If `Sigmax=55,Sigmay=74,Sigmaxy=411,n=10`, then covariance between x and y is

Options

A. `0.5`
B. `-0.5`
C. `-0.4`
D. `0.4`

Answer

To find the covariance between two variables \( x \) and \( y \), we can use the formula for covariance:

\[ \text{Cov}(x, y) = \frac{1}{n} \sum xy - \left( \frac{\sum x}{n} \right) \left( \frac{\sum y}{n} \right) \]

Given the values:
- \( \sum x = 55 \)
- \( \sum y = 74 \)
- \( \sum xy = 411 \)
- \( n = 10 \)

Let's calculate the covariance step by step.

### Step 1: Calculate \( \frac{1}{n} \sum xy \)

\[ \frac{1}{n} \sum xy = \frac{1}{10} \times 411 = 41.1 \]

### Step 2: Calculate \( \frac{\sum x}{n} \)

\[ \frac{\sum x}{n} = \frac{55}{10} = 5.5 \]

### Step 3: Calculate \( \frac{\sum y}{n} \)

\[ \frac{\sum y}{n} = \frac{74}{10} = 7.4 \]

### Step 4: Calculate the product \( \left( \frac{\sum x}{n} \right) \left( \frac{\sum y}{n} \right) \)

\[ \left( \frac{\sum x}{n} \right) \left( \frac{\sum y}{n} \right) = 5.5 \times 7.4 = 40.7 \]

### Step 5: Calculate the covariance

Now we can substitute these values into the covariance formula:

\[ \text{Cov}(x, y) = 41.1 - 40.7 = 0.4 \]

Thus, the covariance between \( x \) and \( y \) is \( 0.4 \).

### Final Answer:
\[ \text{Cov}(x, y) = 0.4 \]

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