The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1,3 and 8, then a ratio of other two observations is

Options

A. `4 : 9`
B. `6 : 7`
C. `5 : 8`
D. `10 : 3`

Answer

To solve the problem step by step, we will follow the given information about the mean and variance of the observations.

### Step 1: Understanding the Mean
We know that the mean of five observations is 5. The formula for the mean is given by:

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

where \( n \) is the number of observations. Here, \( n = 5 \).

Let the five observations be \( 1, 3, 8, x_1, x_2 \). Therefore, we can write:

\[ \frac{1 + 3 + 8 + x_1 + x_2}{5} = 5 \]

### Step 2: Calculate the Sum
Calculating the sum of the known observations:

\[ 1 + 3 + 8 = 12 \]

Substituting this into the mean equation:

\[ \frac{12 + x_1 + x_2}{5} = 5 \]

### Step 3: Solve for \( x_1 + x_2 \)
Multiplying both sides by 5:

\[ 12 + x_1 + x_2 = 25 \]

Subtracting 12 from both sides gives:

\[ x_1 + x_2 = 13 \quad \text{(Equation 1)} \]

### Step 4: Understanding the Variance
The variance is given as \( 9.20 \). The formula for variance \( \sigma^2 \) is:

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

where \( \bar{x} \) is the mean. We know \( \bar{x} = 5 \) and \( n = 5 \).

### Step 5: Set Up the Variance Equation
Using the known observations, we can write:

\[ 9.20 = \frac{(1 - 5)^2 + (3 - 5)^2 + (8 - 5)^2 + (x_1 - 5)^2 + (x_2 - 5)^2}{5} \]

Calculating the squared differences for the known observations:

\[ (1 - 5)^2 = 16, \quad (3 - 5)^2 = 4, \quad (8 - 5)^2 = 9 \]

Thus:

\[ 9.20 = \frac{16 + 4 + 9 + (x_1 - 5)^2 + (x_2 - 5)^2}{5} \]

### Step 6: Simplifying the Variance Equation
Adding the squared differences:

\[ 16 + 4 + 9 = 29 \]

So the equation becomes:

\[ 9.20 = \frac{29 + (x_1 - 5)^2 + (x_2 - 5)^2}{5} \]

Multiplying both sides by 5:

\[ 46 = 29 + (x_1 - 5)^2 + (x_2 - 5)^2 \]

### Step 7: Solve for the Squared Differences
Subtracting 29 from both sides gives:

\[ 17 = (x_1 - 5)^2 + (x_2 - 5)^2 \quad \text{(Equation 2)} \]

### Step 8: Substitute \( x_2 \)
From Equation 1, we have \( x_2 = 13 - x_1 \). Substitute this into Equation 2:

\[ 17 = (x_1 - 5)^2 + ((13 - x_1) - 5)^2 \]

This simplifies to:

\[ 17 = (x_1 - 5)^2 + (8 - x_1)^2 \]

### Step 9: Expand and Simplify
Expanding the squares:

\[ (x_1 - 5)^2 = x_1^2 - 10x_1 + 25 \]
\[ (8 - x_1)^2 = 64 - 16x_1 + x_1^2 \]

Combining these:

\[ 17 = x_1^2 - 10x_1 + 25 + 64 - 16x_1 + x_1^2 \]
\[ 17 = 2x_1^2 - 26x_1 + 89 \]

### Step 10: Rearranging the Equation
Rearranging gives:

\[ 2x_1^2 - 26x_1 + 72 = 0 \]

### Step 11: Solving the Quadratic Equation
Dividing through by 2:

\[ x_1^2 - 13x_1 + 36 = 0 \]

Factoring gives:

\[ (x_1 - 9)(x_1 - 4) = 0 \]

So, \( x_1 = 9 \) or \( x_1 = 4 \). Thus, \( x_2 = 4 \) or \( x_2 = 9 \).

### Step 12: Finding the Ratio
The ratio of \( x_1 \) to \( x_2 \) is:

\[ \frac{x_2}{x_1} = \frac{4}{9} \quad \text{or} \quad \frac{9}{4} \]

Thus, the ratio of the two observations is:

\[ \text{Ratio} = 4:9 \]