A school district has 6,000 high school students and 3,000 middle school students. A random sample of high school students indicates that of high school students participate in an after-school arts program, with a margin of error of percentage points. Independently, a random sample of middle school students indicates that of middle school students participate in the program, with a margin of error of percentage points.
Which choice lists all and only the appropriate conclusions?
I. It is plausible that of all high school students in the district participate in the program.
II. Based on the two samples, it is appropriate to estimate that of all students in the district participate in the program.
III. It is plausible that of all students in the district participate in the program.
When samples come from groups of different sizes, first use each stated margin of error to identify plausible values within that group. Then, for a claim about the entire population, weight each group by its population size rather than taking a simple average of the sample percentages. This lets you distinguish a reasonable overall estimate from one that ignores how many people are in each group.
Hints
Use each stated margin of error
Determine the lowest and highest plausible participation percentage for each school level.
Do not treat the groups equally
There are twice as many high school students as middle school students. Consider how that affects an estimate for the whole district.
Test the districtwide claim
To see whether a districtwide percentage is plausible, look for plausible percentages for both groups that produce that overall percentage when weighted by their population sizes.
Desmos Guide
Use arithmetic as the primary method
A direct calculation is faster than graphing for this question. Use Desmos to verify the weighted average, but use the stated margins of error to reason about plausibility.
Verify the weighted sample estimate
Enter ((6000)(0.18)+(3000)(0.10))/9000. Desmos shows the estimate for all district students when each group is weighted by its number of students.
Check a plausible districtwide value
Enter ((6000)(0.22)+(3000)(0.13))/9000. This checks whether values within the two groups' plausible ranges can produce the districtwide percentage in conclusion III.
Step-by-step Explanation
Find the plausible ranges for each group
For high school students, the estimate is with a margin of error of percentage points, so the plausible range is from to .
For middle school students, the plausible range is from to .
Evaluate conclusion I
The value is the upper end of the plausible range for high school students. Therefore, conclusion I is appropriate.
Evaluate conclusion II
The two groups do not have equal numbers of students: there are 6,000 high school students and 3,000 middle school students. Therefore, the overall estimate should give the high school result twice as much weight as the middle school result.
The estimate from the samples is
This is about , not . Conclusion II is not appropriate.
Evaluate conclusion III
A districtwide proportion of is plausible. For example, if the high school proportion were and the middle school proportion were , both values would be within their respective plausible ranges. The resulting districtwide proportion would be
Therefore, conclusion III is appropriate.