In separate random surveys of registered voters in two states, a research group estimated that of registered voters in State X support a proposed law, with a margin of error of percentage points. The group estimated that of registered voters in State Y support the law, also with a margin of error of percentage points.
Which of the following conclusions is most appropriate based on the estimates and their margins of error?
When comparing two survey estimates with margins of error, first make an interval for each estimate by adding and subtracting its margin of error. Then compare the intervals: if one interval lies entirely above the other, the estimates support a difference, but phrase the conclusion as supported rather than guaranteed.
Hints
Make an interval for each state
For each estimate, subtract and add the stated margin of error.
Compare the two ranges
After finding both intervals, check whether the ranges overlap or whether one range lies entirely above the other.
Use careful statistical language
Choose a conclusion that is supported by the survey estimates without treating either estimate as guaranteed or exact.
Desmos Guide
Calculate both intervals
In Desmos, enter , , , and on separate lines to find the endpoints of the two intervals.
Compare the endpoints
Compare the upper endpoint for State X with the lower endpoint for State Y. If State Y's lower endpoint is greater, the reported intervals support a higher percentage for State Y.
Step-by-step Explanation
Find the interval for State X
For State X, subtract and add the -percentage-point margin of error to the estimate:
The interval for State X is to .
Find the interval for State Y
For State Y, subtract and add the -percentage-point margin of error to the estimate:
The interval for State Y is to .
Compare the intervals
Every value in State Y's interval is greater than every value in State X's interval because is greater than . Thus, the estimates support the conclusion that support is higher in State Y.
Avoid claims of certainty
Margins of error describe plausible ranges based on survey data; they do not guarantee population values or show that the estimates are exact. Therefore, choice D is the most appropriate conclusion.