Question 35·Medium·Inference from Sample Statistics and Margin of Error

The sample proportion is the fraction of surveyed voters who favored the expansion. The $280$ favorable responses are out of the $500$ surveyed, so the fraction is $\frac{280}{500}$. Don't divide by $62{,}000$: that's the county total, not the number surveyed.

Type $280/500$ into Desmos. It shows $0.56$, or $56\%$. A random sample is selected by chance, so this percentage can estimate the share of county registered voters who favor the expansion, not establish it exactly.

The margin of error is $4$ percentage points on either side of the $56\%$ estimate. Subtract for the lower end:

Add for the upper end:

Don't take $4\%$ of $56\%$; that would be a percent change, not a difference of $4$ points.

Apply the lower endpoint to all $62{,}000$ county registered voters, not the $500$ surveyed. Type $52\%\operatorname{of}62000$ into Desmos; it shows $32{,}240$ as the lower plausible count.

Apply the upper endpoint to the same county total. Add $60\%\operatorname{of}62000$ below the earlier Desmos lines; it shows $37{,}200$ as the upper plausible count.

The endpoints round to roughly $32{,}000$ and $37{,}000$. Because the poll estimates rather than counts everyone, the appropriately cautious conclusion is that between approximately $32{,}000$ and $37{,}000$ county registered voters likely favor the park expansion. Choice B.