If the mean and the variance of 6,4,a,8,b,12,10,13 are 9 and 9.25 respectively, then a+b+ab is equal to :
Solutions · 1
Official solution
Let's set up two equations from the given mean and variance:
Mean = 9 \Rightarrow total sum = 8\cdot 9 = 72
Known values sum to 6+4+8+12+10+13 = 53, so
a+b=72−53=19.
Population variance = 9.25 \Rightarrow
\[\frac{\sum x_i^2}{8} - 9^2 = 9.25\quad\Longrightarrow\quad \sum x_i^2 = 8\,(81 + 9.25) = 722.\]
Known squares sum to 6\^2+4\^2+8\^2+12\^2+10\^2+13\^2 = 529, so
a2+b2=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).