Question 37·Easy·Inference from Sample Statistics and Margin of Error
A city council randomly surveyed 500 adult residents about a proposal to build a new community park. Of those surveyed, 270 said they would vote in favor of the proposal. Using the survey results, the council estimated that 54% of all adult residents would vote in favor, with a margin of error of 4 percentage points. Which of the following is the best interpretation of this estimate?
For margin-of-error questions, first identify the estimated percentage (or mean) and the margin of error. Form an interval by subtracting the margin from the estimate and adding the margin to the estimate, then interpret that interval in the context of the entire population, not just the sample. Quickly eliminate answer choices that (1) claim an exact value instead of a range, (2) talk about a specific number of sampled individuals instead of a population percentage, or (3) describe only one side of the range (such as only less than or only greater than the estimate).
Hints
Focus on the margin of error
The council gives an estimate of with a margin of error of 4 percentage points. How do you turn a single estimate with a margin of error into a range of possible values?
Think about plus and minus
A margin of error tells you how far above or below the estimate the true value might be. Try subtracting 4 percentage points from 54 and adding 4 percentage points to 54.
Pay attention to what group is being described
The survey is based on 500 residents, but the estimate and margin of error are about all adult residents in the city, not just the 500 people surveyed. Which choice talks about all adult residents and uses a range of percentages, not an exact single number?
Watch the wording in each option
Look carefully for words like 'exactly', 'between', and 'fewer than'. Which of these best matches the idea that the true value is somewhere within a range around 54%?
Desmos Guide
Use Desmos to compute the endpoints of the margin-of-error interval
In Desmos, type 54-4 on one line and 54+4 on another line. The two outputs are the lower and upper percentage bounds of the estimate. Then look for the answer choice that describes a range of percentages from the smaller of these two values to the larger one, for all adult residents.
Step-by-step Explanation
Understand what the 54% represents
The survey found that 270 out of 500 adults said they would vote in favor.
- The sample proportion is , or .
- This is an estimate of the proportion of all adult residents who would vote in favor, based on the sample.
Interpret the margin of error
A margin of error of 4 percentage points means the actual population percentage could reasonably be as much as 4 percentage points lower or higher than the estimate.
- So we form an interval from
- a lower bound of percentage points, and
- an upper bound of percentage points.
- This interval gives the range of plausible values for the true percentage of all adult residents who would vote in favor.
Compute the interval and match it to the wording
Now compute the endpoints:
- Lower bound: percentage points ()
- Upper bound: percentage points ()
So the best interpretation is that between and of all adult residents will likely vote in favor of the proposal, which corresponds to answer choice C.