If `{:(x:,3,4,8,6,2,1),(y:,5,3,9,6,9,2):}`
The coefficient of correrlation will be approximately

Options

A. `0.49`
B. `-0.49`
C. `0.40`
D. `-0.40`

Answer

To find the coefficient of correlation (r) for the given data points \(x = (3, 4, 8, 6, 2, 1)\) and \(y = (5, 3, 9, 6, 9, 2)\), we will use the formula for the Pearson correlation coefficient:

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

### Step 1: Calculate the necessary sums
We need to calculate the following values:
- \(n\): the number of data points
- \(\sum{x}\): the sum of all x values
- \(\sum{y}\): the sum of all y values
- \(\sum{xy}\): the sum of the product of corresponding x and y values
- \(\sum{x^2}\): the sum of the squares of x values
- \(\sum{y^2}\): the sum of the squares of y values

### Step 2: Create a table
Let's create a table to organize our calculations:

| \(x\) | \(y\) | \(x^2\) | \(y^2\) | \(xy\) |
|-------|-------|---------|---------|--------|
| 3 | 5 | 9 | 25 | 15 |
| 4 | 3 | 16 | 9 | 12 |
| 8 | 9 | 64 | 81 | 72 |
| 6 | 6 | 36 | 36 | 36 |
| 2 | 9 | 4 | 81 | 18 |
| 1 | 2 | 1 | 4 | 2 |
| **Total** | | **130** | **236** | **155** |

### Step 3: Calculate the sums
From the table, we can calculate:
- \(n = 6\) (number of pairs)
- \(\sum{x} = 3 + 4 + 8 + 6 + 2 + 1 = 24\)
- \(\sum{y} = 5 + 3 + 9 + 6 + 9 + 2 = 34\)
- \(\sum{xy} = 15 + 12 + 72 + 36 + 18 + 2 = 155\)
- \(\sum{x^2} = 9 + 16 + 64 + 36 + 4 + 1 = 130\)
- \(\sum{y^2} = 25 + 9 + 81 + 36 + 81 + 4 = 236\)

### Step 4: Substitute into the formula
Now we can substitute these values into the correlation formula:

\[ r = \frac{6 \cdot 155 - 24 \cdot 34}{\sqrt{6 \cdot 130 - 24^2} \cdot \sqrt{6 \cdot 236 - 34^2}} \]

### Step 5: Calculate the numerator
Calculating the numerator:

\[ 6 \cdot 155 = 930 \]
\[ 24 \cdot 34 = 816 \]
So, the numerator is:

\[ 930 - 816 = 114 \]

### Step 6: Calculate the denominator
Calculating the first part of the denominator:

\[ 6 \cdot 130 = 780 \]
\[ 24^2 = 576 \]
So, the first part is:

\[ 780 - 576 = 204 \]

Calculating the second part of the denominator:

\[ 6 \cdot 236 = 1416 \]
\[ 34^2 = 1156 \]
So, the second part is:

\[ 1416 - 1156 = 260 \]

Now, we can find the square roots:

\[ \sqrt{204} \approx 14.28 \]
\[ \sqrt{260} \approx 16.12 \]

### Step 7: Final calculation
Now we can calculate the denominator:

\[ \text{Denominator} = 14.28 \cdot 16.12 \approx 230.5 \]

Finally, we can calculate \(r\):

\[ r \approx \frac{114}{230.5} \approx 0.494 \]

### Conclusion
The coefficient of correlation \(r\) is approximately **0.494**.

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