Find Cov (x,y) for the following pair of observations, given `bar(x) = 30, bar(y) = 40`, (15,44), (20,43), (25, 45), (30, 37), (40,34), (50,37)

Answer

To find the covariance \( \text{Cov}(x, y) \) for the given observations, we will follow these steps:

### Step 1: Write down the given data
We have the following pairs of observations:
- \( (15, 44) \)
- \( (20, 43) \)
- \( (25, 45) \)
- \( (30, 37) \)
- \( (40, 34) \)
- \( (50, 37) \)

We are also given:
- \( \bar{x} = 30 \)
- \( \bar{y} = 40 \)

### Step 2: Calculate \( x - \bar{x} \) and \( y - \bar{y} \)
For each observation, we will calculate \( x - \bar{x} \) and \( y - \bar{y} \):

- For \( (15, 44) \):
- \( x - \bar{x} = 15 - 30 = -15 \)
- \( y - \bar{y} = 44 - 40 = 4 \)

- For \( (20, 43) \):
- \( x - \bar{x} = 20 - 30 = -10 \)
- \( y - \bar{y} = 43 - 40 = 3 \)

- For \( (25, 45) \):
- \( x - \bar{x} = 25 - 30 = -5 \)
- \( y - \bar{y} = 45 - 40 = 5 \)

- For \( (30, 37) \):
- \( x - \bar{x} = 30 - 30 = 0 \)
- \( y - \bar{y} = 37 - 40 = -3 \)

- For \( (40, 34) \):
- \( x - \bar{x} = 40 - 30 = 10 \)
- \( y - \bar{y} = 34 - 40 = -6 \)

- For \( (50, 37) \):
- \( x - \bar{x} = 50 - 30 = 20 \)
- \( y - \bar{y} = 37 - 40 = -3 \)

### Step 3: Calculate \( (x - \bar{x})(y - \bar{y}) \)
Now, we will multiply \( (x - \bar{x}) \) and \( (y - \bar{y}) \) for each pair:

- For \( (15, 44) \):
- \( (-15)(4) = -60 \)

- For \( (20, 43) \):
- \( (-10)(3) = -30 \)

- For \( (25, 45) \):
- \( (-5)(5) = -25 \)

- For \( (30, 37) \):
- \( (0)(-3) = 0 \)

- For \( (40, 34) \):
- \( (10)(-6) = -60 \)

- For \( (50, 37) \):
- \( (20)(-3) = -60 \)

### Step 4: Sum the products
Now, we sum all the products calculated in the previous step:
\[ \text{Sum} = -60 - 30 - 25 + 0 - 60 - 60 = -235 \]

### Step 5: Calculate the covariance
The formula for covariance is:
\[ \text{Cov}(x, y) = \frac{\sum (x - \bar{x})(y - \bar{y})}{n} \]
Where \( n \) is the number of observations. Here, \( n = 6 \).

Substituting the values:
\[ \text{Cov}(x, y) = \frac{-235}{6} \approx -39.17 \]

### Final Answer
Thus, the covariance \( \text{Cov}(x, y) \) is approximately \( -39.17 \).

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