Question 12·Hard·Inference from Sample Statistics and Margin of Error

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During the same week, a transit agency conducted two independent random surveys of adult residents of a city. Each survey was intended to estimate the proportion of all adult residents who had ridden a city bus during the previous month. The margins of error were determined using the same method.

Survey	Estimated proportion	Margin of error
A	$43\%$	$5$ percentage points
B	$47\%$	$3$ percentage points

Which choice identifies all conclusions that are supported by the survey results?

I. The proportion of all adult residents who rode a city bus during the previous month must be between $44\%$ and $48\%$ .

II. A proportion of $46\%$ is plausible based on the results of both surveys.

III. The two survey estimates do not by themselves show incompatible results because their plausible ranges overlap.

IV. Survey B provides a more precise estimate than Survey A.

A margin of error gives a plausible range around a survey estimate: subtract and add the stated number of percentage points. For two surveys of the same population, compare their ranges to see whether a value fits both and whether the results conflict. A smaller margin means greater precision. The trap is treating a plausible range as a guarantee about the true proportion.

Hints

Hint 1
A margin of error measured in percentage points is added to and subtracted from the estimate directly. For Survey A, what endpoints do you get by moving $5$ points below and above $43\%$ ?
Hint 2
An overlap consists of values that lie in both plausible ranges. Find the higher lower endpoint and the lower upper endpoint. Does $46\%$ fall between them?
Hint 3
An estimate's precision describes how tightly it narrows down the population percentage. Compare the two margins of error, not the two estimated percentages: which survey has the smaller margin?

Step-by-step

Compare the surveys' plausible ranges
Step 1
Find Survey A's plausible range
The population proportion is the percent of all adult residents who rode a city bus. Survey A estimates it at $43\%$ . Its margin of error gives a plausible range by moving $5$ percentage points each way, not by taking $5\%$ of $43\%$ . Type $43-5$ and $43+5$ in Desmos. It shows $38$ and $48$ , so A's range is $38\%$ to $48\%$ .
Step 2
Find Survey B's plausible range
Apply the margin of error to B's $47\%$ estimate. Below A's lines, type $47-3$ and $47+3$ . Desmos shows $44$ and $50$ . These are percentage-point endpoints, so B's plausible range is $44\%$ to $50\%$ .
Step 3
Check the shared values
The overlap contains values in both ranges. It starts at the higher lower endpoint, $44\%$ , and ends at the lower upper endpoint, $48\%$ . Since $46\%$ is in that overlap, II is supported. Since the ranges overlap at all, III is supported: the different estimates alone don't show incompatible results.
Step 4
Test the word “must”
The word must is the trap in I. A plausible interval gives reasonable possibilities, not a guarantee that it contains the true proportion; even two surveys can miss it. So an overlap gives plausible values, not a certain boundary, and I isn't supported.
Step 5
Compare precision and choose
Precision means how tightly an estimate narrows down the population proportion. B's margin is $3$ percentage points, smaller than A's $5$ , so B has the narrower plausible range and IV is supported. The supported conclusions are II, III, and IV. Choice D.

Lessons that teach this

SAT Data AnalysisBeginnerCoreEstimate populations and margin of error

Question 12·Hard·Inference from Sample Statistics and Margin of Error

Strategy

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Step-by-step

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Strategy

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