Question 9·Hard·Inference from Sample Statistics and Margin of Error

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During the same week, two independent surveys randomly selected occupied households from all occupied households in a city. Each survey asked whether the household uses a home composting bin.

Survey A sampled $600$ households and estimated that $47\%$ of all occupied households in the city use a home composting bin, with a margin of error of $5$ percentage points.

Survey B sampled $1{,}200$ households and estimated that $51\%$ of all occupied households in the city use a home composting bin, with a margin of error of $3$ percentage points.

Which choice identifies all appropriate conclusions?

I. A true proportion of $49\%$ is plausible according to both survey results.

II. Because Survey B used a larger sample and has a smaller margin of error, its estimate of $51\%$ must be closer to the true proportion than Survey A's estimate of $47\%$ .

III. Because the plausible ranges for the two survey results overlap, the results alone do not indicate that the estimates are inconsistent.

Two surveys of the same population call for comparing plausible intervals: each estimate plus or minus its margin of error. Find both intervals, then check whether a proposed value fits both and whether the intervals overlap. A larger sample usually gives a narrower interval, but it doesn't guarantee that its estimate is closer to the true value.

Hints

Hint 1
A margin of error measured in percentage points goes directly below and above the estimated percentage. Subtract and add each survey's margin to find its plausible range.
Hint 2
A plausible interval contains values consistent with a survey's result. Is the proposed citywide percentage inside both intervals? Do the intervals share any values?
Hint 3
A smaller margin of error gives a tighter range, not a guarantee of being closer to the truth. Could one plausible true percentage be equally far from the two estimates?

Step-by-step

Compare the surveys' plausible intervals
Step 1
Translate the margins into ranges
Let $p$ be the true citywide percentage of households using a bin. A margin of error tells you how far below or above an estimate $p$ could plausibly be. Survey A gives $47-5$ to $47+5$ ; Survey B gives $51-3$ to $51+3$ . These are percentage points, so don't take $5\%$ of $47\%$ .
Step 2
Find Survey A's interval
Type $47-5$ and $47+5$ in Desmos. It shows $42$ and $52$ , so Survey A's plausible interval is $42\%$ to $52\%$ for the city's true percentage.
Step 3
Find Survey B's interval
Keep those lines and type $51-3$ and $51+3$ . Desmos shows $48$ and $54$ , so Survey B's plausible interval is $48\%$ to $54\%$ .
Step 4
Test statement I
$49\%$ lies in both intervals, so it's plausible according to both surveys. Statement I holds. Plausible means consistent with the results, not proven to be the true percentage.
Step 5
Test statement III
The intervals share $48\%$ through $52\%$ . Because both surveys estimate the same city's percentage, one true value could fit both results. Their different estimates don't, by themselves, show a conflict, so III holds.
Step 6
Check the claim about sample size
Survey B's narrower interval makes it more precise: its range of plausible values is smaller. Greater precision doesn't guarantee an estimate is closer to the truth. If the true percentage is the plausible $49\%$ , both estimates are $2$ percentage points away. II is false, so only I and III are appropriate conclusions about the city's households. Choice C.

Lessons that teach this

SAT Data AnalysisBeginnerCoreEstimate populations and margin of error

Question 9·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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Step-by-step

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