A representative sample should reflect the groups that make up the whole population. When a table gives each group’s share, use Desmos to find how many people to select from each group, then check that selection is random. A larger sample can reduce the margin of error, a measure of uncertainty, but it cannot fix a sample that overcounts one group.
Hints
- Hint 1
The district wants an estimate for all its students. The percentage of all surveyed students will count each response equally, so what campus split should the sample have to reflect the district?
- Hint 2
Use each campus’s share of the district to find its share of a sample. In Desmos, you can type each percent followed by “of” and the sample size.
- Hint 3
A larger sample helps with the margin of error only if it represents the population. Check whether chance selects students from both campuses, rather than letting students volunteer or leaving a campus out.
Step-by-step
Match the district’s campus shares
Step 1Read the shares the sample must reflect
The target population is all district students. The table puts at North and at South, so match each campus’s share of the sample to its share of the district. An even split would count South too heavily when the survey uses the percentage of all selected students.
- Step 2
Check how students enter the sample
A stratified random sample picks students at random within each campus. Randomly selecting students from both campuses gives both campuses a place in the sample. Selecting only North leaves South out; inviting social-media followers lets volunteers choose whether to join.
- Step 3
Compare margins of error
The question says that, using the same calculation method, a representative random sample of has a smaller margin of error than one of . So a representative sample of meets the margin-of-error requirement.
- Step 4
Find each campus’s sample size
For a sample of , type and in Desmos. It prints and . So the proportional split is students from North and from South. Randomly selecting those students meets both requirements: the sample represents the district and has the smaller margin of error. Choice C.