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Zorluk: OrtaResolving Statistical and Proportional Paradoxes

Last year, the city of Granchester reported a 15% increase in the absolute number of workplace injuries compared to the previous year. However, according to the city's labor department, the overall workplace injury rate—calculated as the number of injuries per 100,000 hours worked—actually decreased by 5% during the exact same period.

Which of the following statements, if true, help to resolve the apparent discrepancy described above? (Select all that apply.)

  1. Several massive manufacturing plants opened in Granchester last year, increasing the total number of hours worked by the city's labor force by over 25%.Cevap
  2. Due to a booming regional economy, a substantial number of Granchester residents who previously worked part-time transitioned to overtime-heavy full-time employment last year, vastly expanding the total hours worked citywide.Cevap
  3. C
    The official criteria for what qualifies as a reportable workplace injury were expanded last year to include minor incidents that were previously left unrecorded.
  4. D
    A significant proportion of the newly reported workplace injuries last year occurred in traditionally high-risk industries, such as commercial construction and heavy mining.
  5. E
    Last year, the average number of hours worked per employee in Granchester decreased significantly due to newly enforced maximum-hour labor regulations.

Cevap

The correct statements are those that demonstrate a massive increase in the total number of hours worked by the city's labor force, thereby increasing the denominator of the rate calculation by a larger percentage than the numerator.
The discrepancy relies on the relationship between absolute numbers (a numerator) and a rate (a fraction). The rate is calculated as Total Injuries divided by Total Hours Worked. Because the problem states that the numerator (Injuries) increased by 15%, the only mathematical way for the overall rate to decrease is if the denominator (Total Hours Worked) increased by more than 15%. Both correct statements provide logical, real-world mechanisms—new factories opening and a shift to overtime-heavy full-time work—that would massively inflate the total hours worked in the city, perfectly resolving the mathematical paradox.

Adım Adım Çözüm

1
Identify the two seemingly contradictory facts in the stimulus.
Fact 1: Absolute injuries increased by 15%. Fact 2: The injury rate (injuries per 100,000 hours worked) decreased by 5%.
Understanding the exact metric being measured is essential for resolving statistical paradoxes.
2
Formulate the mathematical relationship between the two facts.
Rate = (Absolute Injuries) / (Total Hours Worked).
Exposing the underlying formula reveals how both facts can be simultaneously true.
3
Determine what condition must be met to resolve the discrepancy.
For the overall fraction (Rate) to decrease while the numerator (Injuries) increases by 15%, the denominator (Total Hours) must increase by more than 15%.
This sets the criteria for the correct answers: we need statements that prove a massive surge in total hours worked.
4
Evaluate the answer choices against the required condition.
The statements detailing the opening of massive manufacturing plants and the transition from part-time to overtime-heavy full-time work both perfectly explain a massive surge in total hours worked.
Identifying the mechanisms that manipulate the denominator correctly isolates the answers.

Anahtar Kavram

To resolve a paradox where an absolute count increases but the proportional rate decreases, you must identify a factor that causes the baseline denominator (e.g., total population, total hours) to increase at an even faster pace than the absolute count.
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Resolving Statistical and Proportional Paradoxes

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