Sure, let's approach this problem step by step to find the IQR (Interquartile Range) and the lower limit for outliers given the set of tree heights:
Step 1: Arrange the data in ascending order:
The given heights are: 273, 366, 482, 368, 650, 216, and 225.
First, let's sort these values in ascending order:
[tex]\[ 216, 225, 273, 366, 368, 482, 650 \][/tex]
Step 2: Find the Median (Q2):
To find the median, we need to determine the middle value in this sorted list. Since there are 7 numbers (an odd number), the median is the 4th value:
[tex]\[ \text{Median (Q2)} = 366 \][/tex]
Step 3: Find the First Quartile (Q1) and Third Quartile (Q3):
The first quartile (Q1) is the median of the first half of the data, and the third quartile (Q3) is the median of the second half.
For Q1, consider the first half: [216, 225, 273]. Since there are an odd number of values, Q1 is the middle value:
[tex]\[ Q1 = 225.0 \][/tex]
For Q3, consider the second half: [368, 482, 650]. Since there are an odd number of values, Q3 is the middle value:
[tex]\[ Q3 = 482.0 \][/tex]
Step 4: Calculate the IQR:
The IQR is the difference between Q3 and Q1.
[tex]\[ IQR = Q3 - Q1 = 482.0 - 225.0 = 257.0 \][/tex]
Step 5: Calculate the Lower Limit for Outliers:
The formula for the lower limit for identifying outliers is:
[tex]\[ \text{Lower limit} = Q1 - 1.5 \times IQR \][/tex]
[tex]\[ \text{Lower limit} = 225.0 - 1.5 \times 257.0 = 225.0 - 385.5 = -160.5 \][/tex]
Result Summary:
– The IQR (Interquartile Range) is [tex]\( 257.0 \)[/tex].
– The lower limit for outliers is [tex]\( -160.5 \)[/tex].
Thus, the correct answers are:
[tex]\[ \text{IQR} = 257 \][/tex]
[tex]\[ \text{Lower limit of the outlier} = -160.5 \][/tex]
