The mean and variance of the 7 observations is 8 and 12 resp. . If two observations are 8 and 6 . find the variance of the remaining observation

Options

A. `132/15`
B. `396/25`
C. `396/50`
D. `792/25`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Given Information
We have 7 observations with a mean of 8 and a variance of 12. Two of these observations are 8 and 6. We need to find the variance of the remaining 5 observations.

### Step 2: Calculate the Total Sum of Observations
The mean of the observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 8 for 7 observations:
\[ 8 = \frac{\text{Sum of observations}}{7} \]
Thus, the total sum of the observations is:
\[ \text{Sum of observations} = 8 \times 7 = 56 \]

### Step 3: Calculate the Sum of the Known Observations
We know two observations are 8 and 6. Therefore, the sum of these two observations is:
\[ 8 + 6 = 14 \]

### Step 4: Calculate the Sum of the Remaining Observations
Let the sum of the remaining 5 observations be \( S \). We can find \( S \) as follows:
\[ S + 14 = 56 \implies S = 56 - 14 = 42 \]

### Step 5: Calculate the Variance
The variance is given by the formula:
\[ \text{Variance} = \frac{\sum (x_i^2)}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
Where \( n \) is the number of observations. We know the variance of all 7 observations is 12:
\[ 12 = \frac{\sum (x_i^2)}{7} - 8^2 \]
Calculating \( 8^2 \):
\[ 8^2 = 64 \]
Substituting this into the variance equation:
\[ 12 = \frac{\sum (x_i^2)}{7} - 64 \]
Rearranging gives:
\[ \frac{\sum (x_i^2)}{7} = 12 + 64 = 76 \]
Thus,
\[ \sum (x_i^2) = 76 \times 7 = 532 \]

### Step 6: Calculate the Sum of Squares of the Known Observations
Now, we calculate the sum of squares of the known observations:
\[ 8^2 + 6^2 = 64 + 36 = 100 \]

### Step 7: Calculate the Sum of Squares of the Remaining Observations
Let \( T \) be the sum of squares of the remaining observations:
\[ T + 100 = 532 \implies T = 532 - 100 = 432 \]

### Step 8: Calculate the Variance of the Remaining Observations
The variance of the remaining 5 observations can be calculated as follows:
\[ \text{Variance} = \frac{T}{5} - \left(\frac{S}{5}\right)^2 \]
Where \( S = 42 \):
\[ \text{Variance} = \frac{432}{5} - \left(\frac{42}{5}\right)^2 \]
Calculating \( \frac{432}{5} \):
\[ \frac{432}{5} = 86.4 \]
Calculating \( \left(\frac{42}{5}\right)^2 \):
\[ \left(\frac{42}{5}\right)^2 = \frac{1764}{25} = 70.56 \]
Thus, the variance becomes:
\[ \text{Variance} = 86.4 - 70.56 = 15.84 \]

### Final Answer
The variance of the remaining observations is **15.84**.

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