First slide

Introduction to statistics

Question

The arithmetic mean of a set of observations is X. If each observation is divided by R and then is increased by 10, then the mean of the new series is

Moderate

A

$\frac{\bar{X}}{α}$
B

$\frac{\bar{X} + 10}{α}$
C

$\frac{\bar{X} + 10 α}{α}$
D

$α \bar{X} + 10$

Solution

Let $x_{1}, x_{2}, \dots, x_{n} be n$ observations. Then
$\bar{X} = \frac{1}{n} \sum x_{i}$
Let $y_{i} = \frac{x_{i}}{α} + 10$ Then
$\begin{aligned} \frac{1}{n} \sum_{i = 1}^{n} y_{i} = \frac{1}{α} (\frac{1}{n} \sum_{i = 1}^{n} x_{i}) + \frac{1}{n} (10 n) \\ \Rightarrow \bar{Y} = \frac{1}{α} \bar{X} + 10 = \frac{\bar{X} + 10 α}{α} \end{aligned}$

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