For the data `14, 7, 13, 12, 13, 17, 8, 10, 6, 15, 18, 21, 20, ` compute `P_(80)`

Answer

To compute the 80th percentile \( P_{80} \) for the given data set \( 14, 7, 13, 12, 13, 17, 8, 10, 6, 15, 18, 21, 20 \), we will follow these steps:

### Step 1: Arrange the data in ascending order
First, we need to sort the data in ascending order.

**Data:**
\[ 6, 7, 8, 10, 12, 13, 13, 14, 15, 17, 18, 20, 21 \]

### Step 2: Determine the number of observations (n)
Count the number of data points in the sorted list.

**Number of observations (n):**
\[ n = 13 \]

### Step 3: Use the formula for the k-th percentile
The formula for finding the k-th percentile is given by:
\[ P_k = \frac{k(n + 1)}{100} \]
For \( P_{80} \):
\[ P_{80} = \frac{80(13 + 1)}{100} = \frac{80 \times 14}{100} = \frac{1120}{100} = 11.2 \]

### Step 4: Identify the position of \( P_{80} \)
Since \( P_{80} = 11.2 \), we need to find the 11th and 12th observations to interpolate between them.

### Step 5: Locate the 11th and 12th observations
From our sorted data:
- 11th observation = 18
- 12th observation = 20

### Step 6: Interpolate to find \( P_{80} \)
To find \( P_{80} \), we will interpolate between the 11th and 12th observations:

Using the formula:
\[ P_{80} = \text{11th observation} + 0.2 \times (\text{12th observation} - \text{11th observation}) \]
Substituting the values:
\[ P_{80} = 18 + 0.2 \times (20 - 18) = 18 + 0.2 \times 2 = 18 + 0.4 = 18.4 \]

### Final Answer
Thus, the 80th percentile \( P_{80} \) is:
\[ \boxed{18.4} \]