The new mean of set of new numbers is 305
Solution:
Given, The mean of a set of numbers is 300
Each number in that set is increased by 5,
We have to find what is the new mean of this new set of numbers
[tex]\text { mean }=\frac{\text { sum of observations }}{\text { number of observations }}[/tex]
Now, let the number of observations be n and observations be [tex]a_{1}, a_{2}, a_{3}, a_{4}, \ldots \ldots \ldots, a_{n}[/tex]
[tex]\text { So, mean }=\frac{a_{1}+a_{2}+a_{3}+\cdots a_{n}}{n}[/tex]
[tex]a_{1}+a_{2}+a_{3}+\ldots \ldots+a_{n}=m e a n \times n[/tex]
[tex]a_{1}+a_{2}+a_{3}+\ldots . a_{n}=300 n[/tex]
Now, after each number is increased by 5
[tex]\text { New numbers are }\left(a_{1}+5\right),\left(a_{2}+5\right), \ldots \ldots \ldots\left(a_{n}+5\right)[/tex]
[tex]\text { And, new mean }=\frac{\text {sum of new observations}}{\text {number of observations}}[/tex]
[tex]=\frac{\left(a_{1}+5\right)+\left(a_{2}+5\right)+\ldots \ldots+\left(a_{n}+5\right)}{n}[/tex]
[tex]=\frac{\left(a_{1}+a_{2}+\cdots a_{n}\right)+(5+5+5+\cdots+5(n\text { times)) } }{n}[/tex]
[tex]=\frac{300 n+5 n}{n}=\frac{305 n}{n}=305[/tex]
Hence, the new mean is 305
