A survey’s margin of error gives a range of plausible population percentages: subtract and add the stated percentage points. With two surveys, find both ranges and check whether they share a value. Different sample percentages can still fit the same population percentage. A larger sample usually gives a narrower range, but it doesn’t make the smaller survey worthless.
Hints
- Hint 1
A margin of error measured in percentage points is added to and subtracted from the estimated percent. It isn’t a percent of that estimate. What expressions give the lower and upper endpoints for each survey?
- Hint 2
An interval is a range from a lower endpoint to an upper endpoint. Find both surveys’ intervals, then look for their overlap: the percentages that fall in both ranges.
- Hint 3
Two random samples can produce different estimates of the same population percentage. Check whether a proposed percentage fits both margins of error before treating the difference between estimates as a change in opinion.
Step-by-step
Compare the surveys’ plausible ranges
Step 1Turn each margin into endpoints
A margin of error tells you how many percentage points to subtract from and add to a sample estimate to make a plausible range for the population percentage. Add the points directly; don’t take a percent of the estimate. Survey A’s endpoints are and . Survey B’s are and .
- Step 2
Find Survey A’s range
Type and in Desmos. It shows and , so Survey A’s plausible range for the percentage of all adult residents in favor is to .
- Step 3
Find Survey B’s range
Keep those lines and type and . Desmos shows and , so Survey B’s plausible range is to .
- Step 4
Check what both ranges support
The ranges overlap from to . Since falls in that shared range, it is plausible that of all adult residents favor the public garden. Choice B.