Estimate Family Proportions and Explain Regression Anomalies

Quick Overview

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.

Estimate Family Proportions and Explain Regression Anomalies

Company: Upstart

Role: Data Scientist

Category: Statistics & Math

Difficulty: medium

Interview Round: Onsite

##### 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.

Quick Answer: 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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Aug 4, 2025, 10:55 AM
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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:

  1. 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.
  1. 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.
  1. 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.

Clarifying Questions to Ask Guidance

  • 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 Guidance

  • 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 Guidance

  • 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?
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