First slide

Introduction to statistics

Question

The mean of a set of observations is $\bar{x}$ . If each observation is divided by, $α \neq 0$ and then is increased by 10, then the mean of the new set 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 $x_{1}, x_{2}, x_{3}, \dots x_{n}$ be n observations. Then
$\begin{array}{l} \bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} \\ ∴ New mean, \bar{X} = \frac{1}{n} \sum_{i = 1}^{n} (\frac{x_{i}}{α} + 10) \\ = \frac{1}{α} (\frac{1}{n} \sum_{i = 1}^{n} x_{i}) + \frac{1}{n} \cdot (10 n) \\ = \frac{1}{α} \bar{x} + 10 = \frac{\bar{x} + 10 α}{α} \end{array}$

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