Question 18·Hard·Inference from Sample Statistics and Margin of Error
A polling organization telephoned 1,248 randomly selected adults in River County and asked whether they would support a new county sales tax to fund public libraries. The poll reported that 52% of the adults sampled said they would support the tax and that the result had a margin of error of 3 percentage points at a 95% confidence level.
Which of the following statements is most consistent with the poll's result and its reported margin of error?
For margin-of-error questions, first identify the sample statistic (here, 52%) and the margin of error (3 percentage points). Quickly form the confidence interval by doing "estimate ± margin of error" to get the lower and upper bounds, then look for the choice that correctly interprets that interval as a plausible range for the true population value. Eliminate answers that either treat the sample statistic as exact, confuse margin of error with response mistakes, or change the sampling method in a way that introduces bias rather than improving accuracy.
Hints
Think about what margin of error means
Focus on the phrase "margin of error of 3 percentage points". How does that relate to the reported 52% from the sample?
Turn the margin of error into a range
Try writing an interval by subtracting and adding 3 percentage points to 52%. What does that interval represent in terms of all adults in River County?
Check what margin of error does not say
Margin of error is about how close the sample result is likely to be to the true population value. Does it tell you anything about people answering incorrectly, or about changing who you sample?
Desmos Guide
Use Desmos to find the lower end of the interval
In an empty line, type 0.52 - 0.03. The output is the lower bound, as a decimal, for the plausible proportion of adults who support the tax.
Use Desmos to find the upper end of the interval
On the next line, type 0.52 + 0.03. The output is the upper bound, as a decimal, for the plausible proportion of adults who support the tax.
Convert to percentages and compare with choices
Convert each decimal output from Desmos into a percentage (by multiplying by 100) to get the plausible range in percent form. Look for the choice that describes this entire range as a plausible set of values for all adults in River County.
Step-by-step Explanation
Interpret the poll result and margin of error
The poll found that 52% of the sampled adults said they would support the tax. A margin of error of 3 percentage points at a 95% confidence level means that, based on this random sample, the true percentage of all adults in River County who support the tax is likely to be within 3 percentage points of 52%.
Write the plausible range using the margin of error
To find the plausible interval for the true population percentage, subtract and add the margin of error to the sample percentage:
- Lower end of interval: percentage points
- Upper end of interval: percentage points
So the plausible interval is from % up to %.
Eliminate options that misuse margin of error
Now check each type of statement:
- Saying "exactly 52%" for all adults ignores the margin of error and treats the sample result as a perfect, exact population value.
- Saying something about "at most 3% answered incorrectly" confuses margin of error (sampling variation) with people making mistakes when answering.
- Claiming the margin of error would be smaller if the poll contacted only library users introduces bias; that group is not representative of all adults, and margin of error is about random sampling, not about focusing on a subgroup.
None of these match the idea of a plausible range for the percentage of all adults who support the tax.
Match the correctly interpreted interval to an answer choice
Compute the interval:
- Lower end: percentage points
- Upper end: percentage points
So it is reasonable ("plausible") that the true percentage of all adults in River County who support the tax is between 49% and 55%. The statement that matches this interpretation is:
D) It is plausible that between 49% and 55% of all adults in River County support the sales tax.