First slide

Introduction to statistics

Question

$The weighted mean of the first n natural numbers whose weights are equal to the squares of the corresponding numbers is$

Easy

A

$\frac{n + 1}{2}$
B

$\frac{3 n (n + 1)}{2 (2 n + 1)}$
C

$\frac{(n + 1) (2 n + 1)}{6}$
D

$\frac{n (n + 1)}{2}$

Solution

$Weighted mean = \frac{{\sum x_{i} W_{i}}_{i = 1}^{n}}{\sum_{i = 1}^{n} W_{i}} \begin{array}{l} = \frac{1 . 1^{2} + 2 \cdot 2^{2} + \dots + n \cdot n^{2}}{1^{2} + 2^{2} + \dots + n^{2}} \\ = \frac{Σ n^{3}}{Σ n^{2}} = \frac{\frac{n (n + 1)}{2} \frac{n (n + 1)}{2}}{\frac{n (n + 1) (2 n + 1)}{6}} = \frac{3 n (n + 1)}{2 (2 n + 1)} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App