Estimate Population Mean and Conversion Rate Accurately
Quick Overview
Evaluates core statistical inference skills across p-values, confidence intervals, standard error reduction, tail probability estimation, efficient conversion-rate sampling, and truncated-normal estimation. Strong answers state assumptions, use appropriate intervals and weights, and avoid common interpretation errors.
Estimate Population Mean and Conversion Rate Accurately
Company: Google
Role: Data Scientist
Category: Statistics & Math
Difficulty: hard
Interview Round: Onsite
##### 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.
Quick Answer: Evaluates core statistical inference skills across p-values, confidence intervals, standard error reduction, tail probability estimation, efficient conversion-rate sampling, and truncated-normal estimation. Strong answers state assumptions, use appropriate intervals and weights, and avoid common interpretation errors.
Estimate Population Mean and Conversion Rate Accurately
Google
Jul 12, 2025, 6:59 PM
hardData ScientistOnsiteStatistics & Math
76
0
Estimate Population Mean and Conversion Rate Accurately
You are asked a series of statistical inference questions covering hypothesis testing, confidence intervals, sampling design, tail probability estimation, conversion-rate estimation, and estimation under truncation.
Constraints & Assumptions
Assume independent sampling unless a question states otherwise.
Use standard large-sample approximations when appropriate, and state when a small-sample or exact method would be better.
Explain interpretations in plain language, not only formulas.
For the truncated normal question, account for truncation explicitly instead of treating the observed sample as an ordinary normal sample.
Clarifying Questions to Ask Guidance
Are sample sizes large enough for normal approximations?
Is the standard deviation known or estimated?
Are the samples simple random samples, stratified samples, or case-control samples?
Is the truncation threshold known and applied consistently?
Part 1 - Interpret a P-Value
You test whether a population mean differs from 0 using a two-sided test. What does a p-value of x% mean?
What This Part Should Cover Guidance
Probability of observing a result at least as extreme as the sample result under the null hypothesis.
Clear rejection of common misinterpretations, such as treating the p-value as the probability the null is true.
Part 2 - Construct a Confidence Interval
Given sample mean x_bar = 1 and standard error SE = 0.1, construct a 95% confidence interval for the population mean and state assumptions.
What This Part Should Cover Guidance
Normal or t critical value depending on assumptions.
The interval
x_bar +/- critical_value * SE
.
Interpretation of the confidence procedure.
Part 3 - Reduce Standard Error
What sample-size factor is needed to reduce standard error from 0.1 to 0.01?
What This Part Should Cover Guidance
Relationship
SE
proportional to
1 / sqrt(n)
when variance is stable.
Squared ratio of old to new standard error.
Implication that sample size must increase by a factor of 100.
Part 4 - Improve Inference Without More Sample
If you cannot increase sample size, what actions can improve inference, precision, or power?
Given independent observations from a distribution X, propose an estimator for P(X > 10), construct a 95% confidence interval, and interpret an interval [a, b].
What This Part Should Cover Guidance
Indicator estimator for exceedance probability.
Standard error and interval options such as normal approximation, Wilson, or exact interval.
Correct frequentist interpretation of confidence intervals.
Part 6 - Estimate Overall Conversion with Many Binary Features
You want to estimate the overall conversion rate in a population where each unit has 1,000 binary features. Describe an efficient, unbiased or approximately unbiased estimation and sampling strategy.
What This Part Should Cover Guidance
Simple random sampling as a baseline.
Stratified sampling, post-stratification, inverse-probability weighting, calibration, or model-assisted estimation.
Importance of known inclusion probabilities and representative population margins.
Design-effect and variance estimation.
Part 7 - Estimate a Truncated Normal
Assume X is normally distributed with mean mu and variance sigma^2, but you only observe Y = X | X > 3. How would you estimate mu and sigma^2, and how would you construct 95% confidence intervals?
What This Part Should Cover Guidance
Truncated-normal likelihood rather than naive sample mean and variance.
Maximum likelihood estimation for
mu
and
sigma
, solved numerically if needed.
Confidence intervals using observed Fisher information, delta method, likelihood profiling, or parametric bootstrap.
What a Strong Answer Covers Guidance
A strong answer explains each inference task accurately, states assumptions, chooses appropriate approximations, and avoids common interpretation errors around p-values, confidence intervals, sampling weights, and truncation.
Follow-up Questions Guidance
When would you use a Wilson interval instead of a normal interval for a proportion?
How would you report uncertainty under a complex sampling design?
What bias occurs if you ignore truncation in the normal-estimation problem?