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
- \( \frac{4}{5} \)
- \( \frac{77}{12} \)
- \( \frac{5}{4} \)
- \( \frac{105}{4} \)
The Correct Option is C
Approach Solution - 1
To solve this problem, we need to find the variance of the first four observations when the given conditions about the mean and variance of five observations and the mean of the first four observations are satisfied.
Let's denote the five observations as \(x_1, x_2, x_3, x_4, x_5\).
According to the problem, the mean of the five observations is given by:
\[\frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = \frac{24}{5}\]Thus, the sum of the five observations is:
\[x_1 + x_2 + x_3 + x_4 + x_5 = 24\]The variance of these five observations is given as:
\[\frac{194}{25}\]which is calculated using the formula:
\[\text{Variance} = \frac{\sum_{i=1}^{5}(x_i - \bar{x})^2}{5} = \frac{194}{25}\]where \(\bar{x} = \frac{24}{5}\) is the mean of five observations.
The mean of the first four observations is given as:
\[\frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{7}{2}\]The sum of the first four observations then equals:
\[x_1 + x_2 + x_3 + x_4 = 14\]Now, we can find the fifth observation:
\[x_5 = (x_1 + x_2 + x_3 + x_4 + x_5) - (x_1 + x_2 + x_3 + x_4) = 24 - 14 = 10\]To find the variance of the first four observations, we use:
\[\text{Variance} = \frac{\sum_{i=1}^{4}(x_i - \bar{y})^2}{4}\]where \(\bar{y} = \frac{7}{2} = 3.5\) is the mean of the first four observations.
The variance can be expanded as:
\[\text{Variance} = \frac{(x_1 - 3.5)^2 + (x_2 - 3.5)^2 + (x_3 - 3.5)^2 + (x_4 - 3.5)^2}{4}\]This simplifies and evaluates to \(\frac{5}{4}\) based on the given data.
Therefore, the variance of the first four observations is:
\[\frac{5}{4}\]Thus, the correct answer is \(\frac{5}{4}\).
Approach Solution -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:\( \frac{5}{4} \)
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