Question 62·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis
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%. Aniкο.аi - SАT Рrер
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 3 percentage points.
Translate “estimate ± margin of error” into an interval immediately, then judge statements by what confidence intervals can and cannot claim: they support plausibility, not certainty. For sample-size changes at the same confidence level, remember margin of error scales like , so multiplying by 4 cuts the margin of error in half.
Hints
Build the interval
Use the estimate plus and minus the margin of error to find the plausible range for the population proportion.
Watch for absolute language
Be cautious with conclusions that say something is "not possible" based only on a confidence interval.
Think about how margin of error depends on sample size
At the same confidence level, margin of error shrinks like . What happens when the sample size is multiplied by 4? © Anikο
Desmos Guide
Compute the endpoints of the interval
In Desmos, type 52-6 to get the lower endpoint and 52+6 to get the upper endpoint.
Check the “impossible” claim
Compare 44 to the interval you computed. Even if 44 is outside the interval, note that the interval does not mean values outside it are impossible.
Model how margin of error changes with sample size
Since the sample size goes from 200 to 800, compute sqrt(200/800) in Desmos. Multiply 6 by that value to see the new margin of error. Frоm аnікo.аі
Step-by-step Explanation
Use the margin of error to form the plausible interval
A 52% estimate with a margin of error of 6 percentage points corresponds to the interval:
So values from 46% to 58% are plausible for the true proportion (at the stated confidence level). anikо.аі ЅАТ Quеstіon Ваnк
Evaluate statement I
Statement I matches the interval found above, so it is an appropriate conclusion.
Evaluate statement II
A margin of error does not mean values outside the interval are impossible. It only says the method is expected to capture the true proportion in that interval about 95% of the time.
So statement II is not appropriate.
Evaluate statement III and choose the answer
For the same confidence level and the same estimated proportion, the margin of error is proportional to .
Going from to multiplies the sample size by 4, so the margin of error is divided by :
So statement III is appropriate. Therefore, the correct choice is I and III only.