Decline in Approval Rates

Solution & Explanation

The decline in overall capital approval rates from 85% to 82%, despite individual product approval rates remaining constant or increasing, is a classic example of Simpson's Paradox. This statistical phenomenon occurs when a trend appears in several different groups of data but disappears or reverses when these groups are combined.

Understanding Simpson's Paradox:

1. Individual Product Trends:

   • Product 1: Improved approval from 84% to 85%.
   • Product 2: Approval rate steady at 77%.
   • Product 3: Improved approval from 81% to 82%.
   • Product 4: Approval rate consistent at 88%.

2. Overall Trend Reversal:

   • Despite improvements or stability in individual products, the overall approval rate decreased.

Explanation of the Paradox:

• Weighted Average Impact:

  • The overall approval rate is not simply an arithmetic mean of the product rates but a weighted average based on the volume of applications for each product.
  • If a product with a lower approval rate (e.g., Product 2) receives a disproportionately higher number of applications, it can pull the overall approval rate down.

• Change in Product Mix:

  • A shift in the distribution of applications towards products with lower approval rates can result in a lower overall approval rate.
  • For example, if more applications were processed for Product 2, which has the lowest approval rate, the overall rate would be adversely affected.

Mathematical Representation:

• Let:

  • $p_i$ be the approval rate for product $i$.
  • $n_i$ be the number of applications for product $i$.
  • $n$ be the total number of applications.
  • $p$ be the overall approval rate.

• The overall approval rate is calculated as: $p = \frac{\sum_{i=1}^{4} (n_i \cdot p_i)}{n}$

• A change in $n_i$ towards products with lower $p_i$, like Product 2, can lower $p$ even if individual $p_i$ remain constant or increase.

Conclusion:

This paradox highlights the importance of understanding distribution and weighting in data analysis. When evaluating overall trends, it's crucial to consider how changes in the composition of data groups can affect aggregate statistics. Understanding Simpson's Paradox helps in making more informed decisions and avoiding misleading conclusions drawn from aggregated data.