Question 11·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A city conducted 2 independent random surveys of its residents to estimate the percentage of residents who support converting a downtown street into a pedestrian-only area.

At the time of the first survey, the city had 251,000 residents. The estimated percentage of residents who support the change was 48.6%, with an associated margin of error of 2.3%.
At the time of the second survey, the city had 253,000 residents. The estimated percentage of residents who support the change was 52.1%, with an associated margin of error of 1.0%.

Assuming each margin of error was calculated in the same way, what is the minimum possible increase in the number of residents who support the change from the time of the first survey to the time of the second survey?

A margin of error gives a range around a survey's estimated population percentage. For a minimum increase, use the first survey's highest plausible percentage and the second survey's lowest. Then apply each percentage to its own city's population and subtract the counts. Subtracting the percentages alone misses the change in population size.

Hints

Hint 1
A margin of error is added or subtracted in percentage points; it isn't taken as a percent of the estimate. Which endpoint gives the highest plausible percentage for the first survey?
Hint 2
To make an increase as small as possible, make the earlier count high and the later count low. Which endpoint of the second survey's percentage range gives the lower count?
Hint 3
A population count is a percentage of a whole group. Apply each chosen percentage to its own survey's population before subtracting; the city had a different number of residents at the two survey times.

Step-by-step

Compare the endpoint counts
Step 1
Find the highest first-survey percentage
A margin of error gives the plausible distance from the estimate in percentage points. To make the starting count as large as possible, add the first margin: $48.6\%+2.3\%=50.9\%$ .
Step 2
Find the lowest second-survey percentage
The smallest increase pairs the highest starting percentage with the lowest ending percentage. Subtract the second margin: $52.1\%-1.0\%=51.1\%$ . These are percentages, but the question asks for a number of residents.
Step 3
Convert the first percentage to a count
Type $50.9\%\operatorname{of}251000$ in Desmos. It prints $127{,}759$ , the highest plausible starting count. Use the first survey's $251{,}000$ residents, not the second survey's population.
Step 4
Convert the second percentage to a count
Add $51.1\%\operatorname{of}253000$ in Desmos. It prints $129{,}283$ , the lowest plausible ending count. This time, the percent is of the second survey's $253{,}000$ residents.
Step 5
Subtract the counts
Add $129283-127759$ in Desmos. It prints $1{,}524$ . So, using the stated margins, the minimum increase is $1{,}524$ residents. Choice C.

Question 11·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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