First slide

Introduction to statistics

Question

If the mean of a set of observations $x_{1}, x_{2}, \dots, x_{n} is \bar{X}$ then the mean of the observations $x_{i} + 2 i; i = 1, 2, \dots, n$ is

Moderate

A

$\bar{x} + 2$
B

$\bar{X} + 2 n$
C

$\bar{X} + (n + 1)$
D

$\bar{X} + n$

Solution

We have,
$\bar{X} = \frac{r_{1} + x_{2} + \dots + x_{n}}{n} \Rightarrow n \bar{X} = x_{1} + x_{2} + \dots + x_{n}$
Let Y be the mean of observations $x_{i} + 2 i; i = 1, 2, \dots, n$ Then
$\bar{Y} = \frac{(x_{1} + 2 \cdot 1) + (x_{2} + 2 \cdot 2) + (x_{3} + 2 \cdot 3) + \dots + (x_{n} + 2 \cdot n)}{n}$
$\begin{array}{l} \Rightarrow \bar{Y} = \frac{\sum_{i = 1}^{n} x_{i} + 2 (1 + 2 + 3 + \dots + n)}{n} \\ \Rightarrow \bar{Y} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} + \frac{2 n (n + 1)}{2 n} = \bar{X} + (n + 1) \end{array}$

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