To determine if the data contains any outliers, we will follow these steps:
1. Sort the Data:
The given lengths of fish in inches are: 15, 18, 9, 22, 7, 15, 10, and 18. Firstly, we sort this data in ascending order:
[tex]\[
7, 9, 10, 15, 15, 18, 18, 22
\][/tex]
2. Find the First Quartile (Q1) and the Third Quartile (Q3):
The first quartile (Q1) is the 25th percentile, and the third quartile (Q3) is the 75th percentile of the data. From the sorted data:
[tex]\[
Q1 = 9.75 \quad \text{(25th percentile)}
\][/tex]
[tex]\[
Q3 = 18.0 \quad \text{(75th percentile)}
\][/tex]
3. Calculate the Interquartile Range (IQR):
The interquartile range (IQR) is calculated as the difference between Q3 and Q1.
[tex]\[
IQR = Q3 - Q1 = 18.0 - 9.75 = 8.25
\][/tex]
4. Determine the Lower and Upper Bounds for Outliers:
The lower bound is calculated as [tex]\( Q1 - 1.5 \times IQR \)[/tex] and the upper bound is calculated as [tex]\( Q3 + 1.5 \times IQR \)[/tex].
[tex]\[
\text{Lower bound} = Q1 - 1.5 \times IQR = 9.75 - 1.5 \times 8.25 = -2.625
\][/tex]
[tex]\[
\text{Upper bound} = Q3 + 1.5 \times IQR = 18.0 + 1.5 \times 8.25 = 30.375
\][/tex]
5. Identify Outliers:
Any values in the data set that are less than the lower bound or greater than the upper bound are considered outliers. In this case:
[tex]\[
\text{Lower bound} = -2.625, \quad \text{Upper bound} = 30.375
\][/tex]
As all the values (7, 9, 10, 15, 15, 18, 18, 22) lie within this range, there are no outliers.
Therefore, the data contains no outliers.
