Knew this cold but still second-guessed my wording mid-answer.
Start by defining the null and alternative hypotheses clearly: H0: μ = 0 vs. Ha: μ ≠ 0. Then explain that a p-value of x% is the probability, assuming H0 is true, of observing a sample mean at least as extreme as the one obtained. Finally, interpret it in context: if x% is less than the significance level (e.g., 5%), you reject H0 and conclude the population mean differs from 0; otherwise, you fail to reject H0.
Pro tip: Emphasize that the p-value is not the probability that H0 is true or that the result is due to chance; it's a measure of evidence against H0 under the assumption that H0 is true. Also, mention that practical significance depends on effect size and context, not just the p-value.
Clearly define the null hypothesis (H0: μ = 0) and the alternative hypothesis (Ha: μ ≠ 0). This sets the foundation for interpreting the p-value.
Explain that the p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample, assuming the null hypothesis is true.
If the p-value is x%, it means that if the true population mean were 0, there is an x% chance of seeing a sample mean at least as far from 0 as the one observed.
Compare the p-value to a pre-determined significance level (α, often 0.05). If p-value < α, reject H0; otherwise, fail to reject H0.
State the practical conclusion: either there is sufficient evidence to suggest the population mean differs from 0, or there is not enough evidence to conclude that.
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Start by stating the formula for a confidence interval: sample mean ± critical value × standard error. For a 95% confidence level, use the z-critical value of 1.96 (assuming a large sample). Then plug in the given values to compute the interval and interpret it in context.
Pro tip: Mention that this assumes the sampling distribution is approximately normal, which typically holds for large samples due to the Central Limit Theorem. Also, briefly note that if the sample size were small, you would use a t-critical value instead.
Recognize that the sample mean is 1, the standard error is 0.1, and the confidence level is 95%.
For a 95% confidence interval, the critical z-value is approximately 1.96 for large samples.
Multiply the critical value by the standard error: 1.96 × 0.1 = 0.196.
Add and subtract the margin of error from the sample mean: 1 ± 0.196, yielding (0.804, 1.196).
Explain that we are 95% confident that the true population mean lies between 0.804 and 1.196.
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The first part is just algebra on se = s divided by root n, so you need 100x the sample size.
Start by explaining the relationship between standard error and sample size: SE = σ/√n, so to reduce SE by a factor of 10, you need 100 times the sample size. Then discuss practical alternatives when increasing sample size isn't feasible, such as reducing variance, using sequential testing, or leveraging Bayesian methods.
Pro tip: Quantify the trade-offs: increasing sample size by 100x may be infeasible due to cost or time, so emphasize variance reduction techniques (e.g., stratification, CUPED) that can achieve similar precision gains without more data.
Use the formula SE = σ/√n to show that reducing SE from 0.1 to 0.01 requires a 100-fold increase in sample size (since SE scales inversely with the square root of n).
Acknowledge that a 100x increase is often impractical due to cost, time, or limited traffic, and that it may not be necessary if the effect size is large enough.
Mention methods like stratification, blocking, using covariates (CUPED), or paired designs that reduce σ and thus SE without increasing n.
Suggest sequential testing, Bayesian methods, or using a more sensitive metric to achieve reliable results with smaller samples.
Conclude that the best approach depends on the context, and recommend a combination of variance reduction and appropriate statistical methods when sample size is fixed.
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Used the plug-in estimator, just the proportion of observations above 10.
Start by defining the estimator as the sample proportion of observations exceeding 10, then use the normal approximation to construct a Wald confidence interval. Alternatively, mention exact methods like Clopper-Pearson for small samples, and interpret the interval in terms of the true probability.
Pro tip: Always check the sample size and number of successes to ensure the normal approximation is valid; if not, use exact or Wilson interval. Also, relate the interval to business impact, e.g., 'We are 95% confident that the true probability lies between X and Y, which means...'
Let p be the probability that X > 10. The natural estimator is the sample proportion: p_hat = (1/n) * sum(I(X_i > 10)).
Verify that the sample size is large enough for the normal approximation: n*p_hat >= 10 and n*(1-p_hat) >= 10. If not, consider exact methods.
For large samples, use the Wald interval: p_hat ± z_{α/2} * sqrt(p_hat*(1-p_hat)/n). For small samples, use Wilson or Clopper-Pearson.
Interpret in context: 'We are 95% confident that the true probability that X exceeds 10 lies between L and U.' Avoid saying 'there is a 95% probability that p is in the interval.'
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This one felt a bit underspecified and I spent maybe 30 seconds clarifying what 'overall' meant before diving in.
Start by clarifying the goal: estimate the overall conversion rate across all features, not per-feature. Then propose a sampling strategy that balances statistical efficiency with practical constraints, such as stratified sampling by feature prevalence or using a model-based approach like logistic regression with regularization to handle high dimensionality.
Pro tip: Emphasize the bias-variance tradeoff and the importance of validating your sampling strategy with a hold-out set or simulation. Mention that in practice, you'd also consider computational cost and interpretability for stakeholders.
Confirm whether the goal is a single overall conversion rate or per-feature rates, and understand data size, computational resources, and business context.
Decide between simple random sampling, stratified sampling by feature combinations, or a model-based approach (e.g., logistic regression with L1/L2 regularization) to estimate the overall rate efficiently.
With 1,000 binary features, many combinations may be sparse. Consider dimensionality reduction (e.g., PCA) or feature hashing, and use regularization to prevent overfitting.
Use cross-validation or bootstrapping to assess the stability of your estimate and provide confidence intervals. Compare against a simple baseline (e.g., overall conversion rate without features).
Summarize the chosen approach, its assumptions, and limitations. Discuss how the estimate would be used and whether a more granular analysis is needed.
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This is where I felt the floor drop out a little.
Start by framing the problem as a truncated normal distribution and explain that naive estimates (sample mean and variance) are biased. Then outline a maximum likelihood estimation (MLE) approach using the truncated normal likelihood, and describe how to construct confidence intervals via the asymptotic normality of MLE or bootstrap. Emphasize the importance of accounting for truncation in both estimation and inference.
Pro tip: Mention that the truncation point (3) is known and fixed, so the likelihood is conditional on X > 3; this simplifies the MLE derivation and avoids identifiability issues. Also, note that if the truncation is severe, the information about the mean and variance is limited, leading to wider confidence intervals.
Clarify that the observed data come from a normal distribution conditional on X > 3, with density f(x; μ, σ²) / (1 - Φ((3-μ)/σ)) for x > 3. This is the likelihood contribution for each observation.
Write the log-likelihood for the sample: sum log f(x_i; μ, σ²) - n log(1 - Φ((3-μ)/σ)). Maximize numerically (e.g., Newton-Raphson) to obtain MLEs for μ and σ².
Use the observed Fisher information matrix to get standard errors for the MLEs, then form Wald-type confidence intervals. Alternatively, use bootstrap or profile likelihood for more accurate intervals, especially with small samples.
Verify normality of the underlying distribution (e.g., via QQ plot of the truncated data) and assess whether the truncation threshold is correctly specified. Consider sensitivity analysis if the threshold is uncertain.
Explain how truncation affects estimation: the sample mean underestimates μ, and the sample variance underestimates σ². Highlight that ignoring truncation leads to biased estimates and invalid inference.
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