Find the value of the 20th and 75th percentiles of the data. 24,56,11,89,37,65,42,3,78,91,15,28,50,6,73,20,44,5,67,82 (2 points) a) 20th percentile: b) 75th percentile:
a) To find the 20th percentile, first arrange the data in ascending order:
3, 5, 6, 11, 15, 20, 24, 28, 37, 42, 44, 50, 56, 65, 67, 73, 78, 82, 89, 91
Now, to calculate the 20th percentile, we use the formula:
P = (n/100) * 20
P = (20/100) * 20
P = 0.2 * 20
P = 4
The 20th percentile falls between the 4th and 5th values in the ordered dataset, which are 15 and 20. To interpolate, we use the formula:
20th percentile = L + (k / f) * w
where:
L = Lower boundary of the interval (15)
k = Position of the 20th percentile in the interval (4)
f = Number of values in the interval (2)
w = Width of the interval (20 - 15 = 5)
20th percentile = 15 + (4 / 2) * 5
20th percentile = 15 + 2 * 5
20th percentile = 15 + 10
20th percentile = 25
So, the 20th percentile of the data is 25.
b) To find the 75th percentile, we repeat the same process:
P = (n/100) * 75
P = (20/100) * 75
P = 0.2 * 75
P = 15
The 75th percentile falls between the 15th and 16th values in the ordered dataset, which are 73 and 78. To interpolate:
75th percentile = 73 + (15 / 2) * 5
75th percentile = 73 + 7.5 * 5
75th percentile = 73 + 37.5
75th percentile = 110.5
Therefore, the 75th percentile of the data is 110.5.