Question 5·Medium·Inference from Sample Statistics and Margin of Error
A polling organization selected a random sample of 500 registered voters in a county and found that 275 said they would support a proposed sales-tax increase. The organization reported that 55 percent of registered voters support the increase, with a margin of error of ±4 percentage points. The county has 120,000 registered voters. According to this report, what is the highest estimated number of registered voters in the county who might support the sales-tax increase?
(Express the answer as an integer)
For questions involving poll results and a margin of error, first interpret the margin of error as a range around the reported percentage (add and subtract the percentage points). Decide whether the problem wants the highest, lowest, or just the central estimate, and pick the appropriate percentage from that range. Then convert that percentage to a decimal and multiply by the total population size (not the sample size) to get the estimated number of people. Always watch the wording: “percentage points” means you add or subtract directly from the percentage, not multiply by it.
Hints
Understand the margin of error
The statement says "55 percent ... with a margin of error of ±4 percentage points." What two percentages does that range cover?
Choose the correct end of the range
You’ll get a lowest possible percentage and a highest possible percentage. Since the question asks for the highest estimated number of supporters, which of those two percentages should you use?
Turn a percentage into a count
Once you pick the correct percentage, how do you use the total of 120,000 registered voters to find how many people that percentage represents?
Desmos Guide
Compute the maximum number of supporters
In Desmos, type (0.55 + 0.04) * 120000 and look at the value Desmos outputs; that number is the highest estimated number of registered voters who might support the sales-tax increase according to the report.
Step-by-step Explanation
Interpret the margin of error
The poll result says 55 percent support with a margin of error of ±4 percentage points.
That means the true percentage of all registered voters who support the increase could reasonably be anywhere from:
- up to
- .
So the highest possible percentage consistent with the report is .
Identify what the question is asking for
The question asks for the highest estimated number of registered voters who might support the increase.
That means you should use the highest possible percentage, which is , and apply it to the entire county’s 120{,}000 registered voters (not the 500-person sample).
Convert the highest percentage to a number of voters
First convert to a decimal: .
Now multiply by the total number of registered voters:
So, the highest estimated number of registered voters who might support the sales-tax increase is 70800.