Question 7·Easy·Inference from Sample Statistics and Margin of Error

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A polling organization randomly selected 500 registered voters in a city to find out whether they approve of constructing a new public park. All 500 voters responded, and the poll reported that 62% of the voters approve of the project, with a margin of error of 4%. Which of the following is the best interpretation of these results?

A margin of error tells you how far a sample estimate could reasonably be from the value for the whole group. First identify the population, the group the random sample came from. Subtract and add the margin in percentage points, then look for a cautious claim about the population percentage. Even if everyone selected responds, the sample estimate isn't an exact citywide result.

Hints

Hint 1
A sample is the smaller group surveyed; the population is the whole group you want to describe. Randomly choosing registered voters lets the poll estimate a percentage for all registered voters in the city. Does it tell you that percentage exactly?
Hint 2
A margin of error in percentage points goes on both sides of the reported percentage. Subtract it for the low end and add it for the high end. What two percentages do you get?
Hint 3
A census surveys every member of the population. Getting answers from everyone in the sample avoids missing responses, but does it mean everyone in the population was surveyed?

Step-by-step

Build and interpret the margin-of-error interval
Step 1
Identify the group the poll estimates
No Desmos needed. Each endpoint takes one operation; the key is interpreting the survey. The $500$ randomly selected voters are a sample, a smaller group chosen from the city’s registered voters. So the reported $62\%$ estimates the percentage for all registered voters in the city, the population.
Step 2
Find both ends of the plausible range
The margin of error is $4$ percentage points on either side of $62\%$ , not $4\%$ of $62\%$ . Subtract $4$ points for the low end:
$62\%-4\%=58\%$
Add $4$ points for the high end:
$62\%+4\%=66\%$
Step 3
Keep percentages separate from counts
These endpoints are percentages, not numbers of voters. Don't treat the $500$ surveyed voters as the city's total number of registered voters; a count calculated from $500$ would describe the sample, not the city.
Step 4
State what the results support
All $500$ selected voters responded, but a complete sample isn't a census, a survey of everyone. So the citywide percentage isn't proved to be exactly $62\%$ . It's plausible that $58\%$ to $66\%$ of all registered voters in the city approve. Choice B.

Question 7·Easy·Inference from Sample Statistics and Margin of Error

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