Question 14·Medium·Inference from Sample Statistics and Margin of Error
A city has 12,000 registered voters. A random sample of 600 of these voters shows that 58% plan to vote for a certain candidate, and the poll reports a margin of error of 4%.
Based on this poll, which of the following is a plausible value for the total number of registered voters in the city who plan to vote for the candidate?
For questions involving polls and margins of error, first build the plausible range for the true proportion by taking the reported percentage plus and minus the margin of error. Then connect that percentage range to the total population: either (1) multiply the percentage bounds by the total to get a range of counts and see which answer lies inside it, or (2) convert each answer choice to a percentage of the total and check whether it falls inside the percentage range. Choose whichever method involves simpler arithmetic so you can decide quickly and confidently.
Hints
Interpret the margin of error
The sample result is 58% with a 4% margin of error. What two percentages form the lower and upper ends of the plausible range for the true percentage of supporters?
Turn percentages into counts of voters
Once you have the lower and upper percentages, how can you use the total of 12,000 voters to find the corresponding lower and upper bounds for the number of supporters?
Test the choices against your range
After you find the plausible range of numbers of supporters, compare each answer choice to that range. Which choice lies inside that interval?
Desmos Guide
Compute the plausible range of voters
In Desmos, type 0.54*12000 on one line and 0.62*12000 on another. These two outputs give the lower and upper bounds for the plausible number of voters who support the candidate.
Compare each choice to the Desmos range
Now type each answer choice as separate expressions: 6250, 6900, 7800, and 8200. Compare these values to the lower and upper bounds you found in step 1, and identify which one lies between those two bounds.
Step-by-step Explanation
Use the margin of error to find a plausible percentage range
The sample shows 58% support with a margin of error of 4%.
That means the true percentage of all registered voters who support the candidate is likely between:
- Lower bound:
- Upper bound:
So a plausible range for the true support is from to of all registered voters.
Convert the percentage range to a range of numbers of voters
There are 12,000 registered voters in total.
Find how many voters correspond to and of 12,000:
- Lower bound:
- Upper bound:
So a plausible number of voters who support the candidate is between 6,480 and 7,440.
Compare the answer choices to the plausible range
Now check which choice lies between 6,480 and 7,440:
- 6,250 is less than 6,480, so it is too low.
- 6,900 is between 6,480 and 7,440, so it fits the plausible range.
- 7,800 is greater than 7,440, so it is too high.
- 8,200 is also greater than 7,440, so it is too high.
Therefore, the plausible value for the total number of registered voters who plan to vote for the candidate is 6,900.