Question

The mean & variance of \(x_1, x_2, x_3, x_4\) is 1 and 13 respectively and the mean and variance of \(y_1, y_2, \dots, y_6\) be 2 and 1 respectively, the variance of \(x_1, x_2, \dots, x_4, y_1, y_2, \dots, y_6\) will be

• A. 6.04
• B. 6.00
• C. 5.85
• D. 5.99

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the combined variance of two sets, we first find the combined mean of all 10 observations. Then, we use the combined variance formula which incorporates the individual variances and the deviations of individual means from the combined mean.

Step 2: Key Formula or Approach:
Combined Mean \(\bar{z} = \frac{n_1\bar{x} + n_2\bar{y}}{n_1 + n_2}\). Combined Variance \(\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}\), where \(d_1 = \bar{x} - \bar{z}\) and \(d_2 = \bar{y} - \bar{z}\).

Step 3: Detailed Explanation:
1. **Combined Mean:** \[ \bar{z} = \frac{4(1) + 6(2)}{10} = \frac{4 + 12}{10} = 1.6 \] 2. **Calculate Deviations:** \[ d_1^2 = (1 - 1.6)^2 = (-0.6)^2 = 0.36 \] \[ d_2^2 = (2 - 1.6)^2 = (0.4)^2 = 0.16 \] 3. **Combined Variance:** \[ \sigma^2 = \frac{4(13 + 0.36) + 6(1 + 0.16)}{10} \] \[ \sigma^2 = \frac{4(13.36) + 6(1.16)}{10} = \frac{53.44 + 6.96}{10} \] \[ \sigma^2 = \frac{60.4}{10} = 6.04 \]

Step 4: Final Answer:
The variance of the combined set is 6.04.