A margin of error gives a plausible interval around each survey's sample mean: subtract it for the low end and add it for the high end. Compare the intervals to see whether they share values, rather than comparing the sample means alone. A confidence level does not guarantee that the true mean lies inside an interval.
Hints
- Hint 1
A margin of error tells you how far a survey's estimate can extend in either direction. For survey X, what do you get by subtracting and adding its margin of error to its mean?
- Hint 2
Make a separate plausible interval for survey Y. Use Y's mean and Y's margin of error, not the margin from survey X.
- Hint 3
Two intervals overlap if they share at least one value. Compare X's upper endpoint with Y's lower endpoint: could the same district mean fit both surveys?
Step-by-step
Compare the surveys' plausible intervals
Step 1Find survey X's interval
A sample mean is the average for the students surveyed. A margin of error extends that estimate in both directions to make a plausible interval for the district mean. Type and in Desmos. It shows and , so X's plausible interval is to hours.
- Step 2
Find survey Y's interval
Use Y's own margin of error with its mean. Add and under your X lines in Desmos. It shows and , so Y's plausible interval is to hours. Using X's margin here would give Y the wrong interval.
- Step 3
Check whether one mean fits both
Both surveys are of the same district, and their intervals share to hours. Overlapping intervals mean the same true mean could fit both estimates; they don't prove its exact value. So it is possible that the two surveys estimate the same true mean weekly screen time. Choice C.