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13. statistics — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS13. statistics1 Mark
MCQ
Let in a series of $2 n$ observations, half of them are equal to $a$ and remaining half are equal to $-a.$ Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become $5$ and $20 ,$ respectively. Then the value of $a^{2}+b^{2}$ is equal to ....... .
  • ✓
    $425$
  • B
    $650$
  • C
    $250$
  • D
    $925$
✓

Answer

Correct option: A.
$425$
a
Let observations are denoted by $x _{i}$ for $1 \leq i< 2 n$

$\bar{x}=\frac{\sum x_{i}}{2 n}=\frac{(a+a+\ldots+a)-(a+a+\ldots+a)}{2 n}$

$\Rightarrow \overline{ x }=0$

and $\sigma_{ x }^{2}=\frac{\sum x _{i}^{2}}{2 n }-(\overline{ x })^{2}=\frac{ a ^{2}+ a ^{2}+\ldots+ a ^{2}}{2 n }-0= a ^{2}$

$\Rightarrow \sigma_{x}=a$

Now, adding a constant $b$ then $\overline{ y }=\overline{ x }+ b =5$

$\Rightarrow b =5$

and $\sigma_{y}=\sigma_{x}$ (No change in S.D.) $\Rightarrow a=20$

$\Rightarrow a^{2}+b^{2}=425$

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