Question 28·Easy·Inference from Sample Statistics and Margin of Error

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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 $\pm 7$ 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?

A poll’s margin of error gives a plausible range for the percentage of the whole population, not a count of people surveyed. Subtract and add the stated percentage points to the sample estimate. Then use Desmos’s % of to turn the endpoints into counts. Use the population size, not the sample size, to find population counts.

Hints

Hint 1
A margin of error of $7$ percentage points puts the population percentage as much as $7$ points below or above the sample estimate. Subtract and add $7$ directly. What are the two endpoints?
Hint 2
The population is all the registered voters in town, not the $200$ people surveyed. Once you have the percent endpoints, take each one as a percent of all $10{,}000$ voters.
Hint 3
A count could be the population total if it falls between your two endpoint counts. It doesn’t have to equal the sample estimate converted to a count.

Step-by-step

Build a plausible count range
Step 1
Find the lower percentage
The margin of error tells you how far the population percentage could plausibly be from the sample estimate. For the lower end, subtract $7$ percentage points from $56\%$ , not $7\%$ of $56\%$ :
$56\%-7\%=49\%$
Step 2
Find the upper percentage
For the upper end, add the same $7$ percentage points:
$56\%+7\%=63\%$
So the plausible population percentage runs from $49\%$ to $63\%$ .
Step 3
Convert the lower percentage to voters
The question asks about all registered voters, so use $10{,}000$ , not the $200$ people surveyed. Type $49\%\operatorname{of}10000$ into Desmos. It prints $4{,}900$ , the lower end of the plausible voter count.
Step 4
Convert the upper percentage to voters
Type $63\%\operatorname{of}10000$ on the next Desmos line. It prints $6{,}300$ , the upper end of the plausible voter count.
Step 5
Choose a count inside the range
The plausible count runs from $4{,}900$ to $6{,}300$ . Of the choices, $4{,}950$ falls inside that range, so $4{,}950$ registered voters could support the proposal. Choice B.

Lessons that teach this

SAT Data AnalysisBeginnerCoreEstimate populations and margin of error

Question 28·Easy·Inference from Sample Statistics and Margin of Error

Strategy

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Step-by-step

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Strategy

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