Analyze Cancellation Change with Statistics

Uber rolls out a minor product change intended to lower cancellations. You see the following:

• Before: $n_1 = 50{,}000$ requested trips, cancellation rate $p_1 = 7.2\%$.
• After: $n_2 = 48{,}000$ requested trips, cancellation rate $p_2 = 6.6\%$. For now, treat each request as an independent Bernoulli outcome.

Answer:

1. Pick a suitable hypothesis test for the shift in cancellation rate and build a 95% confidence interval for $p_{\text{after}} - p_{\text{before}}$. Give $H_0$ and $H_A$, and explain what the interval means in practical terms.
2. Next, account for possible confounding by city and hour-of-day. Write down a logistic regression that adjusts for these factors, including an interaction term you think matters. Provide the model equation, state the main assumptions, and explain how you would assess calibration and overdispersion.
3. Power/MDE: What sample size per arm would be required to detect an absolute difference of $0.5$ percentage points at $\alpha = 0.05$ with 80% power, ignoring clustering for now? Then explain how an intraclass correlation of $0.02$ at the driver level would increase the required $N$ under a cluster-robust design; show the design effect and the updated $N$.
4. Nonparametric check: Outline a bootstrap method for constructing a confidence interval for the rate difference with stratification by city (stratified resampling), and discuss when this interval might be preferred over the z-approximation.

Overview: This question tests statistical inference for binary outcomes, regression adjustment for confounding, experiment design and power analysis, and nonparametric uncertainty estimation via bootstrapping. It is aimed at Data Scientist roles in the Statistics & Math domain.