Question 32·Medium·Inference from Sample Statistics and Margin of Error
A transportation researcher surveyed 500 randomly selected employees at a large company about their daily one-way commute time. The survey found a sample mean of 38.6 minutes and reported a margin of error of 2.4 minutes.
Which of the following is the best interpretation of this estimate?
For questions involving a sample mean and a margin of error, first compute the interval by doing mean minus margin and mean plus margin. Then focus on what that interval describes: it almost always refers to a population parameter (like the average for all people in the group), not to individual values or just the sample. Finally, eliminate choices that talk about percentages of individuals, say “exactly” a certain percent, or incorrectly apply the interval to the sample instead of the whole population.
Hints
Think about what margin of error does
You are given a sample mean and a margin of error. How do you use the margin of error with the mean to create an interval?
Calculate the actual number range
Try computing and . That gives you the lower and upper bounds of the estimate.
Decide what the interval is about
Ask yourself: does this interval describe individual commute times, the average for just the 500 surveyed employees, or the average for all employees at the company?
Check the wording carefully
Look for the answer choice that both uses your calculated interval and clearly refers to the average (mean) for the whole company, not an exact percentage of employees or the sample only.
Desmos Guide
Compute the bounds of the interval
In Desmos, type 38.6-2.4 on one line and 38.6+2.4 on another line. Use the two numerical outputs as the lower and upper bounds of the interval described in the question.
Match the numerical interval to the wording
Compare the two numbers you see in Desmos to the intervals mentioned in the answer choices, and then check which choice correctly describes that interval as applying to the average for all employees (not just the sample or individual commute times).
Step-by-step Explanation
Identify what the numbers represent
The problem gives:
- A sample mean commute time of minutes (average for the 500 surveyed employees).
- A margin of error of minutes.
A margin of error around a sample mean is used to estimate the population mean (the average for all employees at the company), not just the sample.
Use the margin of error to find the interval
To find the estimate interval, subtract and add the margin of error to the sample mean:
- Lower bound: minutes
- Upper bound: minutes
So the estimated interval is from 36.2 minutes to 41.0 minutes.
Decide what the interval applies to
This interval does not say where most individual commute times fall, and it does not describe the sample’s average (we already know the sample average exactly: minutes).
Instead, the interval gives a range of plausible values for the true average (mean) commute time of all employees at the company, based on the sample.
Match the correct interpretation to the interval
We need the choice that:
- Uses the interval from 36.2 to 41.0 minutes, and
- Clearly states it is about the average one-way commute time for all employees at the company (the population mean), not about individual employees or just the 500 surveyed.
The only option that correctly does this is:
C) Plausible values for the average one-way commute time for all employees at the company are between 36.2 minutes and 41.0 minutes.