Question 42·Hard·Inference from Sample Statistics and Margin of Error

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County L has 62,000 households. A random sample of 400 households found that 68 percent reported that their primary internet connection is via fiber-optic cable. The margin of error for this estimate is ±4 percentage points at a 95 percent confidence level.

Which of the following could be the number of households in County L whose primary internet connection is via fiber-optic cable?

A sample percentage with a margin of error gives a plausible range for the whole population, not a guaranteed exact result. Subtract and add the margin in percentage points, then use Desmos’s % of to convert both endpoints to counts. The sample size tells you who was surveyed; multiplying by it instead of the population size answers a different question.

Hints

Hint 1
A margin of error in percentage points goes on either side of the sample estimate. Subtract $4$ points from $68\%$ for the low end and add $4$ for the high end. Don’t take $4\%$ of $68\%$ .
Hint 2
The endpoints describe the whole county, not just the sampled households. Apply each endpoint to $62{,}000$ to get an interval, or range, of plausible household counts.

Step-by-step

Find the plausible count range
Step 1
Build the plausible percent interval
The margin of error is $4$ percentage points, so move $4$ points below and above the sample’s $68\%$ . Subtract for the low end:
$68\%-4\%=64\%$
Add for the high end:
$68\%+4\%=72\%$
So the plausible percentage of all county households is $64\%$ to $72\%$ .
Step 2
Convert percentages to household counts
Use all $62{,}000$ households to convert both percent endpoints into counts. The $400$ sample households supplied the estimate; they aren’t the county total. Type $64\%\operatorname{of}62000$ and $72\%\operatorname{of}62000$ in Desmos. It prints $39{,}680$ and $44{,}640$ , so plausible counts run between those numbers.
Step 3
Choose the count within the range
The only listed count inside $39{,}680$ to $44{,}640$ is $40{,}500$ . So $40{,}500$ could be the number of County L households whose primary connection is fiber-optic cable. Choice B.

Lessons that teach this

SAT Data AnalysisBeginnerCoreEstimate populations and margin of error

Question 42·Hard·Inference from Sample Statistics and Margin of Error

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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