Question 20·Medium·Inference from Sample Statistics and Margin of Error
A researcher surveyed a random sample of 200 college students and recorded the number of hours each student studied during the previous week. The sample mean was 12 hours, with an associated margin of error of 1.3 hours.
Which of the following statements is best supported by these data?
For questions involving a sample mean and a margin of error, first compute the interval by doing mean ± margin of error. Then carefully identify what that interval refers to: almost always it is a plausible range for a population parameter (like the true mean or proportion), not individual data values. Eliminate any choices that talk about specific percentages of individuals, exact values for most people, or impossible values, and pick the one that correctly describes a likely range for the population mean or proportion.
Hints
Interpret the margin of error
Ask yourself: when a problem gives a sample mean and a margin of error, what quantity is that margin of error usually describing—individual data values or a population parameter like a mean or proportion?
Use the numbers given
Combine the sample mean of hours with the margin of error of hours. What two numbers do you get when you subtract and add the margin of error?
Focus on what the interval applies to
Look for the answer choice that talks about a range for the average (mean) for all college students, not about individual students or an exact number of hours.
Desmos Guide
Calculate the interval endpoints
In Desmos, type 12 - 1.3 on one line and 12 + 1.3 on another. Note the two numerical results; these are the lower and upper bounds of the interval defined by the margin of error around the mean.
Connect the interval to the population mean
Remember that this interval represents plausible values for the average study time of all college students. Use this idea to choose the answer option that talks about a range for the mean between the two values you found in Desmos.
Step-by-step Explanation
Use the margin of error to find the interval
The sample mean is hours and the margin of error is hours.
Compute the endpoints of the interval:
- Lower end:
- Upper end:
So the data support an interval from to hours.
Recall what a margin of error tells you
A margin of error around a sample mean is used to estimate where the population mean (the true average for all individuals in the group) is likely to fall.
It does not:
- Describe the range of hours for each individual student.
- Guarantee anything about exact values (like everyone studying exactly 12 hours).
- Make extreme values impossible; it just talks about the average.
Match the correct interpretation to the answer choices
We need the choice that correctly describes a plausible range for the population mean study time, based on the sample mean of hours and margin of error hours.
The only option that talks about the mean for all college students being between and hours is:
It is plausible that the mean weekly study time for all college students is between 10.7 and 13.3 hours.