Question

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations is equal to

• A. $\frac{4}{5}$
• B. $\frac{77}{12}$
• C. $\frac{5}{4}$
• D. $\frac{105}{4}$

Answer 1

Answer: (C)

$\frac{5}{4}$

Answer 2

Solution: Let the first four observations be $x_1, x_2, x_3, x_4$.

Step 1. Given:
$\bar{X} = \frac{24}{5}, \quad \sigma^2 = \frac{194}{25}$
Step 2. The mean of five observations:

$\frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = \frac{24}{5} \implies x_1 + x_2 + x_3 + x_4 + x_5 = 24$
Step 3. The mean of the first four observations:

$\frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{7}{2} \implies x_1 + x_2 + x_3 + x_4 = 14$

Step 4. Subtracting (2) from (1):
$x_5 = 24 - 14 = 10$
Step 5. Using the formula for variance of the first four observations:
$\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n}, \quad \text{where } \bar{x} = \frac{7}{2}$
After calculating, the variance is: $\frac{5}{4}$

The Correct Answer is:$\frac{5}{4}$