Question

The means 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

• A. $90$
• B. $7$
• C. $9$
• D. $81$

Answer 1

Answer: (D)

$n_{1}=40$ , $n_{2}=50$
$\bar{x}_{1}=54$ , $\bar{x}_{2}=63$
$\sigma _{1}=6$ , $\sigma _{2}=9$
$\bar{x}=\frac{n_{1} \bar{x}_{1} + n_{2} \bar{x}_{2}}{n_{1} + n_{2}}=\frac{40 \times 54 + 50 \times 63}{90}=59$
$d_{1}=\bar{x}-\bar{x}_{1}=5$
$d_{2}=\bar{x}_{2}-\bar{x}=4$
Combined $S.D=\sqrt{\frac{n_{1} d_{1}^{2} + n_{2} d_{2}^{2} + n_{1} \sigma _{1}^{2} + n_{2} \sigma _{2}^{2}}{n_{1} + n_{2}}}$
$=\sqrt{\frac{40 \times 25 + 50 \times 16 + 40 \times 36 + 50 \times 81}{90}}$
$=9$
Variance $=81$