Following are the scores of 12 students in a class test of of 30 marks : 18,20,9,15,21,26,14,13,27,22,16,28 Find `D_(7)andP_(33)`.

Answer

To solve the problem of finding the 7th decile (D7) and the 33rd percentile (P33) from the given scores of 12 students, we will follow these steps:

### Step 1: Arrange the Data in Increasing Order
First, we need to arrange the scores in increasing order.

Given scores: 18, 20, 9, 15, 21, 26, 14, 13, 27, 22, 16, 28

Arranging these scores in increasing order gives us:
9, 13, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28

### Step 2: Calculate D7 (7th Decile)
The formula for calculating the \(i\)th decile \(D_i\) when the data is discrete is given by:
\[ D_i = \frac{i}{10} \times (n + 1) \text{th term} \]
where \(i\) is the decile we want to find and \(n\) is the total number of data points.

Here, \(i = 7\) and \(n = 12\).

Calculating \(D_7\):
\[ D_7 = \frac{7}{10} \times (12 + 1) = \frac{7}{10} \times 13 = 9.1 \]

Since 9.1 is not a whole number, we will take the 9th and 10th terms from our ordered list.

### Step 3: Identify the 9th and 10th Terms
From the ordered list:
- 9th term = 22
- 10th term = 26

### Step 4: Interpolate to Find D7
To find \(D_7\), we can interpolate between the 9th and 10th terms:
\[ D_7 = 9th \text{ term} + (0.1 \times (10th \text{ term} - 9th \text{ term})) \]
\[ D_7 = 22 + (0.1 \times (26 - 22)) = 22 + (0.1 \times 4) = 22 + 0.4 = 22.4 \]

### Step 5: Calculate P33 (33rd Percentile)
The formula for calculating the \(i\)th percentile \(P_i\) is given by:
\[ P_i = \frac{i}{100} \times (n + 1) \text{th term} \]
where \(i\) is the percentile we want to find.

Here, \(i = 33\) and \(n = 12\).

Calculating \(P_{33}\):
\[ P_{33} = \frac{33}{100} \times (12 + 1) = \frac{33}{100} \times 13 = 4.29 \]

### Step 6: Identify the 4th and 5th Terms
From the ordered list:
- 4th term = 15
- 5th term = 16

### Step 7: Interpolate to Find P33
To find \(P_{33}\), we can average the 4th and 5th terms:
\[ P_{33} = \frac{4th \text{ term} + 5th \text{ term}}{2} = \frac{15 + 16}{2} = \frac{31}{2} = 15.5 \]

### Final Answers
- \(D_7 = 22.4\)
- \(P_{33} = 15.5\)

---