The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.

Explanation

To solve the problem step by step, we will find the two unknown observations \(x_4\) and \(x_5\) given the mean and variance of five observations, along with three known values.

### Step 1: Calculate the total sum of the observations
Given that the mean of the five observations is 4.4, we can calculate the total sum of the observations using the formula for mean:

\[ \text{Mean} = \frac{\text{Sum of observations}}{n} \]

Where \(n\) is the number of observations. Therefore, we have:

\[ 4.4 = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} \]

Multiplying both sides by 5 gives:

\[ x_1 + x_2 + x_3 + x_4 + x_5 = 4.4 \times 5 = 22 \]

### Step 2: Substitute known values
We know three of the observations: \(x_1 = 1\), \(x_2 = 2\), and \(x_3 = 6\). Thus, we can substitute these values into the equation:

\[ 1 + 2 + 6 + x_4 + x_5 = 22 \]

Calculating the sum of the known observations:

\[ 9 + x_4 + x_5 = 22 \]

### Step 3: Solve for \(x_4 + x_5\)
Now, we can isolate \(x_4 + x_5\):

\[ x_4 + x_5 = 22 - 9 = 13 \]

### Step 4: Use the variance to find another equation
The variance is given as 8.24. The formula for variance is:

\[ \text{Variance} = \frac{1}{n} \sum (x_i - \bar{x})^2 \]

Where \(\bar{x}\) is the mean. We can also express variance in terms of the sum of squares:

\[ \text{Variance} = \frac{1}{n} \left( \sum x_i^2 - n \bar{x}^2 \right) \]

Substituting the known values:

\[ 8.24 = \frac{1}{5} \left( 1^2 + 2^2 + 6^2 + x_4^2 + x_5^2 - 5 \times (4.4)^2 \right) \]

Calculating \(1^2 + 2^2 + 6^2\):

\[ 1 + 4 + 36 = 41 \]

Now substituting back into the variance equation:

\[ 8.24 = \frac{1}{5} \left( 41 + x_4^2 + x_5^2 - 5 \times 19.36 \right) \]

Calculating \(5 \times 19.36\):

\[ 5 \times 19.36 = 96.8 \]

So we have:

\[ 8.24 = \frac{1}{5} \left( 41 + x_4^2 + x_5^2 - 96.8 \right) \]

Multiplying both sides by 5:

\[ 41 + x_4^2 + x_5^2 - 96.8 = 41.2 \]

### Step 5: Rearranging to find \(x_4^2 + x_5^2\)
Rearranging gives:

\[ x_4^2 + x_5^2 = 41.2 + 96.8 - 41 = 97 \]

### Step 6: Set up a system of equations
Now we have two equations:

1. \(x_4 + x_5 = 13\) (Equation 1)
2. \(x_4^2 + x_5^2 = 97\) (Equation 2)

### Step 7: Use the identity for squares
We can use the identity:

\[ (x_4 + x_5)^2 = x_4^2 + x_5^2 + 2x_4x_5 \]

Substituting the values from our equations:

\[ 13^2 = 97 + 2x_4x_5 \]

Calculating \(13^2\):

\[ 169 = 97 + 2x_4x_5 \]

### Step 8: Solve for \(x_4x_5\)
Rearranging gives:

\[ 2x_4x_5 = 169 - 97 = 72 \]

Thus:

\[ x_4x_5 = 36 \]

### Step 9: Solve the system of equations
Now we have:

1. \(x_4 + x_5 = 13\)
2. \(x_4 x_5 = 36\)

Let \(x_4\) and \(x_5\) be the roots of the quadratic equation:

\[ t^2 - (x_4 + x_5)t + x_4x_5 = 0 \]

Substituting the values:

\[ t^2 - 13t + 36 = 0 \]

### Step 10: Factor or use the quadratic formula
Factoring gives:

\[ (t - 9)(t - 4) = 0 \]

Thus, the solutions are:

\[ t = 9 \quad \text{or} \quad t = 4 \]

### Conclusion
The values of the two unknown observations are:

\[ x_4 = 9, \quad x_5 = 4 \]