Question 3·Hard·Inference from Sample Statistics and Margin of Error
799 randomly selected US teens were asked how often they talk on a cell phone and about their texting behavior. The data are summarized in the table below.
| Texting behavior | Talks on cell phone daily | Does not talk on cell phone daily | Total |
|---|---|---|---|
| Light | 110 | 146 | 256 |
| Medium | 139 | 164 | 303 |
| Heavy | 166 | 74 | 240 |
| Total | 415 | 384 | 799 |
Based on the data from the study, an estimate of the percent of US teens who are heavy texters is and the associated margin of error is . Which of the following is a correct statement based on the given margin of error?
For margin-of-error questions, first form the interval by adding and subtracting the margin of error from the estimate, then interpret that interval carefully: it represents a likely range for the population value, not an exact value and not a statement of impossibility outside the range. Next, eliminate any choices that treat the margin of error as a misclassification rate, that claim a single exact population percent, or that use absolute language like "not possible" rather than describing what is plausible or doubtful given the interval.
Hints
Think about what 30% describes
Is the 30% value describing the teens in the sample, or an estimate of all US teens? Make sure you know which group the question is about.
Use the margin of error to make an interval
If the estimate is 30% and the margin of error is 3%, what is the lowest and highest percent that are considered likely based on this study?
What does the interval say about possibility vs likelihood?
Margin of error tells you what values are likely, not what values are absolutely impossible. Keep this in mind when judging strong words like "not possible" or exact claims like "is 33%."
Compare each choice to your interval
After you find the interval, check which statements are consistent with an interval from your lower bound to your upper bound and how they describe values outside that range.
Desmos Guide
Compute the likely interval
In Desmos, enter two expressions: 0.30 - 0.03 and 0.30 + 0.03. These outputs are the decimal endpoints of the likely interval for the true proportion of heavy texters.
Convert to percents and compare to choices
Multiply each result by 100 in Desmos (for example, enter (0.30 - 0.03)*100) to see the interval in percent form, then compare that interval to each answer choice’s percent value to judge which descriptions are consistent with the interval and which refer to values outside it.
Step-by-step Explanation
Understand what the 30% represents
From the table, 240 of the 799 teens are heavy texters.
So 30% is the sample estimate for the percent of all US teens who are heavy texters.
Apply the margin of error
A margin of error of 3% means the true population percent is likely within 3 percentage points of the estimate.
So the plausible interval is:
- Lower end:
- Upper end:
This gives a likely range of to for the true percent of all US teens who are heavy texters.
Interpret what the margin of error means
The margin of error does not:
- Talk about misclassification of teens in the sample.
- Say the true percent must equal a single number.
- Say values outside the interval are impossible.
Instead, it says values inside to are consistent with the study, and values outside this range are less likely (doubtful) given the data.
Match the correct interpretation to an answer choice
Now compare each choice to the interval to and the meaning of margin of error:
- A talks about 3% of sample teens being misclassified; margin of error is not about that.
- B says it is not possible for the percent to be less than 27%; margin of error never says values outside the range are impossible, only less likely.
- C states that the percent is 33%, which is just one endpoint of the interval, not a guaranteed true value.
- D says it is doubtful that the true percent is 35%, and 35% lies outside the 27%–33% range, so it is indeed unlikely.
Therefore, the correct statement is: It is doubtful that the percent of all US teens who are heavy texters is 35%.