Question 34·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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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, $46\%$ report getting at least $8$ hours of sleep on a typical school night. The estimated percentage has a margin of error of $4$ percentage points.

Separately, the counselor randomly selects students from all 10th graders at the school. In this sample, $51\%$ report getting at least $8$ hours of sleep on a typical school night. The estimated percentage has a margin of error of $3$ percentage points.

Which choice identifies all the appropriate conclusions?

I. The results of the first survey support the conclusion that between $42\%$ and $50\%$ of all 10th graders at the school get at least $8$ hours of sleep on a typical school night.

II. It is plausible that between $42\%$ and $50\%$ of 10th graders at the school who participate in at least one school-sponsored activity get at least $8$ 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 $8$ 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.

A margin of error gives a plausible range around a survey estimate, but first check who could be selected. Build each range in Desmos by subtracting and adding the margin in percentage points. Then compare the ranges: overlapping ranges cannot establish which population has a higher percentage. Random selection does not make a survey of a subgroup represent everyone.

Hints

Hint 1
A random sample can support a conclusion about the group from which people were selected. Who could enter the first survey, and is that the same group named in claim I?
Hint 2
A margin of error measured in percentage points is subtracted from and added to the estimate. What range do you get from $46\%$ and $4$ points, and which group does it describe?
Hint 3
Compare the two plausible ranges before deciding whether one population has a lower percentage. If the ranges overlap, could both populations have the same percentage? Then consider what a larger sample does to the margin of error.

Step-by-step

Match each survey to its population, then compare ranges
Step 1
Identify whom the first survey represents
A population is the complete group a random sample can represent. The first sample represents activity participants, not all 10th graders. Claim I uses that survey to describe all 10th graders, so I isn't supported.
Step 2
Find the first survey's plausible range
Type $46-4$ and $46+4$ in Desmos. It shows $42$ and $50$ . The $4$ means percentage points, not $4\%$ of $46\%$ . So $42\%$ to $50\%$ is plausible for activity participants, as II says.
Step 3
Find the second survey's plausible range
For the second survey, add $51-3$ and $51+3$ in Desmos. It shows $48$ and $54$ . Because this sample was drawn from all 10th graders, its plausible range of $48\%$ to $54\%$ describes that whole group.
Step 4
Test the claim that one group sleeps less
The two ranges overlap from $48\%$ to $50\%$ . Both populations could have a value of $48\%$ , so the different sample estimates do not establish that activity participants are less likely to get enough sleep. III isn't supported.
Step 5
Check the effect of a larger sample
A larger random sample from the same population, using the same method, generally gives a smaller margin of error: more observations make the estimate more precise. So IV is supported. The appropriate conclusions are II and IV only. Choice B.

Question 34·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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