Estimate Population Mean and Conversion Rate Accurately

Scenario

General statistical inference tasks: testing a population mean, controlling standard error, estimating tail probabilities, computing conversion rate, and parameter estimation under truncated normal sampling.

Question

You collected a sample and want to test whether the population mean differs from 0. What does a p-value of x% mean in this hypothesis-testing context? Given sample mean x̄ = 1 and standard error 0.1, construct the 95% confidence interval for the population mean. What sample size would you need to achieve a standard error of 0.01 instead of 0.1? What actions can you take if the sample size cannot be increased? Given independent observations X₁,…,Xₙ from distribution X, propose an estimator for p = P(X > 10). Construct a 95% confidence interval for P(X > 10) and interpret a resulting interval [a, b] in terms of the true probability p. You have 1,000 binary features and want to estimate the overall conversion rate. Describe how you would design the estimation or sampling strategy. Assume X ∼ N(μ, σ²) but you only observe Y = X conditioned on X > 3 (a truncated normal). How would you estimate μ and σ²? How would you construct 95% confidence intervals for μ and σ² under this truncation setting?

Hints

Use definitions of p-value, z/t intervals, se = s/√n, plug-in estimator for probability, normal or Wilson CI, sample size formula, MLE for truncated normal, bootstrap/delta method for CI.