Question 47·Medium·Inference from Sample Statistics and Margin of Error
A transportation analyst surveyed a simple random sample of employees at a large company about their one-way commute time. The sample produced a mean commute time of minutes with an associated margin of error of minutes.
Based on this information, which statement is the most appropriate conclusion about the mean one-way commute time for all employees at the company?
For questions about a sample mean and margin of error, immediately interpret the margin of error as “plus or minus” around the sample mean: compute mean − margin and mean + margin to get the plausible interval for the population mean. Then eliminate any choices that: (1) give a single exact value instead of a range, or (2) describe values outside that interval, and select the option that correctly states the full plausible range.
Hints
Focus on "margin of error"
Think about what it means when a survey reports a mean and a margin of error. How does the margin of error relate to the mean?
Think plus and minus
If the sample mean is and the margin of error is , what do you get if you subtract from and if you add to ?
Interpret the interval
Once you have the lower and upper numbers from using the margin of error, look for the answer choice that describes that entire range, not a single exact value or values outside that range.
Desmos Guide
Compute the ends of the margin-of-error interval
In Desmos, type 32-3 on one line and 32+3 on another line. Use the outputs to identify the lower and upper bounds of the plausible range for the mean commute time.
(Optional) Visualize the interval on a number line
You can enter an inequality like 29 <= x and x <= 35 in Desmos to shade the plausible interval. Then compare this shaded range to the wording of each answer choice to see which one matches the interval you found.
Step-by-step Explanation
Understand what the margin of error means
The margin of error tells us how far the true population mean is likely to be from the sample mean, based on the sampling method and sample size. It creates an interval around the sample mean where the true mean is considered plausible.
Use the margin of error to build the interval
The sample mean is minutes and the margin of error is minutes. So the plausible interval for the true mean is:
This means the plausible range for the population mean commute time is from to minutes.
Match the interval to the best conclusion
We cannot say the population mean is exactly minutes; the sample mean is only an estimate. Values far outside the interval, like less than or greater than , are not supported by the given margin of error. The only option that correctly describes the plausible range based on the sample mean and margin of error is: It is plausible that the mean commute time is between 29 minutes and 35 minutes.