JEE Main 2021 (Online) 16th March Evening ShiftQuestion 1
Consider the statistics of two sets of observations as follows :
Observation I : Size $10$, Mean $2$, Variance $2$
Observation II : Size $n$, Mean $3$, Variance $1$
If the variance of the combined set of these two observations is $\frac{17}{9}$, then the value of $n$ is equal to $\_\_\_\_\_\_$ .
Statistics
Answer$5$
Solution
Combined mean $m=\dfrac{20+3n}{10+n}$. Combined variance $=\dfrac{10[2+(2-m)^2]+n[1+(3-m)^2]}{10+n}=\dfrac{17}{9}$. Solving this equation in $n>0$ gives $n=5$. Check: $m=\tfrac{35}{15}=\tfrac73$, variance $=\dfrac{10(2+\frac19)+5(1+\frac49)}{15}=\dfrac{\frac{190+65}{9}}{15}=\dfrac{17}{9}$. So $n=5$.