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Question
The mean of n observation is `overlineX`. If the first observation is increased by 1, the second by 2, the third by 3, and so on, then the new mean is
Options
- `overlineX`\[ + \left( 2n + 1 \right)\]
- `overlineX`\[+ \frac{n + 1}{2}\]
- `overlineX`\[ + \left( n + 1 \right)\]
`overlineX`\[ - \frac{n + 1}{2}\]
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, the second by 2, the third by 3 and so on.
Then, the new observations are
\[x_1 + 1, x_2 + 2, x_3 + 3, . . . , x_n + n\].
New mean = \[\frac{\left( x_1 + 1 \right) + \left( x_2 + 2 \right) + \left( x_3 + 3 \right) + . . . + \left( x_n + n \right)}{n}\]
\[= \frac{x_1 + x_2 + x_3 + . . . + x_n + \left( 1 + 2 + 3 + . . . + n \right)}{n}\]
`= (nx +(n(n+1))/2)/n`
`=overlineX`\[ + \frac{n + 1}{2}\]
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