A school district has two campuses. The table shows the distribution of all students in the district.
| Campus | Share of all district students |
|---|---|
| North | |
| South |
The district wants to estimate the percentage of all its students who support extending library hours. Every student selected for a survey will respond.
For comparison, the district could take a random sample of 250 students from the full district enrollment. Assume that, when the same method is used to calculate margins of error, a representative random sample of 500 students has a smaller margin of error than a representative random sample of 250 students.
Which choice describes a survey procedure that is most likely to produce a representative estimate for all district students and have a smaller margin of error than the sample of 250 students?
For questions that combine sampling and margin of error, first make sure the sample can represent the target population. Check that the selection is random and that important groups appear in the sample in the same proportions as in the population, unless the results will be adjusted. Then compare sample sizes: when the sampling conditions and margin-of-error method are comparable, a larger representative random sample gives a more precise estimate.
Hints
Use the campus shares
For a sample of 500 students to reflect the district, determine how many students should come from each campus.
Check the sampling method
A large sample is not enough by itself. The students should be selected randomly, and the sample should reflect both campuses in their district-wide proportions.
Compare sample sizes
After identifying a representative random sample, compare its total number of students with 250 using the margin-of-error information in the question.
Desmos Guide
Verify the needed campus counts
An algebraic proportion check is faster than using Desmos for this question. As a verification in Desmos, enter and .
Compare the proportions
Desmos displays and , which match the district proportions of from North and from South. Then confirm that this plan has 500 selected students, which is greater than 250.
Step-by-step Explanation
Match the district's campus proportions
A representative sample of 500 students should have from North and from South.
of 500 is 300, and of 500 is 200.
Check whether the selections are random and representative
Selecting 300 students at random from North and 200 students at random from South creates a sample with the same campus distribution as the district. Therefore, the combined percentage can represent all district students.
Compare precision
This procedure uses 500 randomly selected students. The question states that a representative random sample of 500 has a smaller margin of error than a representative random sample of 250, so this procedure meets both requirements.