If Cov(x,y) = -2, `Sigma y = 30 , Sigma x = 25 and Sigma xy = 140 ` , find the number of observations .

Answer

To find the number of observations (n) given the covariance of x and y, we can use the formula for covariance:

\[ Cov(x, y) = \frac{1}{n} \Sigma xy - \frac{1}{n^2} \Sigma x \Sigma y \]

Given:
- \( Cov(x, y) = -2 \)
- \( \Sigma y = 30 \)
- \( \Sigma x = 25 \)
- \( \Sigma xy = 140 \)

We can substitute the known values into the covariance formula:

\[ -2 = \frac{1}{n} (140) - \frac{1}{n^2} (25)(30) \]

Now, simplifying this equation:

1. **Multiply through by \( n^2 \)** to eliminate the denominators:

\[ -2n^2 = 140n - 750 \]

2. **Rearranging the equation** gives us:

\[ 2n^2 + 140n - 750 = 0 \]

3. **Dividing the entire equation by 2** to simplify:

\[ n^2 + 70n - 375 = 0 \]

4. **Now, we will factor the quadratic equation**. We need two numbers that multiply to -375 and add to 70. The numbers are 75 and -5.

5. **Factoring the quadratic**:

\[ (n + 75)(n - 5) = 0 \]

6. **Setting each factor to zero gives us**:

\[ n + 75 = 0 \quad \text{or} \quad n - 5 = 0 \]

This results in:

\[ n = -75 \quad \text{or} \quad n = 5 \]

Since the number of observations cannot be negative, we discard \( n = -75 \).

Thus, the number of observations is:

\[ \boxed{5} \]