Business Context

You’re a data scientist at RideShield, a rideshare + insurance partner that prices pay-as-you-drive policies for ~600k active drivers across the US. The company has a “driver safety score” model that ingests telematics events (hard brakes, rapid accelerations, speeding) and outputs a daily risk score used to set premiums and trigger coaching.

A problem surfaced: drivers in San Francisco (SF) are consistently scoring worse than drivers in Phoenix (PHX), and the operations team suspects this is partly due to environmental differences (dense traffic, hills, lower average speeds, more stop-and-go) rather than true driver skill. Leadership wants a normalization approach so that a “good driver” in SF is comparable to a “good driver” in PHX.

You are given a simplified dataset aggregated at the driver-month level. For each driver-month you have:

• events: number of safety-relevant events (hard brake OR harsh accel OR speeding)
• miles: miles driven that month
• night_miles_share: fraction of miles driven at night (0–1)
• rain_hours: total hours of rain encountered while driving that month

The business wants a single normalized metric that can be used to compare drivers across cities and to decide who gets coaching.

Given Data

Assume you sampled a large set of driver-months from each city and computed the following summary statistics for the raw event rate per 100 miles:

$r = 100 \times \frac{\text{events}}{\text{miles}}$

You also fit (from historical data across all cities) a Poisson GLM for event counts with exposure (miles) and covariates:

$\log(\mathbb{E}[\text{events}]) = \log(\text{miles}) + \beta_0 + \beta_1\,\text{night\_miles\_share} + \beta_2\,\text{rain\_hours}$

Estimated coefficients (treat as known for this question):

For a particular driver-month you want to score:

Use significance level $\alpha = 0.05$.

Problem Statement

Design and compute a normalization method that makes driving performance comparable across cities. You should address both simple distributional normalization and a model-based “expected vs observed” normalization that accounts for driving conditions.

Requirements

1. Compute the driver’s raw event rate $r$ (events per 100 miles).
2. City-level z-score normalization: compute the driver’s z-score relative to SF using $z = (r-\bar r_{SF})/s_{SF}$. Explain what this does and what it fails to control for.
3. Model-based normalization: using the Poisson GLM, compute the driver-month’s expected events $\hat\lambda$ and then compute a normalized score such as O/E ratio $= \text{events}/\hat\lambda$ and deviance residual (or an approximate z-score).
4. Hypothesis test: test whether this driver is significantly worse than expected given conditions using a one-sided Poisson test: $H_0: \lambda = \hat\lambda$ vs $H_1: \lambda > \hat\lambda$. Compute the p-value.
5. Recommend which normalization you would use for (a) pricing and (b) coaching, and justify with at least two practical considerations (fairness, stability, gaming risk, interpretability, drift).

Assumptions and Constraints

• Driver-months are independent for the purpose of the city summary statistics.
• Event counts are approximately Poisson conditional on covariates and exposure.
• The GLM coefficients are stable and were trained without leakage from the target month.
• You are not asked to refit the model; only to compute normalized metrics and interpret them.