Statistics Interview Questions

Sample Answer

You could do this with a Poisson count model over 10 minutes, or with exponential interarrival times and then sum waiting times. Poisson wins here because you are asked for a count in a fixed interval, so $N \sim \text{Poisson}(\lambda t)$ with $\lambda=12/\text{hr}$ and $t=1/6$ hr, giving $\lambda t=2$. Then $$P(N\ge 3)=1-\left[P(N=0)+P(N=1)+P(N=2)\right]=1-e^{-2}\left(1+2+\frac{2^2}{2}\right).$$

Sample Answer

You start with $X \sim \text{Binomial}(n=200,p=0.02)$ because you have independent Bernoulli trials and you are counting failures. Since $np=4$ is small and $n$ is large, you can approximate with $X \approx \text{Poisson}(\lambda=np=4)$. Then you compute $$P(X\ge 8)=1-\sum_{k=0}^{7} e^{-4}\frac{4^k}{k!}.$$ If you want a quick sanity check, $8$ is about double the mean, so the tail should be small but not astronomically small.

Sample Answer

Step by step: the p-value is $P(\text{data as or more extreme} \mid H_0)$, not $P(H_1 \mid \text{data})$. So $p = 0.03$ means that if the true effect were zero, you would see results this extreme about 3% of the time. The CI says the data are consistent with a small but positive lift, but it still does not give a probability the effect is positive without a Bayesian prior. If they want a probability of being positive in production, you point them to a posterior probability or to replication and external validity checks.

Sample Answer

This question is checking whether you can connect business decision rules to error rates under multiple comparisons. If you hunt across 12 metrics and declare victory when any hits $p<0.05$, your family-wise Type I error can be much larger than 5%. You should pre-specify a primary metric tied to the goal, and treat the rest as secondary or diagnostic. If multiple metrics truly drive the launch decision, you adjust for multiplicity, for example Bonferroni, Holm, or a controlled FDR, and you define a coherent success criterion before the test runs.

Sample Answer

The standard move is to define the estimand as the mean difference and use a t-test, since with enough users the CLT often makes the mean approximately normal. But here, skew and zero inflation can slow convergence, and variance estimates can be unstable, which makes both p-values and power projections brittle. You can use variance reduction like CUPED, or switch to a robust approach like a permutation test on the mean, or model-based methods like a two-part model, while keeping the estimand clear. You also sanity check power via simulation using historical data rather than relying only on closed-form formulas.

Sample Answer

This question is checking whether you can separate correlation from causation and articulate what the regression is actually identifying. If heavy watchers trigger the system to show more recommendations, then recommendations is endogenous and OLS is biased, you are partly measuring user intent. You would propose an experiment or an instrument that shifts recommendations without directly affecting watch time, for example randomized quota of modules or an eligibility rule. Absent that, you would at least reframe the coefficient as an association conditional on controls, not a causal lift per additional recommendation.

Sample Answer

The standard move is to use heteroskedasticity robust standard errors so your inference does not rely on constant variance. But here, modeling $E[y\mid x]$ with a linear mean might still be misspecified if variance increases because the relationship is nonlinear or you have unmodeled segments. You would consider transforming the outcome, adding nonlinear terms, or fitting a model that matches the distribution, for example a Gamma with log link if minutes are positive and skewed. If the goal is decision making, you also check whether prediction intervals are miscalibrated for high usage users, not just whether $p$ values change.

Sample Answer

Get this wrong in production and you will optimize the wrong lever, for example lowering quality because the coefficient looks negative after conditioning on a collinear proxy. The right call is to admit the coefficients are not stable or individually interpretable because the model is trying to separate near redundant signals, so small data shifts change attribution. You would decide whether you need interpretability or prediction, then either drop one feature, combine them, or use a regularized model and report grouped effects. If you need a causal story, you must define the estimand, because conditioning on predicted CTR could be post treatment or a collider depending on how it is constructed.

Sample Answer

The standard move is to fit an empirical Bayes prior, for example a Beta distribution whose mean and variance match the historical city-to-city variation, then use it as $\text{Beta}(\alpha,\beta)$ for the new city. But here, covariate shift matters because a new city can differ in pricing, supply, regulations, or user mix, so the historical cities might be overconfident and miscalibrated. You would downweight history by using a weaker prior (smaller $\alpha+\beta$), or a hierarchical model with a city-level random effect and partial pooling. You avoid using the historical info when you have strong evidence the new city is not exchangeable with the old ones, in which case a skeptical prior centered broadly is safer.

Sample Answer

Get this wrong in production and you either ship a change that increases tail losses, or you block a real win because the mean is dominated by noise. The right call is a robust likelihood, for example Student-t on log revenue or a mixture model, then compute the posterior of incremental value per user and the posterior predictive for downside risk. You report something decision-ready like $P(\Delta > 0)$ plus a credible interval for $\Delta$, and also $P(\Delta < -\tau)$ for a business harm threshold $\tau$. With asymmetric risk, you choose the action that maximizes expected utility, for example ship only if $$\mathbb{E}[U(\Delta)] > 0$$ or if $P(\Delta < -\tau)$ is below a preset risk cap.