Let's break down each part of the question and present a detailed, step-by-step solution.
### (a) Greatest Spread
The spread of data refers to the range, which is the difference between the maximum and minimum values in the data set.
1. Adam's Training Times:
- Data: 103, 105, 104, 106, 100, 98, 92, 91, 97, 101
- Maximum Value (Max): 106
- Minimum Value (Min): 91
- Spread: [tex]\( 106 - 91 = 15 \)[/tex]
2. Miguel's Training Times:
- Data: 88, 86, 89, 93, 105, 85, 92, 96, 97, 94
- Maximum Value (Max): 105
- Minimum Value (Min): 85
- Spread: [tex]\( 105 - 85 = 20 \)[/tex]
Conclusion:
Miguel's training times have the greatest spread, as 20 (Miguel's range) is greater than 15 (Adam's range).
### (b) Least IQR Spread
The interquartile range (IQR) is used to measure the spread of the middle 50% of data and is the difference between the third quartile (Q3) and the first quartile (Q1).
1. Adam's Training Times IQR:
- Q3 (75th percentile): Value at the 75th percentile in sorted data
- Q1 (25th percentile): Value at the 25th percentile in sorted data
- Sorted Data: 91, 92, 97, 98, 100, 101, 103, 104, 105, 106
- Q1: 97
- Q3: 103.5
- IQR: [tex]\( 103.5 - 97 = 6.5 \)[/tex]
2. Miguel's Training Times IQR:
- Q3 (75th percentile): Value at the 75th percentile in sorted data
- Q1 (25th percentile): Value at the 25th percentile in sorted data
- Sorted Data: 85, 86, 88, 89, 92, 93, 94, 96, 97, 105
- Q1: 88
- Q3: 95.25
- IQR: [tex]\( 95.25 - 88 = 7.25 \)[/tex]
Conclusion:
The middle 50% of Adam's training times have the least spread, as 6.5 (Adam's IQR) is less than 7.25 (Miguel's IQR).
### (c) Interpretation
From the answers to parts (a) and (b), we can infer some insights into Adam's and Miguel's training times:
- Greatest Spread: Miguel's training times have a greater overall variation compared to Adam's. This means Miguel's training times are more spread out around the central value.
- Least IQR Spread: Adam's middle 50% of training times are more consistent than Miguel's. This suggests that despite Adam having a smaller overall variation, his training times are more tightly packed in the middle range.
Overall, while Miguel has more variation in his training times, Adam shows more consistency in the central portion of his training times. This could imply that Adam might be more reliable during his average performance, whereas Miguel's times might vary more widely.
