The mean of two samples of size 40 and 50 were found to be 54 and 63 respectively. Their standard deviations were 6 and 9 respectively. The variance of the combined sample of size 90 is
Step-by-Step Solution
Key Concept: The combined standard deviation is computed using the formula that accounts for both individual variances and squared deviations of individual means from the combined mean.
Given $n_1 = 40$, $n_2 = 50$, $\bar{x}_1 = 54$, $\bar{x}_2 = 63$, $\sigma_1 = 6$, $\sigma_2 = 9$. The combined mean is $\bar{x} = \frac{40(54) + 50(63)}{90} = 59$. Calculate $d_1 = 54 - 59 = -5$ and $d_2 = 63 - 59 = 4$. The combined standard deviation is $\sigma = \sqrt{\frac{40(36 + 25) + 50(81 + 16)}{90}} = \sqrt{\frac{2440 + 4850}{90}} = \sqrt{81} = 9$.
Correct Answer: 9