Question

Can you design a Simpson's paradox? Two companies are vying for a city's "Best Local Employer" award, to be given to the company most committed to hiring local residents. Although both employers hired 300 new people in the past year, Company A brags that it deserves the award because \(70 \%\) of its new jobs went to local residents, compared to only \(60 \%\) for Company B. Company B concedes that those percentages are correct, but points out that most of its new jobs were full-time, while most of Company A's were part-time. Not only that, says Company \(B\), but a higher percentage of its full-time jobs went to local residents than did Company A's, and the same was true for part-time jobs. Thus, Company B argues, it's a better local employer than Company \(\mathrm{A}\). Show how it's possible for Company B to fill a higher percentage of both full- time and part-time jobs with local residents, even though Company A hired more local residents overall.

Short answer

Company B can hire a higher percentage in each job type, but Company A's job mix results in more local hires overall.

Step-by-step solution

Understanding the Overall Hiring Numbers

Both companies hired 300 new people. Company A hired 70% local residents which amounts to \(0.7 \times 300 = 210\) local residents. Company B hired 60% local residents which amounts to \(0.6 \times 300 = 180\) local residents.

Setting Up Full-Time and Part-Time Jobs

Let's assume Company A hired 100 full-time and 200 part-time employees, and Company B hired 200 full-time and 100 part-time employees.

Assigning Residents to Full-Time Jobs

Suppose Company A had 50% of its full-time jobs (50 out of 100) go to local residents, and Company B had 80% of its full-time jobs (160 out of 200) go to local residents.

Assigning Residents to Part-Time Jobs

Assume Company A hired 80% of its part-time jobs (160 out of 200) as local residents, and Company B hired 20% of its part-time jobs (20 out of 100) as local residents.

Verifying Total Hires

For Company A, total local hires: 50 full-time + 160 part-time = 210 local residents. For Company B, total local hires: 160 full-time + 20 part-time = 180 local residents.

Analyzing on Job Type

Company B hired a higher percentage of local residents both full-time (80% compared to A's 50%) and part-time (20% compared to A's 80%). However, the mix of job types affects the overall percentage.

Key concepts

Statistical Analysis

Simpson's Paradox is a fascinating phenomenon in statistical analysis where aggregated data can reveal trends that seem to contradict the trends present within the separate groups involved. In the context of our current example, it manifests when Company B manages to hire a higher percentage of local residents for both full-time and part-time jobs than Company A, yet Company A still winds up with a higher overall percentage of local hires.

This paradox arises because of the differing sizes of the groups being analyzed independently. Statistical analysis in this scenario involves considering both the individual percentages within each job category as well as how they combine. The combined effect can indeed lead to misleading interpretations if one does not carefully examine the full breakdown of the data.

• The primary takeaway in statistical analysis with Simpson's Paradox is to always disaggregate data if possible to uncover hidden patterns.
• When analyzing data where such a paradox might occur, assess the weighted averages or contributions from each subgroup to understand the full picture.

Employment Statistics

Employment statistics often provide insights into hiring patterns and job market dynamics. In the exercise with the companies vying for the "Best Local Employer" award, evaluating stats involves dissecting the data into categories like full-time and part-time employment. These categories can reveal different percentages of hiring than overall figures might suggest.

When analyzing employment statistics in this context:

• It's essential to consider not just the total number of hires but the types of jobs being offered.
• Understanding the distinctions between full-time and part-time roles can significantly change the perceived effectiveness of hiring strategies.
• These statistics also serve to highlight how job types can impact company standing as an employer of choice for local residents, which is central to the award in question.

This kind of breakdown can enlighten discussions around job quality and the benefits each job type provides to a community, beyond just the numbers presented.

Data Interpretation

Interpreting data accurately is crucial to making informed decisions and arguments, especially when numbers might initially seem contradictory. In this example, proper data interpretation requires understanding how the composition of groups affects overall statistics.

Company A seemed, at first glance, to prioritize local residents more based solely on aggregated percentages. However, a deeper dive shows Company B's success in each job category:

• For full-time roles, Company B outperformed with 80% local hires, versus Company A's 50%.
• For part-time positions, while Company A had 80%, the total impact was lesser due to the small number of B's part-time roles.

These subtleties underscore the importance of not taking aggregated data at face value. Understanding the fundamentals of data interpretation can prevent miscommunication and support better decision-making. It aids in uncovering genuine insights rather than simplistic or misleading conclusions.