Estimate and Derive Regression Coefficient for X on y
Problem Description
This interview consists of two distinct statistical problems, designed to assess your foundational understanding of linear regression, sampling theory, and both frequentist and Bayesian estimation techniques.
Part 1: Linear Regression Coefficient Derivation
You are given a simple linear data-generating process defined by the equation y = X + ε. In this model, X and ε are independent standard normal random variables (i.e., mean 0, variance 1).
Your task is to:
- Regress
Xonyusing ordinary least squares (OLS), including an intercept term. - Determine the regression coefficient
β. - Provide a clear derivation for this coefficient.
Part 2: Addressing Sampling Bias and Bayesian Estimation
Consider a village where every family has either 1, 2, or 3 children. You conduct a survey by randomly sampling 100 children (not families) from this village. The observed distribution of these children by their family size is as follows:
- 50 children are from 1-child families.
- 30 children are from 2-child families.
- 20 children are from 3-child families.
Let π = (π1, π2, π3) represent the true proportions of families in the village with 1, 2, and 3 children, respectively. Due to the sampling method (sampling children rather than families), the observed proportions of children from each family size are not equal to π.
Your tasks are to:
- Estimate the proportion
π1of 1-child families in the village. - Construct a 95% confidence interval for
π1. - Describe how to obtain an “exact” Bayesian credible interval for
π1by using a Dirichlet prior and deriving the posterior distribution.
AI interview prep
Core dilemma
Accurately inferring underlying population parameters despite biased sampling mechanisms, coupled with demonstrating a robust understanding of fundamental statistical derivations and contrasting frequentist vs. Bayesian estimation paradigms.
2 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Estimate and Derive Regression Coefficient for X on y
Problem Description
This interview consists of two distinct statistical problems, designed to assess your foundational understanding of linear regression, sampling theory, and both frequentist and Bayesian estimation techniques.
Part 1: Linear Regression Coefficient Derivation
You are given a simple linear data-generating process defined by the equation y = X + ε. In this model, X and ε are independent standard normal random variables (i.e., mean 0, variance 1).
Your task is to:
- Regress
Xonyusing ordinary least squares (OLS), including an intercept term. - Determine the regression coefficient
β. - Provide a clear derivation for this coefficient.
Part 2: Addressing Sampling Bias and Bayesian Estimation
Consider a village where every family has either 1, 2, or 3 children. You conduct a survey by randomly sampling 100 children (not families) from this village. The observed distribution of these children by their family size is as follows:
- 50 children are from 1-child families.
- 30 children are from 2-child families.
- 20 children are from 3-child families.
Let π = (π1, π2, π3) represent the true proportions of families in the village with 1, 2, and 3 children, respectively. Due to the sampling method (sampling children rather than families), the observed proportions of children from each family size are not equal to π.
Your tasks are to:
- Estimate the proportion
π1of 1-child families in the village. - Construct a 95% confidence interval for
π1. - Describe how to obtain an “exact” Bayesian credible interval for
π1by using a Dirichlet prior and deriving the posterior distribution.