Question 17·Medium·Inference from Sample Statistics and Margin of Error
A random sample of college students was surveyed about the number of hours they studied during the previous week. The sample mean was found to be hours, with an associated margin of error of hour. Based on this information, which of the following is the most appropriate conclusion about the average number of hours studied last week by all college students in the population?
For questions involving a sample mean and a margin of error, immediately form the interval , where is the sample mean and is the margin of error. This interval represents the plausible range for the population mean. Then, eliminate any choices that go outside this interval or that claim the population mean is exactly equal to the sample mean. Finally, select the option that correctly describes the mean as lying somewhere within the computed range.
Hints
Relate margin of error to the sample mean
Think about how the margin of error combines with the sample mean. What two operations can you do with and to get a low and high value?
Find the interval endpoints
Calculate and . These give you the endpoints of an interval of plausible values for the true average.
Interpret the interval in words
Once you have the lower and upper bounds, look for the choice that describes the true average as lying somewhere between those two numbers, without claiming it must be exactly any one value.
Desmos Guide
Compute the bounds of the interval
In Desmos, type 6.3-0.7 on one line and 6.3+0.7 on another. Note the two numerical outputs; these are the lower and upper bounds of the plausible range for the true average.
Use Desmos results to pick an answer
Interpret the two values from Desmos as the endpoints of the interval of plausible population means, then choose the answer option that describes the true average as lying between those two endpoints (without saying it must equal the sample mean).
Step-by-step Explanation
Understand what margin of error means
A sample mean is an estimate of the true population mean. The margin of error tells us how far, at most, the true population mean is likely to be from the sample mean.
So if the sample mean is and the margin of error is , then the plausible values for the population mean are all values from to .
Compute the plausible interval
Here, the sample mean is hours and the margin of error is hour.
Find the lower and upper bounds:
- Lower bound:
- Upper bound:
So, based on the sample and its margin of error, the true population average study time is likely to lie somewhere between 5.6 and 7.0 hours.
Match the statistical conclusion to the answer choice
From the margin of error, we conclude that it is reasonable (plausible) that the true average number of hours all college students studied last week is between 5.6 and 7.0 hours, but not confidently below 5.6 or above 7.0, and certainly not guaranteed to be exactly 6.3.
The only option that correctly states this is:
D) It is plausible that the average is between 5.6 and 7.0 hours.