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Zorluk: ZorResolving Statistical and Proportional Paradoxes

An industry report compared employee advancement at two major technology firms, ApexTech and Zenith Dynamics. According to the report, both companies categorize all of their employees into exactly two departments: software engineering and administrative staff.

The report noted that ApexTech promoted 15%15\% of its total workforce last year, while Zenith Dynamics promoted only 8%8\% of its total workforce. However, the data also revealed that Zenith Dynamics promoted a higher percentage of its software engineers than ApexTech did, and Zenith Dynamics also promoted a higher percentage of its administrative staff than ApexTech did.

Which of the following statements, if true, jointly provide a complete mathematical resolution to the apparent discrepancy described above? (Select exactly two options)

  1. Software engineers are promoted at a substantially higher rate than administrative staff at both tech companies.Cevap
  2. Software engineers make up the overwhelming majority of employees at ApexTech, while administrative staff make up the overwhelming majority of employees at Zenith Dynamics.Cevap
  3. C
    ApexTech has a significantly larger overall workforce, employing thousands more total workers than Zenith Dynamics.
  4. D
    Zenith Dynamics maintains a company culture that heavily prioritizes external hiring over internal promotions for all advanced roles.
  5. E
    The human resources department at ApexTech requires that the percentage of software engineers promoted in a given year exactly matches the percentage of administrative staff promoted.

Cevap

The correct options are the statement establishing that engineers are promoted at a higher rate than administrative staff, and the statement establishing that ApexTech's workforce is heavily concentrated with engineers while Zenith's is concentrated with administrative staff.
This question tests a classic statistical phenomenon known as Simpson's Paradox. An overall percentage is a weighted average of its subgroups. Even if Zenith has higher promotion rates in every individual subgroup, its overall rate can be lower if its workforce is heavily concentrated in a subgroup that inherently has a lower promotion rate. To fully resolve this, we need two pieces of information: first, that one department inherently has a lower promotion rate (administrative staff), and second, that Zenith's workforce is heavily concentrated in this low-rate department while ApexTech's workforce is heavily concentrated in the high-rate department. Together, these facts mathematically explain how ApexTech's overall average is pulled up toward the high engineer rate while Zenith's is pulled down toward the low administrative rate.

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1
Identify the apparent contradiction in the stimulus.
Zenith has higher promotion rates in both subgroups (engineers and admin), but a lower overall promotion rate (8%8\%) compared to Apex (15%15\%).
Clearly defining the conflicting facts is required before finding a resolution, which in this case involves the mathematical constraints of weighted averages.
2
Analyze how an overall percentage rate relates to subgroup rates.
An overall rate is a weighted average of the subgroup rates. It is heavily influenced by the relative sizes of the subgroups within the population.
Recognizing that overall percentages depend on population distribution is key to resolving rate-based statistical paradoxes.
3
Determine the mathematical conditions required to resolve the discrepancy.
For Zenith to have a lower overall rate despite higher subgroup rates, one subgroup must naturally have a higher promotion rate than the other, AND Apex must employ mostly the high-rate group while Zenith employs mostly the low-rate group.
This specific combination pulls Apex's overall average up and Zenith's overall average down, allowing the paradox to exist logically.
4
Evaluate the given choices against these required conditions.
One correct option provides the difference in inherent subgroup rates (engineers promoted more than admins). The other correct option provides the skewed workforce distributions (Apex has mostly engineers, Zenith has mostly admins).
Together, these two facts perfectly execute the mathematical requirements of Simpson's Paradox.

Anahtar Kavram

Simpson's Paradox (Weighted Averages)
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Konular

Resolving Statistical and Proportional Paradoxes

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