Question

Let the observations $x_i$ $(1\le i\le 10)$ satisfy $\sum _{i=1}^{10}(x_i-5)=10$ and $\sum _{i=1}^{10}(x_i-5{)}^2=40$. For the observations $(x_1-3),(x_2-3),...,(x_{10}-3)$, which of the following is/are correct?

• A. Mean $=3$
• B. Mean $=5$
• C. Variance $=3$
• D. Variance $=5$

Answer 1

Answer: (A), (C)

Variance $=3$

Given : $\sum _{i=1}^{10}(x_i-5)=10$ and $\sum _{i=1}^{10}(x_i-5{)}^2=40$
Now, $\sum _{i=1}^{10}{x}_i-5\times 10=10$
$\Rightarrow \frac{\sum _{i=1}^{10}{x}_i}{10}=6$
Mean of the observations $x_i-3$ is
$\overline{x}=\frac{\sum _{i=1}^{10}(x_i-3)}{10}\therefore \overline{x}=3$

Variance is unchanged when a constant is added or subtracted from each observation, so
${\sigma }^2=\text{Var}(x_i-3)=\text{Var}(x_i-5)\Rightarrow {\sigma }^2=\frac{\sum _{i=1}^{10}(x_i-5{)}^2}{10}-{\left(\frac{\sum _{i=1}^{10}(x_i-5)}{10}\right)}^2\therefore {\sigma }^2=\frac{40}{10}-1^2=3$