Question 15·Hard·Inference from Sample Statistics and Margin of Error
Survey Results
| Response | Number of respondents |
|---|---|
| Yes | 624 |
| No | 576 |
| Total | 1,200 |
The polling organization reports a margin of error of ±3 percentage points for this survey.
In the upcoming election, 25,000 registered voters with similar demographics are expected to cast ballots. Based on the survey results and its margin of error, what is the greatest number of these 25,000 voters who could reasonably be expected to vote No on Proposition A?
For survey and margin‑of‑error questions, first convert the raw counts in the table into a percentage for the group of interest (here, the No responses). Then adjust that percentage by the margin of error, choosing the upper bound if the question asks for the greatest possible number and the lower bound if it asks for the least. Finally, convert the adjusted percentage to a decimal and multiply by the total number in the larger population; be careful that a margin of error in “percentage points” (like ±3) means you add or subtract 0.03 from the proportion, not 3% of the population size directly.
Hints
Turn the counts into a percentage
First focus only on the survey table. What fraction of the 1,200 respondents answered No, and what percentage is that?
Use the margin of error correctly
The margin of error is ±3 percentage points. Starting from the No percentage you just found, what is the highest percentage of No voters that is still consistent with the survey?
Scale up to 25,000 voters
Once you have the highest reasonable No percentage, convert it to a decimal and multiply by 25,000 to get the corresponding number of voters.
Desmos Guide
Compute the maximum expected number of No votes
In Desmos, type the expression
Here, is the sample proportion who said No, is the 3‑percentage‑point margin of error written as a decimal, and multiplying by scales it to the full group. Use the output of this expression as the greatest reasonable number of No votes.
Step-by-step Explanation
Find the sample percentage who said No
From the table, 576 out of 1,200 respondents answered No.
So, in the sample, 48% of respondents said No.
Apply the margin of error to get the maximum No percentage
The margin of error is ±3 percentage points. This means the true percentage of No voters could reasonably be as low as 48% − 3% or as high as 48% + 3%.
For the greatest number of No votes, use the highest possible percentage:
So up to 51% of similar voters could reasonably be expected to vote No.
Apply that percentage to 25,000 voters
Now find 51% of the 25,000 expected voters:
So the greatest number of the 25,000 voters who could reasonably be expected to vote No is 12,750.