Answer :
To determine which sets contain one or more outliers, we use the concept of the Interquartile Range (IQR). An outlier in a dataset is typically defined as a value that lies more than 1.5 times the IQR below the first quartile (Q1) or above the third quartile (Q3).
Let's proceed step by step for each dataset:
### Set 1: [tex]\(187, 298, 239, 1, 984, 202, 191\)[/tex]
1. Arrange the data: [tex]\(1, 187, 191, 202, 239, 298, 984\)[/tex]
2. Find Q1 (25th percentile) and Q3 (75th percentile).
3. Calculate IQR = Q3 - Q1.
4. Determine lower bound = Q1 - 1.5 IQR.
5. Determine upper bound = Q3 + 1.5 IQR.
6. Identify any values outside these bounds as outliers.
### Set 2: [tex]\(5, 3, 8, 6, 3, 6, 1, 2, 0\)[/tex]
1. Arrange the data: [tex]\(0, 1, 2, 3, 3, 5, 6, 6, 8\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
### Set 3: [tex]\(15, 19, 21, 16, 25, 13, 17\)[/tex]
1. Arrange the data: [tex]\(13, 15, 16, 17, 19, 21, 25\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
### Set 4: [tex]\(56, 1, 5, 72, 67, 59, 74, 60\)[/tex]
1. Arrange the data: [tex]\(1, 5, 56, 59, 60, 67, 72, 74\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
### Set 5: [tex]\(88, 7, 32, 31, 34, 39, 34, 35, 33\)[/tex]
1. Arrange the data: [tex]\(7, 31, 32, 33, 34, 34, 35, 39, 88\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
After carrying out the above steps for each set, we reach the following conclusions:
- Set 1 contains outliers.
- Set 2 does not contain outliers.
- Set 3 does not contain outliers.
- Set 4 contains outliers.
- Set 5 contains outliers.
Thus, the sets that contain one or more outliers are:
[tex]\[ \{ \mathbf{187,298,239,1,984,202,191}, \mathbf{56,1,5,72,67,59,74,60}, \mathbf{88,7,32,31,34,39,34,35,33} \} \][/tex]
Let's proceed step by step for each dataset:
### Set 1: [tex]\(187, 298, 239, 1, 984, 202, 191\)[/tex]
1. Arrange the data: [tex]\(1, 187, 191, 202, 239, 298, 984\)[/tex]
2. Find Q1 (25th percentile) and Q3 (75th percentile).
3. Calculate IQR = Q3 - Q1.
4. Determine lower bound = Q1 - 1.5 IQR.
5. Determine upper bound = Q3 + 1.5 IQR.
6. Identify any values outside these bounds as outliers.
### Set 2: [tex]\(5, 3, 8, 6, 3, 6, 1, 2, 0\)[/tex]
1. Arrange the data: [tex]\(0, 1, 2, 3, 3, 5, 6, 6, 8\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
### Set 3: [tex]\(15, 19, 21, 16, 25, 13, 17\)[/tex]
1. Arrange the data: [tex]\(13, 15, 16, 17, 19, 21, 25\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
### Set 4: [tex]\(56, 1, 5, 72, 67, 59, 74, 60\)[/tex]
1. Arrange the data: [tex]\(1, 5, 56, 59, 60, 67, 72, 74\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
### Set 5: [tex]\(88, 7, 32, 31, 34, 39, 34, 35, 33\)[/tex]
1. Arrange the data: [tex]\(7, 31, 32, 33, 34, 34, 35, 39, 88\)[/tex]
2. Find Q1 and Q3.
3. Calculate IQR.
4. Determine lower and upper bounds.
5. Identify any outliers.
After carrying out the above steps for each set, we reach the following conclusions:
- Set 1 contains outliers.
- Set 2 does not contain outliers.
- Set 3 does not contain outliers.
- Set 4 contains outliers.
- Set 5 contains outliers.
Thus, the sets that contain one or more outliers are:
[tex]\[ \{ \mathbf{187,298,239,1,984,202,191}, \mathbf{56,1,5,72,67,59,74,60}, \mathbf{88,7,32,31,34,39,34,35,33} \} \][/tex]
