Question 23·Medium·Inference from Sample Statistics and Margin of Error
A random sample of 500 high school students was surveyed. Based on the sample, it is estimated that 68% of all high school students send at least 20 text messages per day, with an associated margin of error of 4%.
Which of the following is the most reasonable conclusion about the texting habits of all high school students?
For margin-of-error questions, quickly identify the sample estimate and the margin of error, then compute the interval: estimate minus margin of error (lower bound) and estimate plus margin of error (upper bound). Translate that interval into a verbal statement ("between ___% and ___%") and pick the choice that matches the entire range; avoid answers that claim an exact value or ignore the margin of error.
Hints
Focus on the margin of error
Look at the phrase "margin of error of 4%." How does a margin of error relate to the estimate of 68%?
Think of an interval around the estimate
Try subtracting 4% from 68% to get a lower bound and adding 4% to 68% to get an upper bound. What range do you get?
Compare the range to the options
Once you have the lower and upper bounds, see which answer choice describes that entire range, not just a single value or a value outside that range.
Desmos Guide
Use Desmos to calculate the bounds of the interval
In Desmos, type 0.68 - 0.04 on one line and 0.68 + 0.04 on another line.
Look at the two decimal outputs; they represent the lower and upper bounds of the likely percentage. Convert each decimal to a percent and then choose the option that describes that full percentage interval.
Step-by-step Explanation
Interpret the estimate and margin of error
The 68% is the sample estimate of the percentage of all high school students who send at least 20 texts per day.
A margin of error of 4% means the true population percentage is expected to be within 4 percentage points above or below this estimate.
Compute the lower and upper bounds of the interval
Use the margin of error to build an interval around 68%:
- Lower bound: subtract the margin of error:
- Upper bound: add the margin of error:
So the plausible interval is from to .
Match the interval to the answer choice
We concluded the reasonable range for the true percentage of all high school students who send at least 20 texts per day is from 64% to 72%.
The only choice that states this interval is:
B) Between 64% and 72% of all high school students send at least 20 text messages per day.