A sample consisting of 720 adults who own televisions was selected at random for a study. Based on the sample,...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A sample consisting of \(\mathrm{720}\) adults who own televisions was selected at random for a study. Based on the sample, it is estimated that \(\mathrm{32\%}\) of all adults who own televisions use their televisions to watch nature shows, with an associated margin of error of \(\mathrm{3.41\%}\). Which of the following is the most plausible conclusion about all adults who own televisions?
More than \(\mathrm{35.41\%}\) of all adults who own televisions use their televisions to watch nature shows.
Between \(\mathrm{28.59\%}\) and \(\mathrm{35.41\%}\) of all adults who own televisions use their televisions to watch nature shows.
Since the sample included adults who own televisions and not just those who use their televisions to watch nature shows, no conclusion can be made.
Since the sample did not include all the people who watch nature shows, no conclusion can be made.
1. TRANSLATE the statistical information
- Given information:
- Sample: 720 adults who own televisions (randomly selected)
- Point estimate: 32% use televisions to watch nature shows
- Margin of error: 3.41%
- What this tells us: We need to find the range of plausible values for the true population percentage.
2. INFER what margin of error means
- Margin of error creates a confidence interval around our point estimate
- The true population parameter is plausibly somewhere within:
(point estimate - margin of error) to (point estimate + margin of error) - We need to calculate both bounds of this interval
3. Calculate the confidence interval bounds
- Lower bound: \(32\% - 3.41\% = 28.59\%\)
- Upper bound: \(32\% + 3.41\% = 35.41\%\)
4. APPLY CONSTRAINTS to interpret the interval
- The interval 28.59% to 35.41% represents plausible values for the true population percentage
- This means it's reasonable to conclude that between 28.59% and 35.41% of all adults who own televisions use them to watch nature shows
- Values outside this range are less plausible (but not impossible)
Answer: B. Between 28.59% and 35.41% of all adults who own televisions use their televisions to watch nature shows.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not recognize that "estimate with margin of error" describes a confidence interval concept. They might think the margin of error is just additional information rather than a tool for creating an interval of plausible values.
This leads to confusion about what the numbers mean and may cause them to select Choice A (thinking the estimate means "at least 32%") or abandon systematic reasoning and guess.
Second Most Common Error:
Conceptual confusion about sampling: Students may not understand that the sample was appropriately selected from the target population (adults who own televisions) and think the sampling method invalidates any conclusions.
This misconception may lead them to select Choice C (thinking the sampling was inappropriate) or Choice D (misunderstanding what population we're making inferences about).
The Bottom Line:
This problem tests whether students understand the fundamental concept of statistical inference: how we use sample data with margin of error to make plausible statements about population parameters. The key insight is recognizing that margin of error creates a symmetric interval around the point estimate.
More than \(\mathrm{35.41\%}\) of all adults who own televisions use their televisions to watch nature shows.
Between \(\mathrm{28.59\%}\) and \(\mathrm{35.41\%}\) of all adults who own televisions use their televisions to watch nature shows.
Since the sample included adults who own televisions and not just those who use their televisions to watch nature shows, no conclusion can be made.
Since the sample did not include all the people who watch nature shows, no conclusion can be made.