Estimate Family Proportions and Explain Regression Anomalies
Problem Description
You are presented with a population of families, where each family has either 1, 2, or 3 children. From this population, you draw a sample of 100 children. For each child in your sample, you are able to observe the total number of children in their respective family.
Address the following three distinct statistical problems:
1. Estimating Family-Type Proportions
- From the collected child sample, estimate the true proportions of 1-child, 2-child, and 3-child families within the overall population of families (not the population of children).
- Construct 95% confidence intervals for each of these estimated family-type proportions.
2. OLS Asymmetry and Causality
- Explain the fundamental statistical reasons why the Ordinary Least Squares (OLS) regression slope obtained from regressing Y on X (i.e., Y ~ X) typically differs from the slope obtained from regressing X on Y (i.e., X ~ Y).
- Relate this observed asymmetry to the critical distinction between statistical association and causal direction.
3. Prediction Strength vs. Coefficient Significance
- Consider a scenario where a regression model demonstrates strong predictive performance, yet all of its individual coefficients are found to be statistically insignificant.
- Provide a comprehensive statistical explanation for this seemingly contradictory outcome and propose concrete methods to address or "fix" this situation.
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Core dilemma
Accurately inferring population parameters from biased samples and robustly interpreting statistical model outputs under non-ideal conditions.
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Estimate Family Proportions and Explain Regression Anomalies
Problem Description
You are presented with a population of families, where each family has either 1, 2, or 3 children. From this population, you draw a sample of 100 children. For each child in your sample, you are able to observe the total number of children in their respective family.
Address the following three distinct statistical problems:
1. Estimating Family-Type Proportions
- From the collected child sample, estimate the true proportions of 1-child, 2-child, and 3-child families within the overall population of families (not the population of children).
- Construct 95% confidence intervals for each of these estimated family-type proportions.
2. OLS Asymmetry and Causality
- Explain the fundamental statistical reasons why the Ordinary Least Squares (OLS) regression slope obtained from regressing Y on X (i.e., Y ~ X) typically differs from the slope obtained from regressing X on Y (i.e., X ~ Y).
- Relate this observed asymmetry to the critical distinction between statistical association and causal direction.
3. Prediction Strength vs. Coefficient Significance
- Consider a scenario where a regression model demonstrates strong predictive performance, yet all of its individual coefficients are found to be statistically insignificant.
- Provide a comprehensive statistical explanation for this seemingly contradictory outcome and propose concrete methods to address or "fix" this situation.