A research team conducted two surveys in a state using the same survey method. One random sample included 800 registered voters who cast a ballot in the most recent general election. The other random sample included 800 registered voters who did not cast a ballot in that election.
In the first sample, 52% of voters supported a proposed transportation plan, with a margin of error of 3 percentage points. In the second sample, 46% supported the plan, with a margin of error of 3 percentage points. The team did not determine what proportion of all registered voters belongs to each group.
Which choice identifies all conclusions that are supported by the information given?
I. It is plausible that between 49% and 55% of registered voters who cast a ballot in the most recent general election support the plan.
II. The survey results establish that a greater proportion of registered voters who cast a ballot supports the plan than of registered voters who did not cast a ballot.
III. The survey results alone cannot determine whether a majority of all registered voters in the state supports the plan.
IV. If a future study uses the same survey method and surveys more than 800 registered voters from each group, the margin of error for each group will be smaller than 3 percentage points.
For margin-of-error questions, first turn each estimate into a plausible low-to-high range by subtracting and adding the stated margin of error. To claim that one population proportion is definitely greater than another, the ranges must not overlap. Then check the population being discussed: results for separate groups cannot determine a combined-population result unless the relative sizes of the groups are known. Finally, remember that, with the same survey method, larger random samples have smaller margins of error.
Hints
Find each plausible range
For each survey, subtract the margin of error from the estimate and add the margin of error to the estimate.
Check for overlap
To decide whether one group's true proportion is established as greater, compare the two plausible ranges. Consider whether they have any values in common.
Think about the whole population
An overall percentage for all registered voters would depend on the relative sizes of the two groups. Check whether that information is provided.
Recall the effect of sample size
When the survey method stays the same, consider how increasing a random sample size affects the margin of error.
Desmos Guide
Use conceptual reasoning first
Desmos is not needed to decide the conclusions about the unknown group proportions or the effect of a larger sample. It can quickly verify the endpoint calculations.
Calculate the interval endpoints
Enter 52-3, 52+3, 46-3, and 46+3 on separate lines. These calculations show the low and high ends of the two plausible ranges.
Compare the endpoints
Notice whether the two displayed ranges share an endpoint or overlap. A shared possible value means the survey results do not establish that one group's population proportion is greater than the other's.
Step-by-step Explanation
Interpret the first margin of error
For registered voters who cast a ballot, the estimate is 52% with a margin of error of 3 percentage points. The plausible range is from to , so statement I is supported.
Compare the two groups carefully
For registered voters who did not cast a ballot, the plausible range is from to . The two ranges share 49%, so the results do not establish that the first group's true proportion is greater. Therefore, statement II is not supported.
Consider all registered voters
The overall proportion of registered voters who support the plan depends on how many registered voters are in each group. Because those group proportions are unknown, the survey results alone cannot determine whether support among all registered voters is above 50%. Thus, statement III is supported.
Use the sample-size relationship
With the same survey method, a larger random sample produces a smaller margin of error. Therefore, statement IV is supported.