For the given data, the calculation corresponding to all values of varsities (x, y) is following:
`Sigma (x - barx)^(2) = 36, Sigma(y-bary)^(2)=25, Sigma(x-barx), (y-bary)=20`
Karl Pearson's correlation coefficient is

Options

A. 0.66
B. `-0.66`
C. `-0.5`
D. `1.66`

Answer

To calculate Karl Pearson's correlation coefficient (R) using the provided data, we will follow these steps:

### Step 1: Understand the given data
We have the following values:
- \( \Sigma (x - \bar{x})^2 = 36 \)
- \( \Sigma (y - \bar{y})^2 = 25 \)
- \( \Sigma (x - \bar{x})(y - \bar{y}) = 20 \)

### Step 2: Identify the formula for Karl Pearson's correlation coefficient
The formula for Karl Pearson's correlation coefficient (R) is given by:

\[ R = \frac{\Sigma (x - \bar{x})(y - \bar{y})}{\sqrt{\Sigma (x - \bar{x})^2 \cdot \Sigma (y - \bar{y})^2}} \]

### Step 3: Substitute the known values into the formula
From the given data:
- \( \Sigma (x - \bar{x})(y - \bar{y}) = 20 \)
- \( \Sigma (x - \bar{x})^2 = 36 \)
- \( \Sigma (y - \bar{y})^2 = 25 \)

Now substituting these values into the formula:

\[ R = \frac{20}{\sqrt{36 \cdot 25}} \]

### Step 4: Calculate the denominator
First, calculate \( 36 \cdot 25 \):

\[ 36 \cdot 25 = 900 \]

Now, take the square root:

\[ \sqrt{900} = 30 \]

### Step 5: Complete the calculation for R
Now substitute back into the equation for R:

\[ R = \frac{20}{30} = \frac{2}{3} \]

### Step 6: Convert to decimal form
To express \( \frac{2}{3} \) in decimal form:

\[ R \approx 0.66 \]

### Final Answer
Thus, Karl Pearson's correlation coefficient is:

\[ R \approx 0.66 \]

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