Statistics
Combined Mean and Variance of Shifted Sets
nta_pyq_2023_apr
Grade 11
Question:
Let sets A and B each have 5 elements, with mean 5 and variance 12, and mean 8 and variance 20 respectively. If mean of $(x_i-3)$ and $(y_i+2)$ combined is $\mu$ and variance is $\sigma^2$, then $\mu+\sigma^2$ is
$\dfrac{7}{2}$
$\dfrac{9}{2}$
$32$
$34$
Step-by-Step Solution
Key Concept: Shift $A$ by $-3$: new mean $2$, same variance $12$. Shift $B$ by $+2$: new mean $10$, same variance $20$. Combined mean $=\frac{10+50}{10}=6$, combined variance $=\frac{80+600}{10}-36=32$.
Combined variance $=\frac{80+120\cdot5}{10}-36=32$.
Correct Answer: 3