Question 44·Medium·Inference from Sample Statistics and Margin of Error
From a university population of 4,800 students, a random sample of 200 students was surveyed about how many hours they slept the previous night. The survey reported a mean of 8.5 hours with a margin of error of 0.4 hour.
Based on these results, which of the following is a plausible value for the mean number of hours all 4,800 students slept the previous night?
For margin-of-error questions, immediately form the interval "sample mean ± margin of error" by doing one subtraction and one addition. Then, quickly compare the answer choices to this interval and choose the value(s) that fall inside it, including the endpoints. This avoids overthinking and keeps the problem to two small calculations and a simple check.
Hints
Use what “margin of error” means
The margin of error tells you how far above or below the sample mean the true population mean is likely to be. How can you use that to form an interval around 8.5?
Find one endpoint of the interval
Try subtracting the margin of error, 0.4, from the sample mean, 8.5. What does that number represent?
Find the other endpoint and check choices
Now add the margin of error, 0.4, to the sample mean, 8.5. Then, look at all four answer choices and see which ones fall between your two endpoints (including the endpoints).
Remember what “plausible value” means
A plausible value for the population mean should lie inside the interval determined by the sample mean ± margin of error.
Desmos Guide
Compute the endpoints of the interval
In Desmos, type 8.5 - 0.4 on one line and 8.5 + 0.4 on another line. Note the two numerical outputs; these are the lower and upper bounds of the plausible interval for the population mean.
Compare answer choices to the interval
Check each answer choice against the two bounds you found in Desmos and see which choice lies within (or exactly on) that interval.
Step-by-step Explanation
Interpret the margin of error
The survey reports a sample mean of 8.5 hours with a margin of error of 0.4 hour. This means the true mean for all 4,800 students is expected to be within 0.4 hour above or below 8.5 hours.
Find the lower end of the plausible interval
Subtract the margin of error from the sample mean to find the lower bound:
So, the plausible population mean must be at least 8.1 hours.
Find the upper end and compare with the choices
Add the margin of error to the sample mean to find the upper bound:
So the plausible interval for the true mean is from 8.1 to 8.9 hours, inclusive. Among the answer choices 8.0, 8.9, 9.1, and 9.3, the only value inside this interval is 8.9, so that is the correct answer.