If `sumx_(i) = 50, sumy_(i)= -30, sumx_(i)y_(j)= -115` and n= 10, then covariance between x and y is

Options

A. `3.5`
B. `-3.5`
C. `2.5`
D. `4.5`

Answer

To find the covariance between x and y using the given data, we can follow these steps:

### Step 1: Write down the formula for covariance
The formula for covariance between two variables x and y is given by:

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

### Step 2: Substitute the given values into the formula
From the problem, we have:
- \(\sum x_i = 50\)
- \(\sum y_i = -30\)
- \(\sum (x_i y_i) = -115\)
- \(n = 10\)

Substituting these values into the covariance formula:

\[ \text{Cov}(x, y) = \frac{-115}{10} - \left(\frac{50}{10}\right) \left(\frac{-30}{10}\right) \]

### Step 3: Calculate each term
First, calculate \(\frac{-115}{10}\):

\[ \frac{-115}{10} = -11.5 \]

Next, calculate \(\frac{50}{10}\) and \(\frac{-30}{10}\):

\[ \frac{50}{10} = 5 \quad \text{and} \quad \frac{-30}{10} = -3 \]

Now, multiply these two results:

\[ 5 \times -3 = -15 \]

### Step 4: Combine the results
Now substitute these results back into the covariance formula:

\[ \text{Cov}(x, y) = -11.5 - (-15) \]

This simplifies to:

\[ \text{Cov}(x, y) = -11.5 + 15 = 3.5 \]

### Final Answer
Thus, the covariance between x and y is:

\[ \text{Cov}(x, y) = 3.5 \]

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