First, clarify the sampling assumptions: the 100 kids are a random sample of children, so the distribution of family sizes is size-biased when sampling households. Then, compute the probability of selecting a household with exactly 1 child using the formula P(1 child) = (number of children in 1-child families) / (total number of children), which is 50 / (50*1 + 20*2 + 30*3) = 50/180 = 5/18 ≈ 0.278. Finally, discuss potential biases and whether the school sample is representative of the town.
Pro tip: Explicitly state that you're assuming the 100 kids are a random sample of all children in the town and that family sizes are stable; this shows you understand the size-biased sampling issue and avoids overcomplicating with unnecessary assumptions.
State that the 100 kids are assumed to be a simple random sample of children from the town, and that each child's response accurately reflects their family size.
Explain that when you knock on a random house, you're more likely to land on a family with more children because larger families occupy more houses (if each family has one house) or have more children to be sampled.
Calculate the probability as the proportion of children who live in 1-child families: 50 / (50*1 + 20*2 + 30*3) = 50/180 = 5/18 ≈ 0.278.
Mention that this assumes each family occupies exactly one house and that the school sample is representative of the town; otherwise, the estimate may be biased.
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