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Question
The mean of n observation is `overlineX` . If the first item is increased by 1, second by 2 and so on, then the new mean is
Options
`overlineX+n`
`overlineX+n/2`
`overlineX+(n+1)/2`
None of these
Solution
Let \[x_1 , x_2 , x_3 , . . . , x_n\] be the n observations.
Mean
= `overlineX`\[= \frac{x_1 + x_2 + . . . + x_n}{n}\]
\[\Rightarrow x_1 + x_2 + x_3 + . . . + x_n = n\]`overlineX`
If the first item is increased by 1, second by 2 and so on.
Then, the new observations are
\[x_1 + 1, x_2 + 2, x_3 + 3, . . . , x_n + n\]
New mean = \[\frac{\left( x_1 + 1 \right) + \left( x_2 + 2 \right) + \left( x_3 + 3 \right) + . . . + \left( x_n + n \right)}{n}\]
\[= \frac{x_1 + x_2 + x_3 + . . . + x_n + \left( 1 + 2 + 3 + . . . + n \right)}{n}\]
`=(noverlineX + (n(n+1)/2))/n`
=`overlineX`\[ + \frac{n + 1}{2}\]
Hence, the correct answer is (c).
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