Question 57·Hard·Inference from Sample Statistics and Margin of Error

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A polling organization surveyed a random sample of 500 registered voters in a county. Of those surveyed, 280 said they would vote in favor of a proposed bond measure.

The organization reports its result with a margin of error of ±4 percentage points at the 95 percent confidence level. The county has 120,000 registered voters. Based on this poll, what is the smallest reasonable estimate of the number of registered voters in the county who support the bond measure?

In a poll, the sample proportion estimates the percentage of the whole population with a given view. A stated margin of error gives a range around that estimate. For the smallest reasonable count, subtract the margin in percentage points, then take that percent of the population in Desmos. Don't subtract the margin twice.

Hints

Hint 1
A sample proportion compares the surveyed supporters with everyone surveyed. Treat the supporters as the part and the whole sample as the denominator. Which two numbers form that fraction?
Hint 2
The margin of error is in percentage points, not a percent of the poll's rate. For the smallest reasonable rate, subtract the stated number of points from the estimated percent once.
Hint 3
To turn a percentage into a countywide count, take it of the county's voter total. Once you have the lower percentage, which number should go after “of” in a Desmos percent calculation?

Step-by-step

Find the lower percent, then the count
Step 1
Write the sample support rate
The sample proportion is the fraction of surveyed voters who support the measure. The $280$ supporters are the part of the sample, and all $500$ surveyed voters are the whole, so:
$\text{sample proportion}=\frac{280}{500}.$
Don't divide by $120{,}000$ here; those voters weren't all surveyed.
Step 2
Find the estimated percentage
Type $280/500$ in Desmos. It shows $0.56$ , so the poll's estimated support rate for the county is $56\%$ . An estimate isn't an exact count of county voters.
Step 3
Find the lowest reasonable rate
The margin of error is $4$ percentage points. That means the low end is $4$ points below $56\%$ , not $4\%$ of $56\%$ . Subtract the margin once:
$56\%-4\%=52\%.$
Subtract the margin in percentage points before scaling to the whole population.
Step 4
Convert the lower rate to voters
The question asks for a count among all registered voters, so take $52\%$ of $120{,}000$ . Type $52\%\operatorname{of}120000$ in Desmos; it prints $62{,}400$ . The smallest reasonable estimate is $62{,}400$ registered voters who support the bond measure. Choice B.

Question 57·Hard·Inference from Sample Statistics and Margin of Error

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