Question 62·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A random sample of $200$ registered voters in County Z was surveyed about whether they support a proposed library bond measure. Based on the sample, the estimated proportion of all registered voters in County Z who support the measure is $52\%$ with a margin of error of $6$ percentage points at a $95\%$ confidence level.

Which of the following conclusions are appropriate?

I. It is plausible that the proportion of all registered voters in County Z who support the measure is between $46\%$ and $58\%$ .

II. It is not possible that the proportion of all registered voters in County Z who support the measure is $44\%$ .

III. If a new random sample of $800$ registered voters in County Z were surveyed using the same method and the estimated proportion were still $52\%$ , the margin of error at the $95\%$ confidence level would be less than $6$ percentage points.

A margin of error in percentage points gives a plausible interval around a survey estimate. Subtract and add the margin, using Desmos for the endpoints, then check each claim against that interval. A 95% confidence level does not make the interval a certainty. A value outside the interval is not automatically impossible. A larger sample using the same method gives a smaller margin of error.

Hints

Hint 1
A margin of error measured in percentage points tells you how far to go on each side of the estimate. Subtract it for the low end and add it for the high end, rather than taking a percent of the estimate.
Hint 2
A confidence interval describes values supported by the survey; it isn't an absolute boundary on the true percentage. If a value is outside the interval, can you call it impossible?
Hint 3
With the same survey method and confidence level, a larger random sample gives a more precise estimate. What happens to the margin of error when the sample size grows?

Step-by-step

Build the interval and test each claim
Step 1
Translate the margin into endpoints
A margin of error is the distance on each side of the estimate. Here it's $6$ percentage points, not $6\%$ of the estimate, so use the percentage numbers to write the lower and upper endpoints:
$52-6\quad\text{and}\quad52+6.$
Step 2
Find the plausible interval
Type $52-6$ and $52+6$ on separate lines in Desmos. It shows $46$ and $58$ , so the plausible interval for the proportion of County Z registered voters who support the measure is $46\%$ to $58\%$ . Statement I is appropriate.
Step 3
Check the claim about 44%
The $44\%$ in II is below the interval, but outside the interval doesn't mean impossible. A 95% confidence level means this method produces intervals containing the true population proportion about $95\%$ of the time across repeated random samples. This interval could miss it, so II is too strong.
Step 4
Use the larger sample to judge III
The new random sample has $800$ voters instead of $200$ , while the method, estimate, and confidence level stay the same. A larger sample reduces uncertainty, so its margin of error would be less than $6$ percentage points. III is appropriate, leaving I and III only. Choice B.

Question 62·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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