53758311
In a population, \(40\%\) support a proposal. Of the supporters, \(80\%\) participate in an optional online poll. Of the people who do not support the proposal, only \(30\%\) participate.
What percentage of poll participants support the proposal, and why is the poll result biased as an estimate of population support?
Hints
- Find the population proportion in each opinion group that actually participates.
- Add those proportions to find the total participation rate.
- Compare the composition of participants with the composition of the full population.
Solution
1. The proportion of the population who support the proposal and participate is \(0.40\cdot 0.80=0.32\).
2. The proportion who do not support the proposal and participate is \(0.60\cdot 0.30=0.18\).
3. The total participation rate is \(0.32+0.18=0.50\).
4. Therefore, the support rate among participants is \(\frac{0.32}{0.50}=0.64\), or \(64\%\).
5. Supporters are more likely to participate, so the respondent group overrepresents supporters. Because participation is optional and related to opinion, this creates voluntary-response bias.
Answer
\(64\%\) of participants support the proposal. The poll overestimates population support because supporters participate at a much higher rate than non-supporters.
