Question 30·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

0 / 50 Questions completed

A city maintains an address list of all households in the city. To study whether a message about potential savings affects enrollment in a discounted transit program, researchers randomly selected 2,000 households from the list and mailed each household an eligibility survey. Of the selected households, 1,060 returned the survey. Among the households that returned the survey, 700 were eligible for the program.

The researchers randomly assigned 350 eligible households that returned the survey to receive a standard enrollment letter and the other 350 to receive an enrollment letter that included a comparison of the program's cost with typical transit costs. Of the households receiving the standard letter, 147 enrolled; of the households receiving the comparison letter, 189 enrolled.

A researcher claims that, for all eligible households in the city, including the cost comparison causes the enrollment rate to be 12 percentage points higher. Which choice best evaluates this claim?

In a randomized experiment, chance assigning treatments supports a causal conclusion for the people assigned. To judge a citywide claim, trace who was sampled and who actually responded; nonresponse can make participants unrepresentative even after a random invitation. Check the rate gap in Desmos, then keep cause separate from how far the result reaches. Random assignment supports cause, not representativeness.

Hints

Hint 1
Nonresponse means some selected households never replied. A random invitation doesn't guarantee the respondents resemble the nonrespondents. Which households were actually tested, and could they differ from other eligible households?
Hint 2
Random assignment means chance chose which letter each eligible respondent received. That helps separate the effect of the letter from preexisting differences between the two groups. What does it tell you about causation?
Hint 3
An enrollment rate is the number who enrolled divided by the number who received that letter. Find both rates using the two groups of $350$ , then subtract to check the claimed percentage-point gap.

Step-by-step

Separate cause from who the result represents
Step 1
Find who reached the experiment
The researchers randomly selected $2{,}000$ households, but only $1{,}060$ returned the survey. Of those respondents, $700$ were eligible. This nonresponse matters: eligible households that replied might differ from eligible households that didn't, perhaps in their interest in the program. So the $700$ eligible respondents aren't necessarily representative of all eligible city households.
Step 2
Identify what random assignment establishes
The researchers randomly assigned those $700$ eligible respondents to the two letters. That means chance, rather than a household's interest in enrolling, chose its letter. With the letter type as the planned difference between groups, the comparison supports a causal conclusion for these eligible respondents.
Step 3
Find each letter group's enrollment rate
An enrollment rate is enrolled households divided by households receiving that letter. Let $s$ be the standard-letter rate and $c$ the cost-comparison-letter rate. Type $s=\frac{147}{350}$ and $c=\frac{189}{350}$ in Desmos. It shows $s=0.42$ and $c=0.54$ , or $42\%$ and $54\%$ .
Step 4
Check the percentage-point difference
Type $100(c-s)$ below the rates. Desmos shows $12$ : multiplying the decimal-rate gap by $100$ gives a difference of $12$ percentage points. That's a subtraction of the two percentages, not an increase of $12\%$ relative to the standard-letter rate.
Step 5
Match the conclusion to the households studied
The $12$ -percentage-point gap supports a causal conclusion among eligible households that returned the survey. But the researcher claims the effect holds for all eligible households in the city. Because some selected households didn't respond, the study doesn't establish that broader claim. Choice B.

Question 30·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this