Estimate Family Proportions and Explain Regression Anomalies
Overview
This question evaluates applied statistical inference and causal reasoning, covering estimation of population-type proportions from a sample with size-related selection bias, construction and interpretation of confidence intervals, the asymmetry of OLS slopes under reversed regressions, and diagnosis of cases where models predict well but coefficients lack significance. Commonly asked in Statistics & Math interviews for data scientist roles, it tests both conceptual understanding (association versus causal direction) and practical application (sampling design effects and model diagnostic interpretation).
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.
Solution
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