You survey 100 children at a school and ask how many children are in their family. The responses are: - 50 children say their family has 1 child. - 20 children say their family has 2 children. - 30 children say their family has 3 children. Now you go to a random house in the same town and ask how many children live there. Assuming the school sample is representative and family sizes larger than 3 do not occur, estimate the probability that the randomly chosen house has exactly 1 child. ### Constraints & Assumptions - The school survey samples children, not households. - Larger families are more likely to appear in the school sample because they contribute more children. - Assume 1-, 2-, and 3-child households are the only possibilities. - Explain the sampling bias and do not simply use 50%. ### Clarifying Questions to Ask - Are we sampling a random child or a random household in the final question? - Are all children in the town equally likely to attend the sampled school? - Are households with more than 3 children impossible or just unobserved? - Are we ignoring households with zero children? ### Part 1 - Identify The Sampling Bias Why is the school survey not a direct estimate of household probabilities? #### What This Part Should Cover - Child-sampled proportions versus household-sampled proportions. - Size-biased sampling: a 3-child family contributes three children to the school sample. ### Part 2 - Convert Child Proportions To Household Proportions How do you estimate the household distribution? #### What This Part Should Cover - Let `r_k` be the probability a random child comes from a `k`-child household. - Let `q_k` be the probability a random household has `k` children. - Use `q_k` proportional to `r_k / k`, then normalize. ### Part 3 - Compute The Final Probability What is the probability that a random house has exactly 1 child? #### What This Part Should Cover - Computation using either proportions or implied family counts. - Final answer of `5/7`, approximately `71.4%`. ### What a Strong Answer Covers - Recognizes size-biased sampling immediately. - Correctly converts from child-level to household-level probabilities. - Explains why the intuitive 50% answer is wrong. - States assumptions about zero-child and larger-than-3-child households. ### Follow-up Questions - How would the answer change if households with zero children were included? - What if family sizes larger than 3 exist but were not observed? - What is the general formula for any maximum family size? - How would uncertainty from only 100 surveyed children affect the estimate?