KVPY SX SB Exam 20141 Question Paper

Let n ≥ 3. A list of numbers ${\mathit{x}}_1,{\mathit{x}}_2,...,{\mathit{x}}_{\mathit{n}}$ has mean µ. and standard deviation σ . A new list o f numbers ${\mathit{y}}_1,{\mathit{y}}_2,...,{\mathit{y}}_{\mathit{n}}$ is made as follows: ${\mathit{y}}_1,=\frac{{\mathit{x}}_1+{\mathit{x}}_2}{2},{\mathit{y}}_2=\frac{{\mathit{x}}_1+{\mathit{x}}_2}{2}$ and ${\mathit{y}}_{\mathit{j}}={\mathit{x}}_{\mathit{j}}\mathrm{for}\mathit{j}=3,4,...,,\mathit{n}$. The mean and the standard deviation o f the new list are $\hat{\mathit{\mu }}\mathrm{and}\hat{\mathit{\sigma }}$ . Then which of the following is necessarily true?