To determine which set of data does not contain an outlier, we can use the Interquartile Range (IQR) method. An outlier is typically defined as a value lower than \( Q1 - 1.5 \times IQR \) or higher than \( Q3 + 1.5 \times IQR \), where \( Q1 \) is the first quartile and \( Q3 \) is the third quartile.
Let's examine each set:
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Set 1: 25, 33, 29, 31, 33, 33, 78, 28
- Sorted: 25, 28, 29, 31, 33, 33, 33, 78
- Q1 = 29, Q3 = 33
- IQR = Q3 - Q1 = 33 - 29 = 4
- Lower Bound = Q1 - 1.5 * IQR = 29 - 6 = 23
- Upper Bound = Q3 + 1.5 * IQR = 33 + 6 = 39
- 78 is above 39, so this set contains an outlier.
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Set 2: 6, 7, 5, 8, 9, 1
- Sorted: 1, 5, 6, 7, 8, 9
- Q1 = 5, Q3 = 7.5 (average of 7 and 8)
- IQR = Q3 - Q1 = 7.5 - 5 = 2.5
- Lower Bound = 5 - 3.75 = 1.25
- Upper Bound = 7.5 + 3.75 = 11.25
- 1 is below 1.25, so this set contains an outlier.
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Set 3: 45, 45, 45, 45, 35, 45
- Sorted: 35, 45, 45, 45, 45, 45
- Q1 = 45, Q3 = 45
- IQR = Q3 - Q1 = 0
- Lower Bound = 45 - 0 = 45
- Upper Bound = 45 + 0 = 45
- All numbers are 45 or 35, and 35 is below 45. This set contains an outlier.
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Set 4: 101, 99, 98, 111, 101, 101, 103
- Sorted: 98, 99, 101, 101, 101, 103, 111
- Q1 = 99, Q3 = 101
- IQR = Q3 - Q1 = 101 - 99 = 2
- Lower Bound = 99 - 3 = 96
- Upper Bound = 101 + 3 = 104
- All numbers lie within the bounds, so this set has no outliers.
Based on the analysis, the 4th set (101, 99, 98, 111, 101, 101, 103) does not contain an outlier.
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