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

The Correct Option is C

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:\( \frac{5}{4} \)