Question 57·Hard·Inference from Sample Statistics and Margin of Error
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?
For margin-of-error questions, first turn the sample counts into a proportion or percentage. Then apply the margin of error to create a range (estimate ± margin), and pay attention to whether the question wants the lowest, highest, or just any reasonable estimate. Finally, convert the relevant percentage from that range into an actual count by multiplying by the total population, and match your result to the closest answer choice.
Hints
Start with the sample result
First, find what fraction or percentage of the 500 surveyed voters said they would vote in favor of the bond measure.
Incorporate the margin of error
The margin of error is ±4 percentage points. Using your sample percentage, what is the lowest percentage that could reasonably represent the true support in the whole county?
Move from percentage to number of voters
Once you know the lowest reasonable percentage, apply it to the total of 120,000 registered voters. What operation should you use to turn a percentage into a number of people?
Connect your calculation to the choices
After you compute the smallest reasonable number of supporters, compare your result to the answer options and choose the closest exact match.
Desmos Guide
Check the sample proportion and lower bound (optional)
In Desmos, type 280/500 to see the sample proportion (it should be 0.56). Then type 280/500 - 0.04 to subtract the 0.04 (which is 4 percentage points in decimal form) and see the lower-bound proportion.
Compute the corresponding number of voters
In Desmos, enter the full expression (280/500 - 0.04)*120000. The output is the smallest reasonable estimate of the number of voters who support the measure; compare this number to the answer choices and select the matching one.
Step-by-step Explanation
Find the sample percentage who support the bond
Out of 500 voters, 280 support the bond.
Compute the sample proportion:
- , which is 56%.
So the poll’s estimate is that 56% of voters support the bond, before considering the margin of error.
Use the margin of error to find the lowest plausible support percentage
The margin of error is ±4 percentage points.
Apply this to the 56% estimate:
- Lower bound:
- Upper bound:
The question asks for the smallest reasonable estimate, so we care about the lower bound, 52%.
Write an expression for the smallest reasonable number of supporters
The county has 120,000 registered voters. The smallest reasonable percentage who support the bond is 52%.
Convert 52% to a decimal and set up the calculation:
- Number of supporters (smallest reasonable)
Keep this expression; we will evaluate it in the next step.
Calculate and match to an answer choice
Now compute the product:
So the smallest reasonable estimate of the number of registered voters who support the bond measure is 62,400, which matches answer choice B.