A city has 20,000 households. In a random sample of 400 of... | SAT Inference from sample statistics and margin of error Question, Answer Explained

Question bank / Inference from sample statistics and margin of error

Math · Problem-Solving and Data AnalysisHard

A city has $20{,}000$ households. In a random sample of $400$ of these households, $30\%$ have solar panels. The margin of error for this estimate is $5$ percentage points. Based on this information, which of the following is the greatest plausible number of households in the city that have solar panels?

$6{,}005$

$6{,}300$

$7{,}000$

$8{,}000$

Show answer and explanation

Answer: C

Plausible values for the percent of all households with solar panels are $30\% \pm 5\%$ , or $25\%$ to $35\%$ . The greatest plausible number of households is $0.35 \times 20{,}000 = 7{,}000$ .

Choice B adds $5\%$ of the estimate $6{,}000$ , but the margin of error is $5$ percentage points of all households, not $5\%$ of the estimate.

Why the other choices are wrong

A. This adds 5 households to the estimate, but the margin of error is 5 percentage points of all households.
B. This adds 5% of 6,000. The margin of error is 5 percentage points of 20,000 households.
D. 8,000 is 40% of the households, which adds the margin of error twice.

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