For the distribution `x_(i)` given 18, 20, 9, 15, 21, 26, 14, 13, 27, 22, 16, 28 the 40th percentile is

Options

A. 16
B. 16.4
C. 15
D. 18

Answer

To find the 40th percentile for the given distribution \( x_i = \{18, 20, 9, 15, 21, 26, 14, 13, 27, 22, 16, 28\} \), we will follow these steps:

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

\[ 9, 13, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28 \]

### Step 2: Determine the number of observations
Count the total number of observations \( n \):

\[ n = 12 \]

### Step 3: Calculate the position of the 40th percentile
The formula for the \( p \)-th percentile is given by:

\[ P_k = \frac{k(n + 1)}{100} \]

For the 40th percentile (\( k = 40 \)):

\[ P_{40} = \frac{40(12 + 1)}{100} = \frac{40 \times 13}{100} = \frac{520}{100} = 5.2 \]

### Step 4: Identify the observations at the calculated position
Since \( P_{40} = 5.2 \), we need to find the 5th and 6th observations in the ordered data:

- The 5th observation is \( 16 \).
- The 6th observation is \( 18 \).

### Step 5: Interpolate to find the 40th percentile
Since \( 5.2 \) is not a whole number, we will use interpolation:

\[ P_{40} = \text{5th observation} + (0.2 \times (\text{6th observation} - \text{5th observation})) \]

Substituting the values:

\[ P_{40} = 16 + (0.2 \times (18 - 16)) = 16 + (0.2 \times 2) = 16 + 0.4 = 16.4 \]

### Final Answer
Thus, the 40th percentile is:

\[ \boxed{16.4} \]

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