Calculate Decile 8 of the following sample: 10 11 12 13 14 15 16 18 18 19 20 20 22 23 25 26 27 28 29 30

Solution:
1. Given sample data:
* 10, 11, 12, 13, 14, 15, 16, 18, 18, 19, 20, 20, 22, 23, 25, 26, 27, 28, 29, 30

2. Order the data set (already ordered):
* 10, 11, 12, 13, 14, 15, 16, 18, 18, 19, 20, 20, 22, 23, 25, 26, 27, 28, 29, 30

3. To find the position of the 8th decile (D8), use the formula:
D_k = \left( \frac{k(n + 1)}{10} \right)^{th} \text{ position}
where k = 8 and n = 20 (number of data points).

4. Calculate the position:
D_8 = \left( \frac{8(20 + 1)}{10} \right) = \left( \frac{168}{10} \right) = 16.8^{th} \text{ position}

5. Since 16.8 isn't a whole number, it means D8 is between the 16th and 17th data points. Using interpolation:
- 16th value is 26 and 17th value is 27.
- Interpolating:
D_8 = 26 + 0.8(27 - 26)
D_8 = 26 + 0.8(1)
D_8 = 26.8

6. Therefore, the 8th decile (D8) is:
26.8