Question

A random sample of bonuses (in millions of dollars) paid by large companies to their executives is shown below.
Class boundary
Frequency

0.5----3.5
11

3.5----6.5
12

6.5----9.5
4

9.5----12.5
2

12.5---15.5
1

Find
P25

Answer 1

To find P25 (the 25th percentile), we need to find the value that separates the lowest 25% of the data from the rest of the data.

First, we calculate the total number of observations:

11 + 12 + 4 + 2 + 1 = 30

Next, we calculate the position of the 25th percentile in the data set:

25/100 * 30 = 7.5

Since the position is not an integer, we need to take the average of the 7th and 8th values in the data set.

Since 7 is in the 0.5-3.5 class, and the boundaries for this class are 0.5 and 3.5, we can calculate the width of this class:

3.5 - 0.5 = 3

To find the value at the 25th percentile, we can use the formula:

P25 = lower boundary of the class + [(position - cumulative frequency of the class before) / frequency of the class] * width of the class

P25 = 0.5 + [(7.5 - 0) / 11] * 3
P25 = 0.5 + (7.5/11) * 3
P25 = 0.5 + 0.6818181818181818 * 3
P25 = 0.5 + 2.0454545454545454
P25 ≈ 2.545

Therefore, the 25th percentile of the data set is approximately 2.545 million dollars.