A scientist is weighing each of 30 fishes. Their mean weight worked out is 30 gm and standard deviation of 2 gm. Later, it was found that the measuring scale was misaligned and always under reported every fish weight by 2 gm. The correct mean and standard deviation (in gm) of fishes are respectively.

A
$32, 4$
B
$28, 2$
C
$28, 4$
D
$32, 2$

Solution:

Let $x_{1}, x_{2}, \dots, x_{30}$ be actual weights of $30$ fishes and $y_{1}, y_{2}, \dots, y_{30}$ be the weights of fishes taken from misaligned increasing scale. Then,

$y_{i} = x_{i} + 2; i = 1, 2, \dots, 30$

$\Rightarrow \bar{Y} = \bar{X} + 2$ and $σ_{Y} = σ_{X}$ $[\begin{matrix} ∵ Standard deviation is in- \\ dependent of change of origin \end{matrix}]$

$\Rightarrow 30 = \bar{X} + 2$ and $σ_{Y} = 2$

$\Rightarrow \bar{X} = 28$ and $σ_{Y} = 2$

Solution:

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