This one got me for a second because my instinct was to just say 0.5 and move on, but they wanted the full derivation.
First, clarify that flipping the regression means treating y as the predictor and X as the response. Then derive the OLS coefficient using the standard formula β = Cov(X,y)/Var(y), and compute the covariance and variance using the given distributions. Finally, interpret the result, noting that the coefficient is 1/2 and discussing implications such as attenuation bias and the difference between regression and causal direction.
Pro tip: Emphasize that the coefficient is not 1, which might seem counterintuitive; explain that this is due to errors-in-variables attenuation, and mention that the R² remains the same (1/2) in both directions, highlighting the asymmetry in regression coefficients.
State that the new regression is X = α + β y + u, where y is the predictor and X is the response. Note that OLS estimates β as Cov(X,y)/Var(y).
Since y = X + ε, with X and ε independent standard normals, Cov(X, y) = Cov(X, X+ε) = Var(X) + Cov(X, ε) = 1 + 0 = 1.
Var(y) = Var(X+ε) = Var(X) + Var(ε) + 2Cov(X, ε) = 1 + 1 + 0 = 2.
Plug in the values: β = Cov(X,y)/Var(y) = 1/2. Also note that the intercept α = E[X] - β E[y] = 0 - (1/2)*0 = 0.
Explain that the coefficient is 1/2, not 1, due to attenuation bias from regressing on a noisy predictor. Mention that the R² is the same (1/2) in both directions, and discuss the asymmetry and potential pitfalls in causal interpretation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying the sampling scheme: if children are sampled uniformly, the family-size distribution is size-biased, so the observed proportions estimate the size-biased distribution, not the true family-size distribution. Then compute the MLE and a standard 95% confidence interval for the proportion of 1-child families under the appropriate model, and finally outline a Bayesian approach using a Dirichlet prior on the true family-size proportions, incorporating the size-biasing likelihood to obtain an exact credible interval.
Pro tip: Explicitly state the sampling assumption and note the size-biased sampling issue; this shows statistical maturity and avoids a common pitfall. Also, mention that the Dirichlet-multinomial model yields a Beta posterior for the proportion of interest, enabling exact credible intervals.
Determine whether the sample is of children or families. If children are sampled uniformly, the observed family sizes follow a size-biased distribution, so the proportion of 1-child families in the sample is not an unbiased estimate of the population proportion. Define the target parameter as the true proportion of 1-child families in the village.
Assuming the sample is a simple random sample of children, the MLE of the size-biased proportion of 1-child families is 0.5. Use the normal approximation to construct a 95% confidence interval: p_hat ± 1.96 * sqrt(p_hat(1-p_hat)/n). If correcting for size bias, derive the relationship between size-biased and true proportions and estimate the true proportion accordingly.
Place a Dirichlet prior on the true family-size proportions (π1, π2, π3). The likelihood for the observed counts, given the sampling scheme, is multinomial with probabilities proportional to kπ_k (size-biased). Combine prior and likelihood to obtain the posterior distribution.
Because the size-biased probabilities are a linear transformation of the true proportions, the posterior for the true proportion of 1-child families may not have a closed form. Use Monte Carlo sampling from the Dirichlet posterior (or numerical integration) to obtain the exact posterior distribution and compute the 95% credible interval.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.