Flight Time Distribution Worksheet
Advanced Quantitative Reasoning • Statistics
Data Distribution & Outlier Analysis
Name:
Date:
Period:
TEKS S(4)(C): The student is expected to analyze the distribution characteristics of quantitative data, including determining the possible existence and impact of outliers.
- An aerospace logistics company is testing the battery endurance of next-generation autonomous delivery drones carrying emergency medical payloads. Flight engineers record the operational duration (in minutes) prior to safe landing battery thresholds across 16 standardized test flights under uniform atmospheric conditions. The recorded flight times are shown in the table below.
Flight Durations (minutes)
<table class="w-full text-center text-xs divide-y divide-slate-300"><tbody class="divide-y divide-slate-200 bg-white"><tr><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">72</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">81</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">66</td><td class="py-1.5 px-2 font-semibold text-slate-800">85</td></tr><tr class="bg-slate-50"><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">79</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">68</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">83</td><td class="py-1.5 px-2 font-semibold text-slate-800">74</td></tr><tr><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">58</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">81</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">75</td><td class="py-1.5 px-2 font-semibold text-slate-800">79</td></tr><tr class="bg-slate-50"><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">84</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">70</td><td class="py-1.5 px-2 border-r border-slate-200 font-semibold text-slate-800">64</td><td class="py-1.5 px-2 font-semibold text-slate-800">81</td></tr></tbody></table>
Sample size: \(n = 16\) test flights
Analyze the data. Which information is false about this distribution?
A. The sample standard deviation is 8.0 minutes
B. The median is 77 minutes
C. The interquartile range is 16 minutes
D. The mean is 75 minutes
Work Space Show your analysis and calculations below.
Record your final answer choice:
A
B
C
D
Standard S(4)(C) Distribution Characteristics Page 1 of 1
Flight Time Distribution Key
Teacher Resource • Worked Solution Key Standard: TEKS S(4)(C)
Flight Time Distribution — Answer Key
Correct Answer: C
Mean (\(\bar{x}\))
75.0 min
Median (\(\tilde{x}\))
77.0 min
True IQR (\(Q_3 - Q_1\))
12.0 min
Sample Std Dev (\(s\))
8.0 min
1
Ordered Data Array (\(n = 16\))
58, 64, 66, 68, 70, 72, 74, 75, 79, 79, 81, 81, 81, 83, 84, 85
2
Median & Quartiles
- Median (\(\tilde{x}\)): Average of 8th and 9th items:
\(\frac{75 + 79}{2} = 77\text{ min}\) → Option B is TRUE
- Lower Half (\(n=8\)): 58, 64, 66, 68, 70, 72, 74, 75
\(Q_1\) = \(\frac{68 + 70}{2} = 69\text{ min}\)
- Upper Half (\(n=8\)): 79, 79, 81, 81, 81, 83, 84, 85
\(Q_3\) = \(\frac{81 + 81}{2} = 81\text{ min}\)
- Interquartile Range (\(\text{IQR}\)):
\(\text{IQR} = Q_3 - Q_1 = 81 - 69 = \mathbf{12\text{ min}}\)
Option C claims \(\text{IQR} = 16\text{ min}\), which is FALSE!
3
Mean & Standard Deviation
Sample Mean (\(\bar{x}\)):
\(\bar{x} = \frac{\sum x_i}{n} = \frac{1200}{16} = \mathbf{75\text{ min}}\)
→ Option D is TRUE
Sum of Squared Deviations \(\sum (x_i - 75)^2\):
\((-17)^2 + (-11)^2 + (-9)^2 + (-7)^2 + (-5)^2 + (-3)^2 + (-1)^2 + 0^2 + 4^2 + 4^2 + 6^2 + 6^2 + 6^2 + 8^2 + 9^2 + 10^2 = \mathbf{960}\)
Sample Standard Deviation (\(s\)):
\(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} = \sqrt{\frac{960}{15}} = \sqrt{64} = \mathbf{8.0\text{ min}}\)
→ Option A is TRUE
4
Outlier Analysis (\(1.5 \times \text{IQR}\) Rule)
Fence Step:
\(1.5 \times \text{IQR} = 1.5 \times 12 = 18\)
Lower Fence:
\(Q_1 - 18 = 69 - 18 = \mathbf{51\text{ min}}\)
Upper Fence:
\(Q_3 + 18 = 81 + 18 = \mathbf{99\text{ min}}\)
Conclusion: Since minimum value (\(58 \ge 51\)) and maximum value (\(85 \le 99\)) are within the fences, no mathematical outliers exist in this distribution.
5
Detailed Option Rationale & Distractor Audit
A. True: Sample standard deviation \(s = \sqrt{960/15} = 8.0\text{ min}\). (A student confusing population \(\sigma = \sqrt{960/16} = 7.75\) might second-guess, but sample standard deviation is precisely 8.0).
