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Introduction to statistics

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Question

If the mean and standard deviation of 10 observations $x_{1}, x_{2}, \dots, x_{10}$ are 2 and 3 respectively, then mean of ${(x_{1} + 1)}^{2}, {(x_{2} + 1)}^{2}, \dots, {(x_{10} + 1)}^{2}$ is equal to

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Solution

We have, Mean = 2 and S. D. = 3
$\Rightarrow \frac{1}{10} \sum_{i = 10}^{10} x_{i} = 2$ and $\frac{1}{10} \sum_{i = 1}^{10} x_{i}^{2} - 2^{2} = 3^{2}$
$\Rightarrow \sum_{i = 1}^{10} x_{i} = 20$ and $\sum_{i = 1}^{10} x_{i}^{2} = 130$
Let $\bar{X}$ be the mean of ${(x_{1} + 1)}^{2}, {(x_{2} + 1)}^{2}, \dots, {(x_{10} + 1)}^{2}$ . Then,
$\begin{array}{l} \bar{X} = \frac{1}{10} \sum_{i = 1}^{10} {(x_{i} + 1)}^{2} \\ \Rightarrow \bar{X} = \frac{1}{10} (\sum_{i = 1}^{10} x_{i}^{2}) + \frac{2}{10} (\sum_{i = 1}^{10} x_{i}) + \frac{1}{10} \sum_{i = 1}^{10} 1 \\ \Rightarrow \bar{X} = \frac{1}{10} \times 130 + \frac{2}{10} \times 20 + \frac{10}{10} = 18 \end{array}$

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