Answer :
To determine if there are any outliers in the data set when creating a modified boxplot, follow these steps:
1. List the Data: The data set is: 71, 74, 78, 79, 79, 80, 82.
2. Find the Quartiles:
- Q1 (First Quartile): This is the 25th percentile of the data set. In this data set, [tex]\( Q1 = 76.0 \)[/tex].
- Q3 (Third Quartile): This is the 75th percentile of the data set. In this data set, [tex]\( Q3 = 79.5 \)[/tex].
3. Calculate the Interquartile Range (IQR):
[tex]\[
\text{IQR} = Q3 - Q1 = 79.5 - 76.0 = 3.5
\][/tex]
4. Determine the Outlier Boundaries:
- Lower Bound: This is calculated as [tex]\( Q1 - 1.5 \times \text{IQR} \)[/tex].
[tex]\[
\text{Lower Bound} = 76.0 - 1.5 \times 3.5 = 70.75
\][/tex]
- Upper Bound: This is calculated as [tex]\( Q3 + 1.5 \times \text{IQR} \)[/tex].
[tex]\[
\text{Upper Bound} = 79.5 + 1.5 \times 3.5 = 84.75
\][/tex]
5. Identify the Outliers:
- Any data points below the lower bound or above the upper bound would be considered outliers.
- In this data set, all the values (71, 74, 78, 79, 79, 80, 82) are within the calculated boundary range of 70.75 to 84.75.
Therefore, there are no outliers in this data set. The correct answer is "no outliers."
1. List the Data: The data set is: 71, 74, 78, 79, 79, 80, 82.
2. Find the Quartiles:
- Q1 (First Quartile): This is the 25th percentile of the data set. In this data set, [tex]\( Q1 = 76.0 \)[/tex].
- Q3 (Third Quartile): This is the 75th percentile of the data set. In this data set, [tex]\( Q3 = 79.5 \)[/tex].
3. Calculate the Interquartile Range (IQR):
[tex]\[
\text{IQR} = Q3 - Q1 = 79.5 - 76.0 = 3.5
\][/tex]
4. Determine the Outlier Boundaries:
- Lower Bound: This is calculated as [tex]\( Q1 - 1.5 \times \text{IQR} \)[/tex].
[tex]\[
\text{Lower Bound} = 76.0 - 1.5 \times 3.5 = 70.75
\][/tex]
- Upper Bound: This is calculated as [tex]\( Q3 + 1.5 \times \text{IQR} \)[/tex].
[tex]\[
\text{Upper Bound} = 79.5 + 1.5 \times 3.5 = 84.75
\][/tex]
5. Identify the Outliers:
- Any data points below the lower bound or above the upper bound would be considered outliers.
- In this data set, all the values (71, 74, 78, 79, 79, 80, 82) are within the calculated boundary range of 70.75 to 84.75.
Therefore, there are no outliers in this data set. The correct answer is "no outliers."
