be XX and this year's be YY. The anchor concludes 0.40Y>0.20X0.40Y > 0.20X. For this to be true, we must assume Y>0.5XY > 0.5X.
Establishing the exact mathematical boundary shows what condition is strictly required for the conclusion to hold.
4
Evaluate the options against this mathematical requirement.
The options stating that total crime did not decrease by 50 percent or more, or drop significantly, perfectly match the Y>0.5XY > 0.5X requirement.
An assumption is a missing premise that is required for the conclusion to be valid.

Key Concept

Percentages vs. Numbers Flaws
Estimated Time:1m 0s
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Topics

Percentages vs. Numbers Flaws

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