To find the interquartile range (IQR), we use the formula:
\[ \text{IQR} = Q_3 - Q_1 \]
where \(Q_1\) is the first quartile and \(Q_3\) is the third quartile.
From the provided values:
- \(Q_1 = 54\)
- \(Q_3 = 110\)
So, we can calculate the IQR as follows:
\[ \text{IQR} = 110 - 54 = 56 \]
Thus, the correct IQR is indeed 56, as you noted.
Now, let's examine the mistake the student might have made:
- He incorrectly calculated the IQR to be 142.
- Since the IQR is found by subtracting \(Q_1\) from \(Q_3\), if the student calculated a much larger number like 142 instead of 56, he likely did not perform the subtraction correctly.
Let's analyze the options:
A. He found the sum of the first and third quartiles of the data set instead of the difference.
- This would yield \(54 + 110 = 164\), which is still not 142.
B. He found the difference between the maximum and minimum values instead of the first and third quartiles of the data set.
- This would yield \(153 - 11 = 142\), which matches the student's incorrect answer.
C. He found the sum of the maximum and minimum values instead of the first and third quartiles of the data set.
- This would yield \(153 + 11 = 164\), which is incorrect.
D. He found the sum of the maximum and minimum values instead of the difference.
- This would not apply, as the process of summing them does not relate to IQR calculation.
The correct choice is B. He found the difference between the maximum and minimum values instead of the first and third quartiles of the data set.