The coefficient of correlation for the following data will be approximately
`{:(x:,2,4,5,6,3,6,8,10),(y:,5,6,6,8,4,8,12,15):}`

Options

A. 90
B. 96
C. `0.90`
D. none of these

Answer

To find the coefficient of correlation for the given data, we will follow these steps:

### Step 1: Organize the Data
We have two sets of data:
- \( x: 2, 4, 5, 6, 3, 6, 8, 10 \)
- \( y: 5, 6, 6, 8, 4, 8, 12, 15 \)

### Step 2: Calculate Necessary Sums
We need to calculate the following:
- \( \sum x \)
- \( \sum y \)
- \( \sum xy \)
- \( \sum x^2 \)
- \( \sum y^2 \)

Let's calculate these values step by step.

#### Calculation of \( \sum x \):
\[ \sum x = 2 + 4 + 5 + 6 + 3 + 6 + 8 + 10 = 44 \]

#### Calculation of \( \sum y \):
\[ \sum y = 5 + 6 + 6 + 8 + 4 + 8 + 12 + 15 = 64 \]

#### Calculation of \( \sum xy \):
\[ \sum xy = (2 \cdot 5) + (4 \cdot 6) + (5 \cdot 6) + (6 \cdot 8) + (3 \cdot 4) + (6 \cdot 8) + (8 \cdot 12) + (10 \cdot 15) \]
\[ = 10 + 24 + 30 + 48 + 12 + 48 + 96 + 150 = 418 \]

#### Calculation of \( \sum x^2 \):
\[ \sum x^2 = 2^2 + 4^2 + 5^2 + 6^2 + 3^2 + 6^2 + 8^2 + 10^2 \]
\[ = 4 + 16 + 25 + 36 + 9 + 36 + 64 + 100 = 290 \]

#### Calculation of \( \sum y^2 \):
\[ \sum y^2 = 5^2 + 6^2 + 6^2 + 8^2 + 4^2 + 8^2 + 12^2 + 15^2 \]
\[ = 25 + 36 + 36 + 64 + 16 + 64 + 144 + 225 = 640 \]

### Step 3: Apply the Correlation Coefficient Formula
The formula for the coefficient of correlation \( r \) is given by:
\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{n \sum x^2 - (\sum x)^2} \sqrt{n \sum y^2 - (\sum y)^2}} \]
Where \( n \) is the number of pairs of data points. Here, \( n = 8 \).

Substituting the values we calculated:
\[ r = \frac{8 \cdot 418 - 44 \cdot 64}{\sqrt{8 \cdot 290 - 44^2} \sqrt{8 \cdot 640 - 64^2}} \]

### Step 4: Calculate Each Component
1. Calculate \( 8 \cdot 418 \):
\[ 8 \cdot 418 = 3344 \]

2. Calculate \( 44 \cdot 64 \):
\[ 44 \cdot 64 = 2816 \]

3. Calculate \( n \sum x^2 - (\sum x)^2 \):
\[ 8 \cdot 290 - 44^2 = 2320 - 1936 = 384 \]

4. Calculate \( n \sum y^2 - (\sum y)^2 \):
\[ 8 \cdot 640 - 64^2 = 5120 - 4096 = 1024 \]

### Step 5: Substitute and Simplify
Now substituting back into the formula:
\[ r = \frac{3344 - 2816}{\sqrt{384} \cdot \sqrt{1024}} \]
\[ = \frac{528}{\sqrt{384} \cdot 32} \]

Calculating \( \sqrt{384} \):
\[ \sqrt{384} \approx 19.6 \]

Thus:
\[ r \approx \frac{528}{19.6 \cdot 32} \approx \frac{528}{627.2} \approx 0.842 \]

### Final Result
The coefficient of correlation \( r \) is approximately \( 0.842 \).