For the data
`{:(x:,4,7,8,3,4),(y:,5,8,6,3,5):}`
The Karl Pearson coefficient is

Options

A. `(63)/(sqrt(94xx66))`
B. 63
C. `(63)/(sqrt(94))`
D. none of these

Answer

To find the Karl Pearson coefficient of correlation (denoted as \( r \)) for the given data, we will follow these steps:

### Step 1: Organize the Data
We have the following data points:
- \( x: 4, 7, 8, 3, 4 \)
- \( y: 5, 8, 6, 3, 5 \)

Let's create a table to organize this data along with the calculations we need for \( x^2 \), \( y^2 \), and \( xy \).

| \( x \) | \( y \) | \( x^2 \) | \( y^2 \) | \( xy \) |
|---------|---------|-----------|-----------|----------|
| 4 | 5 | 16 | 25 | 20 |
| 7 | 8 | 49 | 64 | 56 |
| 8 | 6 | 64 | 36 | 48 |
| 3 | 3 | 9 | 9 | 9 |
| 4 | 5 | 16 | 25 | 20 |
| **Total** | | **154** | **159** | **153** |

### Step 2: Calculate Required Sums
- \( n = 5 \) (the number of data points)
- \( \Sigma x = 4 + 7 + 8 + 3 + 4 = 26 \)
- \( \Sigma y = 5 + 8 + 6 + 3 + 5 = 27 \)
- \( \Sigma x^2 = 16 + 49 + 64 + 9 + 16 = 154 \)
- \( \Sigma y^2 = 25 + 64 + 36 + 9 + 25 = 159 \)
- \( \Sigma xy = 20 + 56 + 48 + 9 + 20 = 153 \)

### Step 3: Use the Formula for Karl Pearson Coefficient
The formula for the Karl Pearson coefficient \( r \) is given by:

\[ r = \frac{n \Sigma xy - \Sigma x \Sigma y}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}} \]

### Step 4: Substitute the Values
Now we will substitute the calculated values into the formula:

\[ r = \frac{5 \cdot 153 - 26 \cdot 27}{\sqrt{[5 \cdot 154 - 26^2][5 \cdot 159 - 27^2]}} \]

Calculating each part:

1. \( n \Sigma xy = 5 \cdot 153 = 765 \)
2. \( \Sigma x \Sigma y = 26 \cdot 27 = 702 \)
3. \( n \Sigma x^2 = 5 \cdot 154 = 770 \)
4. \( (\Sigma x)^2 = 26^2 = 676 \)
5. \( n \Sigma y^2 = 5 \cdot 159 = 795 \)
6. \( (\Sigma y)^2 = 27^2 = 729 \)

Now substituting these values:

\[ r = \frac{765 - 702}{\sqrt{(770 - 676)(795 - 729)}} \]

Calculating the differences:

1. \( 765 - 702 = 63 \)
2. \( 770 - 676 = 94 \)
3. \( 795 - 729 = 66 \)

Now substituting back:

\[ r = \frac{63}{\sqrt{94 \cdot 66}} \]

Calculating \( 94 \cdot 66 = 6204 \):

\[ r = \frac{63}{\sqrt{6204}} \approx \frac{63}{78.8} \approx 0.799 \]

### Final Result
Thus, the Karl Pearson coefficient \( r \) is approximately \( 0.799 \).