"If the mean and the variance of $6,4, a, 8, b, 12,10,13$ are 9 and 9.25 respectively, then $a+b+a b$ is equal to :"

Question

Accepted Answer

"

Let’s set up two equations from the given mean and variance:

Mean = 9 ⇒ total sum = 8·9 = 72

Known values sum to 6+4+8+12+10+13 = 53, so

$a + b = 72 - 53 = 19.$

Population variance = 9.25 ⇒

$$\frac{\sum x_i^2}{8} - 9^2 = 9.25\quad\Longrightarrow\quad \sum x_i^2 = 8\,(81 + 9.25) = 722.$$

Known squares sum to 6²+4²+8²+12²+10²+13² = 529, so

$a^2 + b^2 = 722 - 529 = 193.$

Now use

$(a + b)^2 = a^2 + 2ab + b^2 \quad\Longrightarrow\quad 19^2 = 193 + 2ab$

so

$361 = 193 + 2ab\quad\Longrightarrow\quad ab = 84.$

Finally,

$a + b + ab = 19 + 84 = 103.$

Answer: 103 (Option A).

"

If the mean and the variance of $6,4, a, 8, b, 12,10,13$ are 9 and 9.25 respectively, then $a+b+a b$ is equal to :