##### Scenario On-site statistics round ##### Question Population contains one-, two- and three-child families. Estimate the proportion of each family type from a sample of 100 children. Construct a 95 % confidence interval. Explain why the OLS coefficient of Y~X differs from X~Y and relate to causal direction. A regression model shows all coefficients statistically insignificant yet high predictive performance. Provide a statistical explanation and propose a fix. ##### Hints Multinomial proportions, bootstrap CI; reverse causality; multicollinearity and LASSO/ridge.

Estimate Family Proportions and Explain Regression Anomalies evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

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What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Onsite rounds at Upstart.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Upstart during technical interviews.

Estimate Family Proportions and Explain Regression Anomalies

On-site Statistics Round

Task Overview

You are given a population of families that have either 1, 2, or 3 children. You sample 100 children (i.e., the sampling unit is a child, not a family). For each sampled child, you can observe the size of their family.

Answer the following:

Estimating family-type proportions

From the child sample, estimate the proportions of 1-child, 2-child, and 3-child families in the population of families (not in the population of children).
Construct 95% confidence intervals for those family-type proportions.

OLS asymmetry and causality

Explain why the OLS slope from Y ~ X generally differs from the slope from X ~ Y.
Relate this to the distinction between association and causal direction.

Prediction strong, coefficients insignificant

A regression shows all coefficients are statistically insignificant, yet the model predicts well. Provide a statistical explanation and propose fixes.

Hints: Multinomial proportions with size-bias correction and bootstrap CIs; regression asymmetry and reverse causality; multicollinearity and ridge/LASSO.

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.
If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask

Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
Show enough derivation for the interviewer to follow the reasoning.
Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers

A correct setup with definitions, formulas, and boundary conditions.
A step-by-step derivation or estimation plan.
Interpretation of the result, including uncertainty and practical limitations.
Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions

How would the result change if the assumptions were relaxed?
Can you verify the answer with a simulation?
What is the most likely source of estimation error?