A survey’s margin of error tells you how far its estimate could reasonably be from the population percentage, but first check who could have been sampled. A random sample from the whole target group can speak for that group; a voluntary, topic-focused poll may not. To support a majority claim, a representative study’s entire plausible interval must be above . Subtract percentage points directly, not as a percent of the estimate.
Hints
- Hint 1
The target population is the whole group the department wants to describe. A random draw from a list of that entire group can support an estimate about it. Which studies used that method?
- Hint 2
In a voluntary-response sample, people choose whether to answer. People following transit-focused accounts may be especially interested in a bus route, so a small stated margin cannot remove that sampling bias.
- Hint 3
A margin of error measured in percentage points is subtracted directly from the estimate. Compare each representative study’s lowest plausible percentage with . Could half or fewer of all commuters still use the route?
Step-by-step
Check the samples, then the margins
Step 1Identify which studies can represent the county
No Desmos needed. Each interval endpoint takes one subtraction or addition; the key question is who was sampled.
The target population is all commuters in the county. Chance selected commuters from a list of that whole group for Studies I and III, so those random samples can support estimates about all county commuters.
- Step 2
Set aside the voluntary poll
Study II posted a link on cycling and transit social media. People chose whether to answer, making this a voluntary-response sample. Its audience may be more interested in transit than county commuters overall, so the small stated margin cannot fix that sampling bias.
- Step 3
Find Study I’s plausible range
A margin of error tells you how far the county percentage could reasonably be from the estimate. Subtract and add Study I’s percentage points directly, rather than taking of :
Study I’s plausible interval, or range of reasonable values, includes percentages below half.
- Step 4
Find Study III’s plausible range
Apply the same rule to Study III. Subtract and add its -point margin:
Even the lowest plausible percentage, , is above .
- Step 5
Match the evidence to the claim
The intervals for I and III overlap: both allow values from to , so their different estimates do not conflict. Only Study III’s entire interval is above among the studies that can represent all county commuters. So only Study III supports concluding that more than half of them would use the route. Choice D.