Question

The numbers $3,5,7$ and $9$ have their respective frequencies $x-2,x+2,x-3$ and $x+3$. If the arithemetic mean is $6.5$, then the value of $x$ is

• A. More than $3$
• B. Less than $8$
• C. $5$
• D. None

Answer 1

Answer: (A), (B), (C)

The correct options are
A More than $3$
B Less than $8$
C $5$

Given that arithmetic mean is $\overline{x}=6.5$.

The numbers $x_i$ are $3,5,7,9$ and their corresponding frequencies are $x-2,x+2,x-3,x+3$.

Mean is given by $\overline{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

$⟹\overline{x}=\frac{3(x-2)+5(x+2)+7(x-3)+9(x+3)}{(x-2)+(x+2)+(x-3)+(x+3)}$

$⟹6.5=\frac{3x-6+5x+10+7x-21+9x+27}{4x}$

$⟹6.5\left(4x\right)=10+24x$

$⟹26x=10+24x$

$⟹2x=10$

$⟹x=5$.

$⟹x=5<8$.

$⟹x=5>3$.

Therefore, options $A,B$ and $C$ are correct.