Question 7·Easy·Inference from Sample Statistics and Margin of Error
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?
For margin-of-error questions, first identify what statistic is reported (usually a sample percent) and what the margin of error is. Interpret the margin of error as "within this many percentage points" of the reported percent, then compute the lower and upper bounds using simple addition and subtraction. Finally, choose the answer that describes a range of plausible values for the population percent, and avoid options that claim an exact value, confuse percent with number of individuals, or say the poll "proves" something about the whole population.
Hints
Sample percent vs. population percent
Ask yourself: Is 62% describing only the 500 people in the sample, or is it being used to say something about all registered voters in the city? Which do the answer choices talk about?
What does a margin of error mean?
A margin of error of 4% means the true value is likely within 4 percentage points of the reported percent. Think about how you would express that as a range around 62%.
Percent or number of voters?
Look closely at the wording of each option: some talk about a percent, others about a number of voters, and some use strong words like "exactly" or "proves." Which type of statement usually goes with a margin of error in a poll like this?
Desmos Guide
Use Desmos to compute the margin-of-error endpoints
In Desmos, type 62-4 and 62+4 on separate lines to find the lower and upper bounds of the percent range. Read the two numerical results; these are the endpoints of the plausible percent interval around 62% with a 4% margin of error.
Step-by-step Explanation
Understand what the poll measured
The poll asked 500 randomly selected registered voters whether they approve of the project, and 62% of those 500 said yes. This 62% is a sample statistic (what happened in the sample), and it is being used to estimate the percent of all registered voters in the city (the population) who approve.
Interpret the margin of error
A margin of error of 4% means the polling organization believes the true population percent is likely within 4 percentage points of the reported 62%. In symbols, the plausible range for the population percent is from to . This is a range of percents, not a claim about an exact value and not directly a range of numbers of people.
Convert the symbolic range and match the interpretation
Compute the endpoints: and . So the results are best interpreted as saying that plausible values of the percent of all registered voters in the city who approve of the project are between 58% and 66%. That is exactly what the correct answer choice states.