A researcher surveyed randomly selected high school seniors to estimate the mean number of text messages they send per day. The sample produced a mean of messages with an associated margin of error of messages. Which of the following is a reasonable statement about the mean number of text messages sent per day by all high school seniors?
For SAT questions involving a sample statistic and a margin of error, immediately translate the information into an interval by taking the sample estimate and going the margin of error amount below and above it ("estimate plus or minus margin"). Compute both endpoints quickly, then choose the option that correctly describes the entire interval for the population parameter, being careful not to pick answers that only talk about values below the lower bound, above the upper bound, or outside the interval.
Hints
Use the margin of error with the sample mean
You are told the sample mean is 82 and the margin of error is 6. How can you combine these two numbers to create a range of plausible values for the true mean?
Think "plus or minus" the margin of error
Start at the sample mean of 82. Go down by 6 to get the lower bound, and up by 6 to get the upper bound. What two numbers do you get?
Match your range to a choice
Once you have the two endpoint values, look for the answer choice that talks about values lying within that range, not only below one endpoint or only above the other.
Desmos Guide
Compute the endpoints of the margin-of-error interval
In Desmos, enter 82-6 on one line and 82+6 on another line. Use the two numerical results as the lower and upper bounds of the plausible interval for the population mean, then select the answer choice that describes that interval.
Step-by-step Explanation
Understand what the margin of error represents
When a survey reports a sample mean and a margin of error, the margin of error tells you how far above or below the sample mean the true population mean could reasonably be.
So you form an interval that goes from "sample mean minus margin of error" to "sample mean plus margin of error."
Set up the lower and upper bounds
In this problem, the sample mean is messages per day and the margin of error is messages.
Compute the two bounds using the sample mean and the margin of error:
- Lower bound:
- Upper bound:
Calculate the interval and match it to an answer choice
Now calculate the bounds:
So a reasonable estimate is that the true mean number of text messages per day for all high school seniors is between and messages.
This corresponds to choice D: It is between and messages.