Question 54·Hard·Inference from Sample Statistics and Margin of Error

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A polling firm surveyed a simple random sample of 400 adult residents of a metropolitan area and found that 52 percent support a proposed public–transit expansion. The firm reports a margin of error of ±5 percentage points for this estimate at the 95 percent confidence level.

If the sample is representative of the entire population of 3.2 million adult residents in the metropolitan area, which interval best represents the possible number of residents who support the expansion?

A survey’s margin of error gives a plausible interval for the population percentage. When the margin is in percentage points, subtract and add it directly to the estimate. Then use Desmos’s % of calculation to turn both endpoints into counts of the entire population. Using the number surveyed instead gives counts for the sample, not the population.

Hints

Hint 1
A percentage point is one step from, say, $50\%$ to $51\%$ . Because the margin is 5 percentage points, move five points below and above the estimate. Don’t take $5\%$ of the estimate.
Hint 2
The population is everyone the question asks about; the 400 people surveyed are only a sample. Use the whole population, not 400, in each % of calculation once you have the percent endpoints.

Step-by-step

Find the percent endpoints, then the counts
Step 1
Find the plausible percentages
The margin of error tells how far the population percentage may be from the sample’s $52\%$ . Because the margin is 5 percentage points, subtract and add $5$ points directly, not $5\%$ of $52\%$ :
$52\%-5\%=47\%$ $52\%+5\%=57\%$
The $95\%$ confidence level doesn’t change these calculations.
Step 2
Convert the lower endpoint to a count
The population is all $3.2$ million adult residents, not the sample of 400. Type $47\%\operatorname{of}3200000$ in Desmos. It shows $1{,}504{,}000$ , the lower plausible count of supporters.
Step 3
Convert the upper endpoint to a count
Type $57\%\operatorname{of}3200000$ on the next Desmos line. It shows $1{,}824{,}000$ , the upper plausible count. So the possible number of adult residents who support the expansion runs from $1{,}504{,}000$ to $1{,}824{,}000$ . Choice B.

Lessons that teach this

SAT Data AnalysisBeginnerCoreEstimate populations and margin of error

Question 54·Hard·Inference from Sample Statistics and Margin of Error

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

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