- Two box plots compare marathon completion times (in hours) for runners in different age groups. The younger group has Q1 = 3.2, median = 3.8, Q3 = 4.5. The older group has Q1 = 3.6, median = 4.2, Q3 = 4.8. If both groups have the same number of runners, which comparison of their performances is most justified?
Answer: The younger group performed better because their median time is faster and their IQR shows similar consistency at 1.3 hours
Younger group IQR = 4.5 - 3.2 = 1.3 hours. Older group IQR = 4.8 - 3.6 = 1.2 hours. The younger group has a faster median (3.8 vs 4.2 hours) and nearly the same consistency (IQR of 1.3 vs 1.2). Choice A incorrectly calculates the younger group's IQR. Choice C states both IQRs are 1.3. Choice D is incorrect because Q3 for younger group is 4.5, meaning only 75% finished by 4.5 hours, not 4.2 hours.
- A researcher creates a box plot from survey data about daily screen time (in hours) for teenagers. The box plot shows Q1 = 4, median = 6, Q3 = 8, with whiskers extending to 2 and 11 hours. If the researcher wants to identify teenagers with "unusually high" screen time using the 1.5 × IQR rule, what is the minimum number of hours of daily screen time that would be considered unusually high?
Answer: Any screen time greater than 14 hours should be considered unusually high
Using the 1.5 × IQR rule: IQR = Q3 - Q1 = 8 - 4 = 4 hours. The upper fence = Q3 + 1.5(IQR) = 8 + 1.5(4) = 8 + 6 = 14 hours. Any value above 14 hours is considered unusually high (an outlier). Choice A uses Q3, choice B uses the maximum whisker value, and choice D incorrectly calculates the fence.
- A box plot displays the heights (in inches) of basketball players on a team. The plot shows that Q1 = 70 inches and Q3 = 76 inches. A coach states that "the team has good height consistency since most players are within 6 inches of each other." Evaluate the accuracy of the coach's reasoning.
Answer: The coach's reasoning is incorrect because the IQR of 6 inches represents only 50% of players, not most of the team
IQR = Q3 - Q1 = 76 - 70 = 6 inches, but this represents the range containing the middle 50% of players, not 'most' of the team. The coach incorrectly interprets what the IQR represents. Choice A incorrectly states the IQR covers 75%. Choice C makes a judgment about what constitutes 'low variability' without context. Choice D misses the main error in the coach's reasoning about what percentage the IQR represents.
- Two schools compare their students' performance on a standardized test using box plots. School A has Q1 = 520, median = 580, Q3 = 640. School B has Q1 = 540, median = 580, Q3 = 620. An administrator concludes that "both schools performed equally well since they have the same median." Which analysis of this conclusion is most appropriate?
Answer: The conclusion is incomplete because School B shows more consistent performance with a smaller IQR of 80 compared to School A's IQR of 120
School A IQR = 640 - 520 = 120. School B IQR = 620 - 540 = 80. While both have the same median, School B shows more consistent performance with less variability in the middle 50% of scores. The conclusion is incomplete because it ignores the consistency aspect. Choice A overstates the median's importance. Choice C mentions range but focuses on extremes rather than the relevant IQR comparison. Choice D restates what equal medians mean without addressing the consistency difference.
