Choose tests and solve distribution parameters

You are comparing engagement for new versus existing users over the period 2025-08-05 through 2025-09-01.

1. The data are per-user daily session counts, which are integer-valued, skewed, and contain many zeros. Which method would you choose for comparing the central tendency of the two cohorts, and why: a two-sample t-test, Welch's t-test, Mann–Whitney U, or a GLM-based approach? List the assumptions and the diagnostics you would run.

2. Assume that for existing users, daily sessions per user approximately follow a Negative Binomial distribution with mean $\mu = 2.40$ and variance $\sigma^2 = 6.96$. Use the $(r, p)$ parameterization of the NB with $E[X] = r(1-p)/p$ and $Var[X] = r(1-p)/p^2$. Find $r$ and $p$, then calculate $P(X = 0)$.

3. For new users, the estimated mean is $\mu = 1.85$ and the estimated variance is $\sigma^2 = 4.20$. Using delta-method or GLM reasoning, construct a 95% confidence interval for the difference in mean sessions per user between the cohorts, assuming independent samples with $n_{new} = 5{,}000$ and $n_{exist} = 5{,}000$. State any approximations you make.

4. You run a Welch's t-test and get $p = 0.04$ and Cohen's $d = 0.08$. Interpret the practical versus statistical significance, explain how you would control for multiple testing if the analysis were also split by 5 countries, and give one robust effect-size measure for count data (such as the ratio of means) along with how to estimate its confidence interval.

Overview: This question tests skill in statistical inference for skewed count data, including choosing a test for central tendency, estimating negative binomial parameters and computing the zero-probability, building confidence intervals for mean differences, and interpreting p-values relative to effect sizes in the Statistics & Math area for a Data Scientist position. It is often used to evaluate both conceptual knowledge—assumptions, diagnostics, statistical versus practical significance, and multiple-testing issues—and applied skills such as solving for parameters, justifying a delta-method or GLM approach, and estimating robust effect sizes when working with real-world count data.

Read the full Data Scientist interview experience this question came from.