A sample mean estimates the mean of the larger group the sample came from. Subtract and add the margin of error to find a plausible interval. Keep the conclusion about the mean for the whole group, not each person's time, and don't turn an estimate into a certainty.
Hints
- Hint 1
A sample mean averages the people selected. A population mean averages everyone in the group they came from. The survey can estimate the second mean, but does it measure it exactly?
- Hint 2
A margin of error is subtracted from and added to the sample estimate. The results form a plausible interval for the population mean, not a range that every individual must fall inside.
Step-by-step
Build and interpret the margin-of-error interval
Step 1Identify the mean being estimated
No Desmos needed. Each endpoint uses one operation. The sample mean is the average for the selected voters. Because the voters were randomly sampled from the city, their -minute mean can estimate the population mean, the average for all city voters.
- Step 2
Translate the margin of error
The margin of error puts the plausible population mean up to minutes below or above the sample mean. So the two endpoints come from and .
- Step 3
Find both endpoints
Subtract for the lower end:
Add for the upper end:
So the plausible interval runs from to minutes.
- Step 4
Choose a conclusion the sample supports
The interval is about the mean for all city voters, not every voter's time. It also isn't a guarantee: the sample doesn't prove the city's mean is exactly or certainly inside the interval. So it's plausible that the actual mean is between and minutes. Choice D.