The mean and variance of 8 observations are respectively 9 and 9.25. If six observations are 4,6,7,8,12 and 13 then find the remaining two observations.

Explanation

To solve the problem, we need to find the remaining two observations given the mean and variance of a set of eight observations. Let's break down the solution step by step.

### Step 1: Understand the Given Information
We are given:
- Mean of 8 observations = 9
- Variance of 8 observations = 9.25
- Six observations = 4, 6, 7, 8, 12, 13

Let the remaining two observations be \( x \) and \( y \).

### Step 2: Use the Mean to Find the Sum of Observations
The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Given the mean is 9 and there are 8 observations, we can write:
\[ 9 = \frac{4 + 6 + 7 + 8 + 12 + 13 + x + y}{8} \]
Calculating the sum of the known observations:
\[ 4 + 6 + 7 + 8 + 12 + 13 = 50 \]
Now substituting this into the mean equation:
\[ 9 = \frac{50 + x + y}{8} \]
Multiplying both sides by 8:
\[ 72 = 50 + x + y \]
Thus, we find:
\[ x + y = 72 - 50 = 22 \quad \text{(Equation 1)} \]

### Step 3: Use the Variance to Find Another Equation
The formula for variance is:
\[ \text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n} \]
Given the variance is 9.25 for 8 observations:
\[ 9.25 = \frac{\sum (x_i - 9)^2}{8} \]
Multiplying both sides by 8:
\[ 74 = \sum (x_i - 9)^2 \]
Now we will calculate \( \sum (x_i - 9)^2 \) for the known observations:
\[ (4 - 9)^2 + (6 - 9)^2 + (7 - 9)^2 + (8 - 9)^2 + (12 - 9)^2 + (13 - 9)^2 \]
Calculating each term:
\[ (4 - 9)^2 = 25, \quad (6 - 9)^2 = 9, \quad (7 - 9)^2 = 4, \quad (8 - 9)^2 = 1, \quad (12 - 9)^2 = 9, \quad (13 - 9)^2 = 16 \]
Summing these:
\[ 25 + 9 + 4 + 1 + 9 + 16 = 64 \]
Now we can write:
\[ 74 = 64 + (x - 9)^2 + (y - 9)^2 \]
Thus:
\[ (x - 9)^2 + (y - 9)^2 = 74 - 64 = 10 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
We have two equations:
1. \( x + y = 22 \)
2. \( (x - 9)^2 + (y - 9)^2 = 10 \)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 22 - x \]
Substituting this into Equation 2:
\[ (x - 9)^2 + ((22 - x) - 9)^2 = 10 \]
This simplifies to:
\[ (x - 9)^2 + (13 - x)^2 = 10 \]
Expanding both squares:
\[ (x^2 - 18x + 81) + (169 - 26x + x^2) = 10 \]
Combining like terms:
\[ 2x^2 - 44x + 250 = 10 \]
Rearranging gives:
\[ 2x^2 - 44x + 240 = 0 \]
Dividing through by 2:
\[ x^2 - 22x + 120 = 0 \]

### Step 5: Factor the Quadratic Equation
Factoring the quadratic:
\[ (x - 12)(x - 10) = 0 \]
Thus, \( x = 12 \) or \( x = 10 \).

### Step 6: Find Corresponding Values of \( y \)
Using \( x + y = 22 \):
- If \( x = 12 \), then \( y = 22 - 12 = 10 \).
- If \( x = 10 \), then \( y = 22 - 10 = 12 \).

### Conclusion
The remaining two observations are \( 10 \) and \( 12 \).

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