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 $=81$