I knew the formula but the number still surprised me when I computed it.
Start by clearly stating Bayes' rule and defining each term in the context of the problem. Then plug in the given values (prevalence, sensitivity, false positive rate) and compute the posterior probability step by step, explaining the intuition behind the result.
Pro tip: Emphasize that the low prevalence (1%) makes the base rate dominate, so even with a highly sensitive test, the probability of having the disease given a positive result is surprisingly low. This demonstrates an understanding of the base rate fallacy, which is crucial in data science and product analytics.
Write the formula: P(D|+) = P(+|D) * P(D) / P(+), and define each term: P(D) is prevalence, P(+|D) is sensitivity, P(+|¬D) is false positive rate.
Assign the numbers: P(D) = 0.01, P(+|D) = 0.99, P(+|¬D) = 0.05. Note that P(¬D) = 0.99.
Calculate P(+) = P(+|D)*P(D) + P(+|¬D)*P(¬D) = 0.99*0.01 + 0.05*0.99 = 0.0099 + 0.0495 = 0.0594.
Compute P(D|+) = (0.99 * 0.01) / 0.0594 ≈ 0.1667, or about 16.7%.
Explain that despite a positive test, the probability of having the disease is only ~16.7% due to the low base rate, illustrating the base rate fallacy.
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Talked through omitted variable bias fine, but fumbled a bit on the second part.
Start by explaining that adding controls can change the main coefficient due to confounding or mediation, then discuss how controls can introduce bias through collider stratification or overcontrol. Emphasize the importance of using causal diagrams (DAGs) to decide which variables to control for, and highlight the trade-off between bias and variance.
Pro tip: Mention that in observational studies, the goal is to estimate a causal effect, so you must control for confounders but avoid controlling for mediators or colliders. At Netflix, where experimentation is gold standard, you'd validate observational findings with A/B tests when possible.
Adding controls can change the main coefficient because it adjusts for confounding (removing omitted variable bias) or because it blocks a causal pathway (mediation), altering the interpretation.
Confounders should be controlled to get unbiased estimates; mediators should not be controlled if you want the total effect; colliders should never be controlled as they induce spurious associations.
Controls can introduce bias (e.g., collider bias, overcontrol bias), increase variance, or reduce statistical power, especially with small samples or many controls.
Use DAGs to identify the minimal sufficient adjustment set and avoid controlling for inappropriate variables.
Acknowledge that observational findings should be validated with randomized experiments (A/B tests) when possible, as they eliminate confounding.
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Start by formally stating the Central Limit Theorem (CLT), then immediately pivot to its practical implications for the sampling distribution of the sample mean. Use a concrete example relevant to Netflix (e.g., average watch time per user) to illustrate how the CLT enables inference and hypothesis testing even when the population distribution is unknown.
Pro tip: Emphasize that the CLT applies to the sampling distribution of the mean, not the raw data, and that the approximation improves with sample size—this shows you understand the subtlety and can avoid common misinterpretations.
Define the Central Limit Theorem: For a population with finite mean μ and finite variance σ², the sampling distribution of the sample mean approaches a normal distribution with mean μ and variance σ²/n as the sample size n becomes large, regardless of the population's shape.
Describe how the CLT allows us to use normal-based methods (e.g., confidence intervals, hypothesis tests) for the sample mean even when the population distribution is not normal, provided n is sufficiently large.
Mention that the required n depends on the population's skewness and kurtosis; for highly skewed data, larger n is needed. Typically n ≥ 30 is a rule of thumb, but it's not universal.
Give a Netflix-specific example: e.g., estimating average daily watch time per user. Even if individual watch times are skewed, with a large enough sample, the sample mean will be approximately normal, enabling A/B testing and metric comparisons.
Summarize that the CLT underpins many statistical procedures used in product analytics, such as t-tests, confidence intervals for means, and control charts, making it foundational for data-driven decision-making.
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Start by clearly stating the probability density function (PDF) of the uniform distribution on (a, b). Then derive the mean using the integral of x times the PDF, and the variance using the definition Var(X) = E[X^2] - (E[X])^2. Show each step of the integration and simplify to the final formulas.
Pro tip: Emphasize that the derivation is straightforward but be prepared to explain why the variance formula makes intuitive sense (e.g., it scales with the square of the interval width). Also, mention that in practice, you might use these formulas to quickly compute summary statistics for uniformly distributed data.
Write the probability density function of X: f(x) = 1/(b-a) for a < x < b, and 0 otherwise.
Compute E[X] = ∫_a^b x * f(x) dx = (1/(b-a)) ∫_a^b x dx = (a+b)/2.
Compute E[X^2] = ∫_a^b x^2 * f(x) dx = (1/(b-a)) ∫_a^b x^2 dx = (a^2 + ab + b^2)/3.
Use Var(X) = E[X^2] - (E[X])^2 = (a^2 + ab + b^2)/3 - ((a+b)/2)^2 = (b-a)^2/12.
State the final results: mean = (a+b)/2, variance = (b-a)^2/12. Optionally, discuss the standard deviation and its interpretation.
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State the formula for the one-sample t-statistic clearly: t = (x̄ - μ₀) / (s / √n), defining each term. Then briefly explain the intuition: it measures how many standard errors the sample mean is from the hypothesized mean, and note the t-distribution with n-1 degrees of freedom.
Pro tip: Mention that in A/B testing contexts, this is essentially the same as comparing a treatment mean to a baseline, and that using the sample standard deviation (not population) is why we use t rather than z.
Identify x̄ as the sample mean, μ₀ as the hypothesized population mean, s as the sample standard deviation, and n as the sample size.
State the t-statistic: t = (x̄ - μ₀) / (s / √n). Emphasize that the denominator is the standard error of the mean.
Describe the t-statistic as the number of standard errors the sample mean is away from the hypothesized mean, indicating how unusual the sample would be if the null hypothesis were true.
Note that under the null hypothesis, the t-statistic follows a t-distribution with n-1 degrees of freedom, which accounts for the uncertainty in estimating the standard deviation.
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This one tripped me up more than it should have.
Start by clearly defining effect size and MDE, then explain how statistical power, variance, sample size, and significance threshold each influence MDE. Use a formula or relationship to show the interplay, and conclude with practical implications for experiment design at Netflix.
Pro tip: Emphasize that MDE is a design parameter, not a post-hoc calculation—it should be determined before the experiment to ensure adequate power. Also, mention that reducing variance (e.g., via stratification or CUPED) can lower MDE without increasing sample size.
Effect size quantifies the magnitude of a difference (e.g., relative lift, Cohen's d). MDE is the smallest true effect an experiment can detect with a given power and significance level.
Present the relationship: MDE ∝ (Z_{1-α/2} + Z_{1-β}) * σ / √n, where α is significance threshold, β is Type II error, σ is variance, and n is sample size per group.
Describe how increasing power (1-β) or significance threshold (α) increases MDE; increasing variance increases MDE; increasing sample size decreases MDE.
Explain how to use these relationships to design experiments: e.g., to detect smaller effects, increase sample size, reduce variance, or accept lower power.
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