Question 19·Easy·Inference from Sample Statistics and Margin of Error
A polling agency surveyed a random sample of 500 college students about how many hours they study each day. The sample had a mean study time of 2.4 hours with a margin of error of 0.2 hour. Which of the following statements is most justified by the results of the survey?
For margin-of-error questions, first combine the sample statistic with the margin of error by computing statistic ± margin to get a numerical interval. Then decide whether that interval describes the sample or the population; on the SAT, margin of error with a random sample almost always refers to the population parameter (like the true mean or proportion). Finally, eliminate options that are one-sided (only less than or only greater than a bound), have the wrong endpoints, or mistakenly apply the interval to the sample instead of the population.
Hints
How is margin of error used?
Think about what you do with a margin of error when you have a sample mean. Do you use it to describe the sample itself, or to estimate something about the larger population?
Form the interval
Use the sample mean 2.4 hours and the margin of error 0.2 hour to create a lower and upper bound. What do you get when you subtract 0.2 from 2.4, and when you add 0.2 to 2.4?
Population vs. sample
Once you have that interval, ask: does this interval describe the 500 surveyed students, or does it describe all college students that the sample is supposed to represent?
Check direction and endpoints
Make sure the statement you choose does not go beyond the interval you found and does not use only one side of it (like just less than or just greater than a single number).
Desmos Guide
Compute the endpoints of the interval
In Desmos, type 2.4-0.2 on one line and 2.4+0.2 on another. Note the two numerical results; these are the lower and upper bounds of the interval suggested by the sample mean and margin of error.
Interpret the interval
Ask yourself what this interval is supposed to describe: the known sample mean, or the unknown true mean for the entire population of college students. Then choose the answer choice that both uses these two endpoints and correctly refers to the population mean.
Step-by-step Explanation
Understand the role of margin of error
The poll reports a sample mean of 2.4 hours with a margin of error of 0.2 hour. In this context, the margin of error is used to build an interval that likely contains the true mean for the entire population (all college students), based on the sample data.
Compute the confidence interval
To form the interval, subtract and add the margin of error to the sample mean:
So the estimated interval for the population mean study time is from 2.2 hours to 2.6 hours.
Decide what the interval describes
This interval is about the mean for all college students (the population), because the sample was random and the margin of error is defined for estimating a population parameter. It is not about the 500 students' sample mean itself, which is already known to be exactly 2.4 hours.
Match the interpretation to an answer choice
We need the option that (1) uses the interval from 2.2 to 2.6 hours and (2) clearly refers to the mean daily study time for all college students. That matches choice B) The mean daily study time for all college students is between 2.2 and 2.6 hours.