Question

The mean and variance of the observations \( x_1, x_2, x_3, \dots, x_{15} \) are respectively 2 and 4. If the mean and variance of the observations \( Y_1, Y_2, \dots, Y_{10} \) are respectively 2 and 5, then the variance of the observations \( x_1, x_2, \dots, x_{15}, Y_1, Y_2, \dots, Y_{10} \) is:

• A. \( 6.5 \)
• B. \( 5.3 \)
• C. \( 3.4 \)
• D. \( 4.4 \)

Hint:

When combining two datasets with the same mean, the pooled variance is given by: \[ \sigma^2 = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2}{n_1 + n_2} \] This formula helps in efficiently computing variance across merged samples while maintaining accuracy.

Answer 1

The Correct Option is D

To compute the variance of the combined observations, we use the formula: \[ \sigma^2 = \frac{n_1 \sigma_1^2 + n_2 \sigma_2^2}{n_1 + n_2} \] where: - \( n_1 = 15 \), \( \sigma_1^2 = 4 \) (variance of \( x_i \)) - \( n_2 = 10 \), \( \sigma_2^2 = 5 \) (variance of \( Y_i \)) Substituting the values: \[ \sigma^2 = \frac{(15 \times 4) + (10 \times 5)}{15 + 10} \] \[ = \frac{60 + 50}{25} \] \[ = \frac{110}{25} \] \[ = 4.4 \] Thus, the correct answer is \( 4.4 \).