Question 53·Medium·Inference from Sample Statistics and Margin of Error
In a random survey of 600 registered voters, a research group estimated that of all registered voters in a state support a proposed law. The estimate was reported with a margin of error of 4 percentage points.
Which of the following statements is the most appropriate conclusion based on the estimate and its margin of error?
For margin-of-error questions, first identify the sample estimate, then immediately compute the interval by adding and subtracting the margin of error. Interpret this interval as a likely or plausible range for the true population value, not a certainty. Finally, eliminate answer choices that claim guarantees, exact values, or conclusions (like changes over time) that are not directly supported by the information given in the question.
Hints
Think about what “margin of error” means
When a survey reports an estimate with a margin of error, what does that say about the possible true value for the entire population?
Use the margin of error with the estimate
Start by finding the lowest and highest percentages that are 4 percentage points away from 58%. What two numbers do you get?
Be careful about certainty vs. likelihood
Does a margin of error say the true percentage is guaranteed to be in that range, or that it is likely/plausible based on the sample?
Check what information you actually have
Do you see any information about a previous survey in the problem, or only this one survey with its estimate and margin of error?
Desmos Guide
Compute the interval around the estimate
In Desmos, type 58 - 4 on one line and 58 + 4 on another. Use the two results as the lower and upper bounds of the percentage range implied by the margin of error.
Relate the numbers to the choices
Look at the two values Desmos gives you; they form the interval around 58%. Then, choose the answer that uses this interval and correctly describes it as what the survey results support based on a margin of error (not as a guarantee, exact value, or change over time).
Step-by-step Explanation
Understand the estimate
The survey found that 58% of the sample of 600 registered voters support the proposed law. This 58% is an estimate of the true percentage of all registered voters in the state who support the law.
So 58% is not known to be exact; it is a best guess based on the sample.
Use the margin of error to find the interval
The margin of error is 4 percentage points, which means the true value is expected to be within 4 points of the estimate.
Compute the low and high ends of that range:
- Lower bound: (so 54%)
- Upper bound: (so 62%)
So the estimate plus/minus the margin of error gives a range from 54% to 62%.
Interpret what margin of error really means
In statistics, a margin of error describes a range where the true population value is likely or plausible, based on the sample. It does not mean:
- The true value is guaranteed to be in that range.
- The true value is exactly equal to the sample estimate.
- Anything about how opinions changed over time, unless the problem explicitly gives past data.
So we are looking for an answer that describes the 54%–62% range as a likely or plausible interval, not as certain, exact, or about change.
Match the statistical meaning to the choices
Now compare each option to what the margin of error tells us:
- The correct conclusion must:
- Use the interval from 54% to 62%, and
- Describe it as a plausible/likely range, not a guarantee, exact value, or change over time.
Only choice A) It is plausible that the actual percentage of all registered voters who support the proposed law is between 54% and 62%. matches both the numeric interval and the correct interpretation of margin of error, so A is the correct answer.