Question

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

• A. a = 0, b = 7
• B. a = 5, b = 2
• C. a = 1, b = 6
• D. a = 3, b = 4

Answer 1

Answer: (D)

mean $=\frac{\sum x_{i}}{n}$
$=\frac{a+b+8+5+10}{5}=6$
$a+b+23=30$
$a+b=7$
variance $=\frac{\sum\left(x_{i}-\bar{x}\right)^{2}}{n}$
$=\frac{(a-6)^{2}+(b-6)^{2}+4+1+16}{5}=6.8$
$(a-6)^{2}+(b-6)^{2}+21=34$
$(a-6)^{2}+(b-6)^{2}=13$
$(a-6)^{2}+(1-a)^{2}=13$
$a^{2}-12 a+36+1-2 a+a^{2}=13$
$2 a^{2}-14 a+24=0$
$a^{2}-7 a+12=0$
$(a-4)(a-3)=0$
$a=4$ or 3
and $b=3$ or 4