First slide

Introduction to statistics

Question

The AM of n numbers of a series is $\vec{x}$ . If the sum of first (n - 1) terms is k, then the nth number is

Easy

A

$\vec{x} - k$
B

$n \vec{x} - k$
C

$\vec{x} - nk$
D

$n \vec{x} - nk$

Solution

Let the n numbers be $x_{1}, x_{2}, \dots, x_{n}$ , then
$\begin{array}{l} \bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} \\ \Rightarrow \bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n - 1} + x_{n}}{n} \\ \Rightarrow \bar{x} = \frac{k + x_{n}}{n} (∵ x_{1} + x_{2} + \dots + x_{n - 1} = k) \\ ∴ x_{n} = n \bar{x} - k \end{array}$

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