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. `10:3`
D. `5:8`

Explanation

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
The mean of five observations is given as 5. The formula for the mean is:

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

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

\[ 5 = \frac{1 + 3 + 8 + x + y}{5} \]

Calculating the sum of the known observations:

\[ 1 + 3 + 8 = 12 \]

Substituting this into the mean formula gives:

\[ 5 = \frac{12 + x + y}{5} \]

### Step 2: Setting Up the Equation
Multiplying both sides by 5:

\[ 25 = 12 + x + y \]

Rearranging gives:

\[ x + y = 25 - 12 = 13 \quad \text{(Equation 1)} \]

### Step 3: Understanding the Variance
The variance is given as 9.20. The formula for variance is:

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

Where \( \mu \) is the mean and \( n \) is the number of observations. Substituting the values we have:

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

Calculating the squared differences for the known observations:

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

Adding these values:

\[ 16 + 4 + 9 = 29 \]

Substituting back into the variance formula gives:

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

### Step 4: Setting Up the Variance Equation
Multiplying both sides by 5:

\[ 46 = 29 + (x - 5)^2 + (y - 5)^2 \]

Rearranging gives:

\[ (x - 5)^2 + (y - 5)^2 = 46 - 29 = 17 \quad \text{(Equation 2)} \]

### Step 5: Expanding Equation 2
Expanding Equation 2:

\[ (x^2 - 10x + 25) + (y^2 - 10y + 25) = 17 \]

Combining like terms:

\[ x^2 + y^2 - 10x - 10y + 50 = 17 \]

Rearranging gives:

\[ x^2 + y^2 - 10x - 10y + 33 = 0 \]

### Step 6: Substituting from Equation 1
From Equation 1, we know \( x + y = 13 \). We can express \( y \) in terms of \( x \):

\[ y = 13 - x \]

Substituting this into the expanded variance equation:

\[ x^2 + (13 - x)^2 - 10x - 10(13 - x) + 33 = 0 \]

Expanding:

\[ x^2 + (169 - 26x + x^2) - 10x - 130 + 10x + 33 = 0 \]

Combining like terms gives:

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

### Step 7: Solving the Quadratic Equation
Dividing the entire equation by 2:

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

Factoring gives:

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

Thus, \( x = 9 \) or \( x = 4 \). Consequently, if \( x = 9 \), then \( y = 4 \), and if \( x = 4 \), then \( y = 9 \).

### Step 8: Finding the Ratio
The ratio of \( y \) to \( x \) is:

\[ \frac{y}{x} = \frac{4}{9} \quad \text{or} \quad \frac{9}{4} \]

### Final Answer
The ratio of the other two observations is \( \frac{4}{9} \) or \( \frac{9}{4} \).