Question 38·Medium·Inference from Sample Statistics and Margin of Error
A polling organization surveyed a random sample of 1,200 registered voters. In the sample, 58 % of the voters said they support a proposed environmental law, with an associated margin of error of 2.8 %.
Which of the following is the most reasonable conclusion about all registered voters?
For questions involving a sample percentage and a margin of error, immediately translate the wording into a “plus or minus” calculation: compute (sample − margin of error) and (sample + margin of error) to find the lower and upper bounds of the plausible interval for the population. Then, carefully read each choice and select the one that describes that interval for all individuals in the population, avoiding answers that claim an exact value or that incorrectly dismiss valid conclusions from a well-designed sample.
Hints
Focus on what margin of error tells you
The question gives a sample percentage (58%) and a margin of error (2.8%). Think about how these two numbers are combined to describe a range of possible values for the whole population.
Think in terms of “plus or minus”
A margin of error is usually interpreted as “plus or minus” a certain amount from the reported percentage. How could you use 2.8% together with 58% in this way?
Calculate the endpoints of the range
Try computing and . Then look at the choices and see which one describes that range in words.
Desmos Guide
Compute the lower bound
In Desmos, type 58 - 2.8 and press Enter. Note the result; this is the lower end of the plausible percentage range for all voters.
Compute the upper bound
Type 58 + 2.8 and press Enter. This result is the upper end of the plausible percentage range. Use these two numbers to identify which answer choice correctly describes the interval.
Step-by-step Explanation
Interpret what the margin of error means
A sample result of 58% with a margin of error of 2.8% means the true percentage for all registered voters is likely within 2.8 percentage points of 58%. In other words, we should consider all values that are 2.8% above or below 58% as plausible for the whole population.
Find the lower end of the plausible range
Subtract the margin of error from the sample percentage.
So one end of the plausible interval is 55.2%.
Find the upper end of the plausible range
Add the margin of error to the sample percentage.
So the other end of the plausible interval is 60.8%.
Match this interval to the answer choices
We now know the reasonable conclusion is that the true percentage of all registered voters who support the law is between 55.2% and 60.8%. Among the options, this is exactly what choice B states, so B is correct.