At a high school, a counselor randomly selects students from all 10th graders who participate in at least one school-sponsored activity. Of the students selected, report getting at least hours of sleep on a typical school night. The estimated percentage has a margin of error of percentage points.
Separately, the counselor randomly selects students from all 10th graders at the school. In this sample, report getting at least hours of sleep on a typical school night. The estimated percentage has a margin of error of percentage points.
Which choice identifies all the appropriate conclusions?
I. The results of the first survey support the conclusion that between and of all 10th graders at the school get at least hours of sleep on a typical school night.
II. It is plausible that between and of 10th graders at the school who participate in at least one school-sponsored activity get at least hours of sleep on a typical school night.
III. The two surveys establish that 10th graders who participate in at least one school-sponsored activity are less likely to get at least hours of sleep than all 10th graders at the school.
IV. If the counselor selects a larger random sample from the same population as the first sample, using the same method, the margin of error will generally be smaller.
For inference questions, first identify the exact population from which each random sample was selected. Then interpret each stated margin of error as a plausible range for that population only. When comparing two estimates, check whether their ranges overlap before claiming that one population has a greater or smaller percentage. Finally, remember that increasing a random sample size from the same population generally decreases the margin of error.
Hints
Match each sample to its population
Ask who could have been selected for the first survey. Is that group all 10th graders or only a subgroup of them?
Find each plausible range
Subtract and add the stated margin of error to each survey's estimated percentage.
Check whether the ranges settle the comparison
Before deciding whether one group is less likely than another, see whether the two plausible ranges overlap.
Recall the sample-size principle
Consider what usually happens to a margin of error when a random sample from the same population becomes larger.
Desmos Guide
Use reasoning first
The key issue is identifying the population represented by each sample, so a written comparison is faster than Desmos. Desmos can quickly verify the endpoints of the stated ranges.
Calculate the interval endpoints
Enter , , , and on separate lines. The results give the two plausible ranges.
Compare the ranges
The first range and the second range share values between and . This numerical overlap helps verify that the surveys do not establish the comparison claimed in statement III.
Step-by-step Explanation
Identify the population for each survey
The first sample was selected from 10th graders who participate in at least one school-sponsored activity. Therefore, that group is the population represented by the first estimate.
The second sample was selected from all 10th graders, so it represents all 10th graders.
Interpret the first margin of error
The first estimate is with a margin of error of percentage points. A plausible range for the percentage of activity participants is from to .
Thus, statement II is appropriate, but statement I incorrectly uses the first survey to support a conclusion about the wrong population.
Assess the comparison claim
The second survey's plausible range is from to . This range overlaps the first range from to .
Because the ranges overlap, the surveys do not establish that activity participants are less likely to get at least hours of sleep. Statement III is not appropriate.
Use the sample-size relationship
For the same population and sampling method, a larger random sample generally produces a smaller margin of error. Therefore, statement IV is appropriate.
The appropriate conclusions are II and IV only.