Statistics
Mean/Mode/Variance Statement
MMTS_Full_Test_04
Grade 12
Question:
Let $\bar{x}$, $M$, $\sigma^2$ be mean, mode, variance of $n$ observations $x_1,\ldots,x_n$ and $d_i=-x_i-a$, $i=1,\ldots,n$. Statement 1: variance of $d_1,\ldots,d_n$ is $\sigma^2$. Statement 2: Mean and mode of $d_i$ are $-\bar{x}-a$ and $-M-a$ respectively.
Both false
Both true
S1 true S2 false
S1 false S2 true
Step-by-Step Solution
Key Concept: Variance is translation-invariant; mean and mode shift by $-(a+\bar{x})$ and $-(a+M)$
Variance of $d_i=$ Var$(x_i)=\sigma^2$ ✓ (S1 true). Mean of $d_i=-\bar{x}-a$ ✓; Mode$=-M-a$ ✓ (S2 true). Both true.
Correct Answer: 2