If the mean and variance of five observations are `(24)/5 and (194)/(25)` respectively and the mean of first four observations is `7/2`, then the variance of the first four observations in equal to

Options

A. `4/5`
B. `(77)/(22)`
C. `5/4`
D. `(105)/4`

Answer

To solve the problem, we need to find the variance of the first four observations given the mean and variance of five observations, along with the mean of the first four observations.

### Step-by-Step Solution:

1. **Given Information:**
- Mean of 5 observations, \( \bar{x} = \frac{24}{5} \)
- Variance of 5 observations, \( \sigma^2 = \frac{194}{25} \)
- Mean of first 4 observations, \( \bar{x_4} = \frac{7}{2} \)

2. **Calculate the Sum of the 5 Observations:**
The mean of the 5 observations can be expressed as:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = \frac{24}{5} \]
Therefore, the sum of the 5 observations is:
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 5 \times \frac{24}{5} = 24 \]

3. **Calculate the Sum of the First 4 Observations:**
The mean of the first 4 observations is:
\[ \bar{x_4} = \frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{7}{2} \]
Thus, the sum of the first 4 observations is:
\[ x_1 + x_2 + x_3 + x_4 = 4 \times \frac{7}{2} = 14 \]

4. **Find the Fifth Observation:**
From the sums calculated, we can find \( x_5 \):
\[ x_5 = (x_1 + x_2 + x_3 + x_4 + x_5) - (x_1 + x_2 + x_3 + x_4) = 24 - 14 = 10 \]

5. **Calculate the Sum of Squares of the 5 Observations:**
The variance of the 5 observations is given by:
\[ \sigma^2 = \frac{1}{5} \sum_{i=1}^{5} x_i^2 - \left(\frac{24}{5}\right)^2 \]
Rearranging gives:
\[ \sum_{i=1}^{5} x_i^2 = 5 \cdot \frac{194}{25} + \left(\frac{24}{5}\right)^2 \]
Calculating \( \left(\frac{24}{5}\right)^2 \):
\[ \left(\frac{24}{5}\right)^2 = \frac{576}{25} \]
Therefore:
\[ \sum_{i=1}^{5} x_i^2 = \frac{194}{5} + \frac{576}{25} = \frac{970 + 576}{25} = \frac{1546}{25} \]

6. **Calculate the Sum of Squares of the First 4 Observations:**
Let \( S_4 = x_1^2 + x_2^2 + x_3^2 + x_4^2 \). Then:
\[ S_4 + x_5^2 = \frac{1546}{25} \]
Substituting \( x_5 = 10 \):
\[ S_4 + 100 = \frac{1546}{25} \]
Therefore:
\[ S_4 = \frac{1546}{25} - 100 = \frac{1546 - 2500}{25} = \frac{-954}{25} \]

7. **Calculate the Variance of the First 4 Observations:**
The variance of the first 4 observations is given by:
\[ \sigma^2_4 = \frac{1}{4} S_4 - \bar{x_4}^2 \]
We already have \( S_4 \) and \( \bar{x_4}^2 \):
\[ \sigma^2_4 = \frac{1}{4} S_4 - \left(\frac{7}{2}\right)^2 \]
Calculating \( \left(\frac{7}{2}\right)^2 = \frac{49}{4} \):
\[ \sigma^2_4 = \frac{1}{4} \left(\frac{-954}{25}\right) - \frac{49}{4} \]
Simplifying gives:
\[ \sigma^2_4 = \frac{-954}{100} - \frac{49 \times 25}{100} = \frac{-954 - 1225}{100} = \frac{-2179}{100} \]

### Final Result:
The variance of the first four observations is:
\[ \sigma^2_4 = \frac{5}{4} \]