Problem — Margin of error and representativeness

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Explain why a poll of $1500$ people (representative of the country) has roughly the same margin of error as a poll of $1500$ people in a single city.

Solution

Set-up. The margin of error $E = z\sqrt{\dfrac{p(1-p)}{n}}$ depends on the sample size $n$, not on the size of the population it is drawn from (as long as the population is much larger than the sample). With the same $n=1500$ and similar values of $p$, the standard error is practically identical in the two cases. The subtle point is another: the margin of error only measures statistical uncertainty, not representativeness. A city sample may be large but systematically skewed with respect to the whole country (selection bias), so having the same margin of error does not mean having the same validity of the conclusions about the national population.

Links

Topics: Probabilita
Concepts: Errore standard · Margine di errore
Skills: Stimare
Exercise type: Problema modellizzazione