To find the IQR (interquartile range) for X, we first need to find the first and third quartiles.
The first quartile (Q1) is the median of the lower half of the data, which represents the area of farms that are smaller than the median farm size. To find Q1, we can use the formula:
Q1 = median of (n/2) smallest values
where n is the sample size. In this case, n = 38, so we need to find the median of the 19 smallest values. We can do this by arranging the farm sizes in ascending order and finding the middle value:
... (174-55) (174) (174+55) ...
The median of the lower half is 174-55 = 119, so Q1 = 119 acres.
The third quartile (Q3) is the median of the upper half of the data, which represents the area of farms that are larger than the median farm size. To find Q3, we can use the formula:
Q3 = median of (n/2) largest values
We need to find the median of the 19 largest values, which can be arranged as:
... (174+55) (174) (174-55) ...
The median of the upper half is 174+55 = 229, so Q3 = 229 acres.
Now we can find the IQR:
IQR = Q3 - Q1
= 229 - 119
= 110 acres
Therefore, the IQR for X is from 119 acres to 229 acres.