Question 33·Hard·Inference from Sample Statistics and Margin of Error
A researcher surveyed 1,200 randomly selected college freshmen from a particular state and asked whether they had worked a full-time job during the summer before matriculating. Of those surveyed, 312 students responded that they had worked full-time. Based on the sample results, the researcher reported an estimate that 26% of all college freshmen in the state worked full-time the previous summer, with an associated margin of error of 2.5%.
Which of the following statements is best supported by the reported margin of error?
For margin-of-error questions, translate the reported statistic into an interval using "estimate ± margin of error," compute the lower and upper bounds quickly, and then look for an answer that says the true population value is likely (or with high confidence) in that interval. Eliminate choices that misinterpret margin of error as a response error rate, or that claim something is guaranteed or impossible, since statistical margins of error are about high probability, not absolute certainty.
Hints
Focus on the phrase "margin of error"
Think about what a margin of error tells you when you see a poll result, like "50% ± 3%." What does the "± 3%" part represent?
Connect the estimate to the margin of error
You are told the estimate is 26% and the margin of error is 2.5%. How can you use 26% and 2.5% together to form a range of possible true values?
Interpret the range correctly
Once you have a lower and upper bound, ask: Does the margin of error make something impossible or guaranteed, or does it just tell you what is very likely?
Desmos Guide
Compute the bounds of the margin-of-error interval
In the first expression line, type 26-2.5 and in the second line type 26+2.5. The two outputs give you the lower and upper ends of the plausible percentage range (in percent units).
Step-by-step Explanation
Understand what the 26% represents
The researcher surveyed 1,200 freshmen and found that 312 worked full-time.
- The sample proportion is
This 26% is an estimate of the percentage of all college freshmen in the state who worked full-time the previous summer.
Recall what a margin of error means
A margin of error tells you how far the sample estimate might be from the true population value, in either direction, when using a given method (usually with high confidence, like 95%).
So a margin of error of around an estimate of means the true population percentage is expected to be within percentage points above or below .
Compute the interval given by the margin of error
Use the margin of error to find the lower and upper bounds around the estimate:
- Lower bound:
- Upper bound:
So the margin of error defines an interval from to around the estimate.
Interpret the interval and match it to an answer choice
The margin of error means we can be highly confident, but not absolutely certain, that the true percentage of all college freshmen in the state who worked full-time lies somewhere between and .
This is exactly what choice A) With high confidence, the percent of all college freshmen in the state who worked full-time the previous summer is between 23.5% and 28.5%. states, so A is the best-supported statement.