Separate random samples from two groups call for margins of error: each sample percent estimates its own group's population percent. Subtract and add the given number of percentage points, then check whether the plausible intervals touch before claiming one group has more support. To judge support across both groups, you also need their population sizes, not just their sample sizes.
Hints
- Hint 1
A margin of error in percentage points goes straight onto the sample percentage: subtract it for the low end and add it for the high end. What interval does each sample give for its own group?
- Hint 2
Two intervals overlap if they share a possible value, even at one endpoint. Compare the endpoints of these intervals: what would their relationship tell you about a claim that one whole group has more support?
- Hint 3
A sample size of tells you how many people were surveyed, not how many belong to the group statewide. With the group sizes unknown, can you determine the support rate among all registered voters?
Step-by-step
Check each claim against the two intervals
Step 1Find the interval for voters who cast a ballot
A margin of error of percentage points means moving points below and above the estimate, not taking of . Type and in Desmos. It shows and , so to is plausible for this group. That supports I.
- Step 2
Find the interval for voters who did not cast a ballot
Use the same -point margin around this group's estimate. Add and in Desmos. It shows and , so this group's plausible interval is to .
- Step 3
Test whether one group must have more support
The intervals meet at . That rate is plausible for both groups, so their different sample percentages don't establish which group's statewide support rate is greater. II is not supported. Compare plausible intervals, not just sample estimates, before claiming a difference between populations.
- Step 4
Check whether all registered voters have a majority
The overall proportion depends on both groups' support rates and how many voters belong to each group. Surveying people from each group doesn't mean the groups are equally large statewide. Since one group's support could be above and the other's below it, the unknown group sizes prevent the survey results alone from settling whether all registered voters have a majority. III is supported.
- Step 5
Check the effect of a larger sample
With the same survey method, surveying more people in each group reduces sampling uncertainty, giving a smaller margin of error for each group. That supports IV; it doesn't predict whether either support estimate will rise or fall. The supported conclusions are I, III, and IV. Choice D.