Question 28·Easy·Inference from Sample Statistics and Margin of Error
A polling organization surveyed 200 randomly selected registered voters in a town with 10,000 registered voters about a new proposal. In the sample, 56% said they support the proposal, and the poll’s reported margin of error is percentage points.
Based on these results, which of the following could be the number of all registered voters in the town who support the proposal?
For questions involving polls and margins of error, first turn the reported percentage plus or minus the margin of error into a range of possible true percentages (reported ± margin). Then, if the total population is given, multiply each endpoint of that percentage range (as a decimal) by the total population to get a range of possible counts. Finally, eliminate any answer choices that fall outside this interval; usually, only one will be inside. This approach is quick and avoids overthinking the statistical details.
Hints
Understand the margin of error
Margin of error tells you how far above or below the reported percentage the true percentage might be. Use it to create a low and high possible percentage around 56%.
Find the possible percentage range
Compute 56 minus 7 and 56 plus 7 to find the smallest and largest possible percentages of all voters who may support the proposal.
Convert percentages to counts
Once you have the percentage range, turn those percentages into actual numbers of supporters by multiplying each percentage (as a decimal) by 10,000.
Check which option fits the range
Look to see which of the answer choices lies between the smallest and largest possible numbers of supporters you just calculated.
Desmos Guide
Compute the percentage interval
In Desmos, type 56-7 and 56+7 on separate lines to see the low and high possible percentages of support based on the margin of error. These should be your lower and upper percentage bounds.
Convert percentages to numbers of voters
Next, convert each percentage bound to a decimal and multiply by 10000. In Desmos, type (56-7)/100*10000 on one line and (56+7)/100*10000 on another. These outputs are the minimum and maximum plausible numbers of supporters.
Compare with choices
Compare each answer choice to the two numbers Desmos gave you. Identify which choice lies between the lower and upper bounds.
Step-by-step Explanation
Interpret the margin of error
The poll result is 56% support with a margin of error of ±7 percentage points. That means the true percentage of all registered voters who support the proposal could reasonably be anywhere from
So the plausible percent range is 49% to 63%.
Convert the percentage range to numbers of voters
The town has 10,000 registered voters. To find the possible numbers of supporters, multiply the percentages (as decimals) by 10,000:
- Minimum possible supporters:
- Maximum possible supporters:
So any reasonable estimate of the actual number of supporters should be between 4,900 and 6,300.
Compare the answer choices to the plausible range
Now compare each option to the interval from 4,900 to 6,300:
- 4,600 is less than 4,900, so it is too low.
- 6,500 and 6,700 are greater than 6,300, so they are too high.
- 4,950 lies between 4,900 and 6,300.
Therefore, the number of registered voters in the town who could support the proposal is 4,950.