Mean and variance of 7 observation 2,4,10,12,14,x,y are 8 and 16 respectively. find xy is

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given that the mean and variance of the observations \( 2, 4, 10, 12, 14, x, y \) are \( 8 \) and \( 16 \) respectively.

### Step 1: Calculate the Mean
The mean \( \bar{x} \) is given by the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \( n \) is the number of observations. Here, \( n = 7 \) and \( \bar{x} = 8 \).

Thus, we have:
\[ 8 = \frac{2 + 4 + 10 + 12 + 14 + x + y}{7} \]

Multiplying both sides by \( 7 \):
\[ 56 = 2 + 4 + 10 + 12 + 14 + x + y \]

Calculating the sum of the known values:
\[ 2 + 4 + 10 + 12 + 14 = 42 \]

So, we can rewrite the equation as:
\[ 56 = 42 + x + y \]

This simplifies to:
\[ x + y = 56 - 42 = 14 \quad \text{(Equation 1)} \]

### Step 2: Calculate the Variance
The variance \( \sigma^2 \) is given by the formula:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]
Given \( \sigma^2 = 16 \) and \( \bar{x} = 8 \), we can substitute these values:
\[ 16 = \frac{\sum x_i^2}{7} - 8^2 \]

Calculating \( 8^2 \):
\[ 8^2 = 64 \]

Thus, we have:
\[ 16 = \frac{\sum x_i^2}{7} - 64 \]

Rearranging gives:
\[ \frac{\sum x_i^2}{7} = 16 + 64 = 80 \]

Multiplying both sides by \( 7 \):
\[ \sum x_i^2 = 80 \times 7 = 560 \]

Now, we calculate \( \sum x_i^2 \):
\[ \sum x_i^2 = 2^2 + 4^2 + 10^2 + 12^2 + 14^2 + x^2 + y^2 \]

Calculating the squares of the known values:
\[ 2^2 = 4, \quad 4^2 = 16, \quad 10^2 = 100, \quad 12^2 = 144, \quad 14^2 = 196 \]
Adding these up:
\[ 4 + 16 + 100 + 144 + 196 = 460 \]

So we can write:
\[ 560 = 460 + x^2 + y^2 \]

This simplifies to:
\[ x^2 + y^2 = 560 - 460 = 100 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now we have two equations:
1. \( x + y = 14 \)
2. \( x^2 + y^2 = 100 \)

Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \):
\[ 14^2 = 100 + 2xy \]

Calculating \( 14^2 \):
\[ 196 = 100 + 2xy \]

Rearranging gives:
\[ 2xy = 196 - 100 = 96 \]

Thus,
\[ xy = \frac{96}{2} = 48 \]

### Final Answer
The value of \( xy \) is \( 48 \).

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