Question

The mean of n observations is X . IF the first item is increased by 1, second by 2 and so on, then the new mean is: 1. X+n 2. X+n/2 3. X+(n+1)/2 4. X+(n-1)/2

Answer 1

The correct option is,(3) X+(n+1)/2.

What is mean of observation?

• An act or the power of seeing or taking notice of something His detailed description shows great powers of observation.
• The gathering of information by noting facts or occurrences weather observations.
• An opinion formed or expressed after watching or noticing It's not a criticism, just an observation.

According to the question:

Mean of $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \mathrm{x}_{\mathrm{n}} is \mathrm{x}$

$\begin{aligned}&\frac{\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\ldots+\mathrm{x}_{\mathrm{n}}}{\mathrm{n}}=\mathrm{x} \\&\Rightarrow \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{x}=\mathrm{nx}\end{aligned}$

on adding 1 to $x_{1}$, 2 to $x_{2}$  and so on

we get  $x_{1}+1+x_{2}+2+\ldots+x_{n}+n$

$\begin{aligned}&=\left(\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots+\mathrm{x}_{\mathrm{n}}\right)+(1+2+\ldots+\mathrm{n}) \\&=\sum \mathrm{n}+\frac{\mathrm{n}(\mathrm{n}+1)}{2}\end{aligned}$

New mean $=\frac{\text { New summation }}{\mathrm{n}}$

$=\frac{\sum \mathrm{n}+\frac{\mathrm{n}(\mathrm{n}+1)}{2}}{\mathrm{n}}=\frac{2 \mathrm{nx}+\mathrm{n}(\mathrm{n}+1)}{2 \mathrm{n}}=\mathrm{x}+\left(\frac{\mathrm{n}+1}{2}\right)$

Hence, the new mean is X+(n+1)/2.

Learn more about mean of observation here,

https://brainly.in/question/2882407?msp_poc_exp=5

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