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Let $n \geq 3$.A list of numbers $x_{1}, x_{2}, \ldots \ldots \ldots x_{n}$ has mean $\mu$ and standard deviation $\sigma .$ A new list of numbers $y_{1}, y_{2}, \ldots \ldots \ldots . y_{n}$ is made as follows: $y_{1}=\frac{x_{1}+x_{2}}{2}, y 2=\frac{x_{1}+x_{2}}{2}$ and $y_{j}=x$ for $j=3,4, \ldots \ldots . n$. The mean and the standard deviation of the list are $\hat{\mu}$ and $\hat{\sigma}$. Then which of the following is necessarily true?
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The correct answer is:
$\mu=\hat{\mu}$ and $\sigma \geq \hat{\sigma}$
$\mu=\frac{\sum x_{i}}{n} \quad n \geq 3$
$\sigma=\sqrt{\frac{\sum x_{i}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}}$
$\hat{\mu}=\frac{y_{1}+y_{2}+y_{3}+\ldots \ldots+y_{n}}{n}$
$=\frac{\frac{x_{1}+x_{2}}{2}+\frac{x_{1}+x_{2}}{2}+x_{3}+x_{4}+\ldots \ldots \ldots+x_{n}}{n}$
$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots \ldots . .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}=\mu$
$\hat{\sigma}=\sqrt{\frac{\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}}$
$\begin{array}{l}=\sqrt{\frac{\frac{1}{4}\left(2 x_{1}^{2}+2 x_{2}^{2}+4 x_{1} x_{2}\right)+x_{3}^{2}+\ldots \ldots . .+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}} \\ =\sqrt{\frac{\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)+x_{1} x_{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \leq \sqrt{\frac{x_{1}^{2}+x_{2}^{2}+\ldots \ldots+x_{2}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \hat{\sigma} \leq \sigma & \ldots \ldots \ldots(2) \\ \text { by }(1) \text { and }(2)\end{array}$
$\sigma=\sqrt{\frac{\sum x_{i}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}}$
$\hat{\mu}=\frac{y_{1}+y_{2}+y_{3}+\ldots \ldots+y_{n}}{n}$
$=\frac{\frac{x_{1}+x_{2}}{2}+\frac{x_{1}+x_{2}}{2}+x_{3}+x_{4}+\ldots \ldots \ldots+x_{n}}{n}$
$=\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots \ldots . .+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}=\mu$
$\hat{\sigma}=\sqrt{\frac{\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+\left(\frac{x_{1}+x_{2}}{2}\right)^{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}}$
$\begin{array}{l}=\sqrt{\frac{\frac{1}{4}\left(2 x_{1}^{2}+2 x_{2}^{2}+4 x_{1} x_{2}\right)+x_{3}^{2}+\ldots \ldots . .+x_{n}^{2}}{n}-\left(\frac{\sum y_{1}}{n}\right)^{2}} \\ =\sqrt{\frac{\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)+x_{1} x_{2}+x_{3}^{2}+\ldots \ldots+x_{n}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \leq \sqrt{\frac{x_{1}^{2}+x_{2}^{2}+\ldots \ldots+x_{2}^{2}}{n}-\left(\frac{\sum x_{i}}{n}\right)^{2}} \\ \hat{\sigma} \leq \sigma & \ldots \ldots \ldots(2) \\ \text { by }(1) \text { and }(2)\end{array}$
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