Compare Two Rare-Event Rates with Different Exposure
Company: Waymo
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Technical Screen
# Compare Two Rare-Event Rates with Different Exposure
Two versions were observed over different amounts of mileage, and each produced only a small number of events. Can you treat the estimated event rate of one version as a known value when computing a p-value for the other? Explain how you would compare the rates and report uncertainty. No event counts or mileages are supplied.
### What a Strong Answer Covers
- Uncertainty in both estimated rates and a distinction from an externally fixed benchmark.
- A count/exposure model and a valid two-sample comparison for sparse events.
- Interpretation of zero counts, confidence intervals, and practical effect sizes.
- Comparability of operating conditions and assumptions behind independent Poisson events.
```hint Condition on the total number of events
Under a shared-rate null, ask how the total events should be allocated between two different exposure amounts.
```
### Follow-up Questions
- What if one version has zero observed events?
- How would different mixes of operating conditions affect the comparison?
Overview: Compare sparse event rates across unequal mileage using a two-sample Poisson framework, exact conditional testing, and honest uncertainty.
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