Question

Consider a sample with data values of 27, 24, 21, 16, 30, 33, 28, and 24. Compute the 20th, 25th, 65th, and 75th percentiles. 20th percentile 20 Correct: Your answer is correct. 25th percentile 21 Incorrect: Your answer is incorrect. 65th percentile 27.5 Incorrect: Your answer is incorrect. 75th percentile 28 Incorrect: Your answer is incorrect.

Answer 1

`P_(20) = 20` --- 20th percentile

`P_(25) = 21.75` --- 25th percentile

`P_(65) = 27.85` --- 65th percentile

`P_(75) = 29.5` --- 75th percentile

Explanation:

Given

27, 24, 21, 16, 30, 33, 28, and 24.

N = 8

First, arrange the data in ascending order:

Arranged data: 16, 21, 24, 24, 27, 28, 30, 33

Solving (a): The 20th percentile

This is calculated as:

`P_(20) = 20 * (N +1)/(100)`

`P_(20) = 20 * (8 +1)/(100)`

`P_(20) = 20 * (9)/(100)`

`P_(20) = (20 * 9)/(100)`

`P_(20) = (180)/(100)`

`P_(20) = 1.8th\ item`

This is then calculated as:

`P_(20) = 1st\ Item +0.8(2nd\ Item - 1st\ Item)`

`P_(20) = 16 + 0.8*(21 - 16)`

`P_(20) = 16 + 0.8*5`

`P_(20) = 16 + 4`

`P_(20) = 20`

Solving (b): The 25th percentile

This is calculated as:

`P_(25) = 25 * (N +1)/(100)`

`P_(25) = 25 * (8 +1)/(100)`

`P_(25) = 25 * (9)/(100)`

`P_(25) = (25 * 9)/(100)`

`P_(25) = (225)/(100)`

`P_(25) = 2.25\ th`

This is then calculated as:

`P_(25) = 2nd\ item + 0.25(3rd\ item-2nd\ item)`

`P_(25) = 21 + 0.25(24-21)`

`P_(25) = 21 + 0.25(3)`

`P_(25) = 21 + 0.75`

`P_(25) = 21.75`

Solving (c): The 65th percentile

This is calculated as:

`P_(65) = 65 * (N +1)/(100)`

`P_(65) = 65 * (8 +1)/(100)`

`P_(65) = 65 * (9)/(100)`

`P_(65) = (65 * 9)/(100)`

`P_(65) = (585)/(100)`

`P_(65) = 5.85\th`

This is then calculated as:

`P_(65) = 5th + 0.85(6th - 5th)`

`P_(65) = 27 + 0.85(28 - 27)`

`P_(65) = 27 + 0.85(1)`

`P_(65) = 27 + 0.85`

`P_(65) = 27.85`

Solving (d): The 75th percentile

This is calculated as:

`P_(75) = 75 * (N +1)/(100)`

`P_(75) = 75 * (8 +1)/(100)`

`P_(75) = 75 * (9)/(100)`

`P_(75) = (75 * 9)/(100)`

`P_(75) = (675)/(100)`

`P_(75) = 6.75th`

This is then calculated as:

`P_(75) = 6th + 0.75(7th - 6th)`

`P_(75) = 28 + 0.75(30- 28)`

`P_(75) = 28 + 0.75(2)`

`P_(75) = 28 + 1.5`

`P_(75) = 29.5`