Business Context

PayWave, a digital payments company, noticed that transactions using a VPN are flagged as fraudulent more often than transactions without a VPN. The fraud team wants to know whether VPN usage itself causes fraud risk to increase, or whether the relationship is driven by other factors such as cross-border activity.

Problem Statement

You are given aggregate transaction data split by VPN usage and by cross-border status. First, quantify the observed correlation between VPN usage and fraud. Then evaluate whether that relationship still holds after controlling for cross-border transactions, and explain why correlation alone does not establish causation.

Given Data

Overall transaction totals

Stratified by cross-border status

Assume fraud labels are finalized and transaction assignment to these groups is accurate.

Requirements

1. Compute the overall fraud-rate difference between VPN and No VPN transactions.
2. Run a two-proportion z-test on the overall rates at $\alpha = 0.05$.
3. Compute the fraud-rate difference within each cross-border segment.
4. Explain whether the aggregate correlation supports a causal claim.
5. Give one better approach to estimate causality in this fraud setting.

Assumptions

• Transactions are independent for the purpose of this exercise.
• Sample sizes are large enough for normal approximation.
• Cross-border status is a plausible confounder because it is associated with both VPN usage and fraud risk.