A) increases by 1
B) decreases by 1
C) decreases by 2
D) increases by 2
Correct Answer: A
Solution :
There are 2n observations \[{{x}_{1}},{{x}_{2}},...,{{x}_{2n}}\] So, mean \[=\sum\limits_{i=1}^{2n}{\frac{{{x}_{i}}}{2n}}\] Let these observations be divided into two parts \[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\]and \[{{x}_{n+1}},...,{{x}_{2n}}\] Each in 1st part 5 is added, so total of first part is\[\sum\limits_{i=1}^{n}{{{x}_{i}}+5n.}\] In second part 3 is subtracted from each So, total of second part is\[\sum\limits_{i=n+1}^{2n}{{{x}_{i}}-3n}\] Total of 2n terms are \[\sum\limits_{i=1}^{n}{{{x}_{i}}+5n}+\sum\limits_{i=n+1}^{2n}{{{x}_{i}}}-3n=\sum\limits_{i=1}^{2n}{{{x}_{i}}+2n}\] Mean\[=\sum\limits_{i=1}^{2n}{\frac{{{x}_{i}}+2n}{2n}}=\sum\limits_{i=1}^{2n}{\frac{{{x}_{i}}}{2n}+1}\] So, it increase by 1.
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