Question 10·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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A city has $12{,}000$ occupied households. The number of occupied households did not change during the time of two surveys.

Survey A randomly selected households throughout the city and estimated that $42\%$ of all occupied households use public transit at least once per week. The margin of error for Survey A was $5$ percentage points.

Survey B independently and randomly selected households throughout the city and estimated that $39\%$ of all occupied households use public transit at least once per week. The margin of error for Survey B was $3$ percentage points.

Assuming that both sampling methods were sound and that each stated margin of error describes a plausible range for the population percentage, which choice gives the complete range of numbers of occupied households that is consistent with both survey results?

Two surveys of the same unchanged population give two ranges for one population percentage. A margin of error in percentage points means subtracting and adding that many points to each estimate. Keep only the percentages both ranges allow; taking the widest span includes values that fail one survey. Use Desmos’s percent-of entry to turn the shared endpoints into household counts.

Hints

Hint 1
A percentage point is a direct difference between percentages. Don’t take a percent of the estimate: subtract and add the stated margin to find a lower and upper percentage for each survey.
Hint 2
Because the household total didn’t change, both surveys estimate one citywide percentage. The overlap of their ranges contains the percentages both allow. Which lower endpoint is larger, and which upper endpoint is smaller?
Hint 3
The question wants numbers of households, not percentages. Take each endpoint of the shared range as a percent of all $12{,}000$ occupied households. Those results give the least and greatest possible counts.

Step-by-step

Find the shared percent range, then convert to counts
Step 1
Find Survey A’s plausible range
Survey A’s margin of error is 5 percentage points from its $42\%$ estimate. Subtract and add $5$ points directly, rather than taking $5\%$ of $42\%$ :
$42\%-5\%=37\%$ $42\%+5\%=47\%$
So Survey A allows $37\%$ to $47\%$ .
Step 2
Find Survey B’s plausible range
Apply Survey B’s 3-percentage-point margin to its $39\%$ estimate. Subtract and add $3$ points:
$39\%-3\%=36\%$ $39\%+3\%=42\%$
So Survey B allows $36\%$ to $42\%$ .
Step 3
Keep only percentages both surveys allow
The same unchanged city has one population percentage, so it must fit both ranges. The shared values start at the higher lower end, $37\%$ , and stop at the lower upper end, $42\%$ . Keep only the overlap: percentages that satisfy both surveys. The wider span from $36\%$ to $47\%$ includes values that fail one survey.
Step 4
Convert the lower endpoint to households
Type $37\%\operatorname{of}12000$ in Desmos. It displays $4{,}440$ , the smallest possible count, because $37\%$ is the smallest percentage allowed by both surveys.
Step 5
Convert the upper endpoint and give the range
Below the first line, type $42\%\operatorname{of}12000$ . Desmos displays $5{,}040$ , the greatest possible count. So the complete range is from $4{,}440$ to $5{,}040$ occupied households that use public transit at least once per week. Choice A.

Lessons that teach this

SAT Data AnalysisBeginnerCoreEstimate populations and margin of error

Question 10·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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