Question 62·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis
A random sample of 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 with a margin of error of percentage points at a 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 and .
II. It is not possible that the proportion of all registered voters in County Z who support the measure is .
III. If a new random sample of registered voters in County Z were surveyed using the same method and the estimated proportion were still , the margin of error at the confidence level would be less than percentage points.
Translate the estimate and margin of error into an interval immediately, then distinguish plausible values from values that are impossible. For qualitative sample-size comparisons at the same confidence level, remember that a larger random sample produces a smaller margin of error.
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.
Compare the sample sizes
At the same confidence level, a larger random sample produces a smaller margin of error. Compare the new sample size with the original one.
Desmos Guide
Compute the endpoints of the interval
In Desmos, type to get the lower endpoint and to get the upper endpoint.
Check the “impossible” claim
Compare with the interval you computed. Even though is outside the interval, the interval does not mean that values outside it are impossible.
Compare the sample sizes
No calculation is needed for the sample-size claim. Since is greater than , the new random sample would have a smaller margin of error at the same confidence level.
Step-by-step Explanation
Use the margin of error to form the plausible interval
An estimate of with a margin of error of percentage points corresponds to the interval:
So values from to are plausible for the true proportion at the stated confidence level.
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 indicates that this interval is a plausible range based on the sample and the stated confidence level.
So statement II is not appropriate.
Evaluate statement III and choose the answer
The new sample has voters, which is larger than the original sample of voters. At the same confidence level and using the same method, a larger random sample produces a smaller margin of error.
Therefore, statement III is appropriate, and the correct choice is I and III only.