Let X1-X2-X18 be eighteen observation such that -18i-1Xi-36 and - 18i-1Xi-2-90- where - and - are distinct real numbers- If the standard deviation of these observations is 1- then the value of - is

Given, ∑^18_i=1(X_i α)=36 ⇒∑ x_i 18α=36 ⇒∑ x_i=18(α+2) ...(1) Also, ∑^18_i=1(X_i β)^2=90 ⇒∑ x^2_i+18β^2 2β∑ x_i=90 ⇒∑ x^2_i+18β^2 2β×18(α+2)=90 nbsp; nbsp; ...

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Byju's Answer

Standard XII

Mathematics

Standard Deviation about Mean

Let X1,X2,......

Question

Let $X_{1}, X_{2}, . . . X_{18}$ be eighteen observation such that $18 \sum i = 1 (X_{i} - α) = 36$ and $18 \sum i = 1 (X_{i} - β)^{2} = 90$ , where $α$ and $β$ are distinct real numbers. If the standard deviation of these observations is $1$ , then the value of $| α - β |$ is

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Solution

Given, $18 \sum i = 1 (X_{i} - α) = 36$
$\Rightarrow \sum x_{i} - 18 α = 36$
$\Rightarrow \sum x_{i} = 18 (α + 2) . . . (1)$
Also, $18 \sum i = 1 (X_{i} - β)^{2} = 90$
$\Rightarrow \sum x_{i}^{2} + 18 β^{2} - 2 β \sum x_{i} = 90$
$\Rightarrow \sum x_{i}^{2} + 18 β^{2} - 2 β \times 18 (α + 2) = 90$ (using equation $(1)$ )
$\Rightarrow \sum x_{i}^{2} = 90 - 18 β^{2} + 36 β (α + 2)$
Now
$σ^{2} = 1 \Rightarrow \frac{1}{18} \sum x_{i}^{2} - {(\frac{\sum x_{i}}{18})}^{18} = 1$
( $∵ σ = 1,$ given)
$\Rightarrow \frac{1}{18} (90 - 18 β^{2} + 36 α β + 72 β) - {(\frac{18 (α + 2)}{18})}^{2} = 1$
$\Rightarrow 5 - β^{2} + 2 α β + 4 β - (α + 2)^{2} = 1$
$\Rightarrow 5 - β^{2} + 2 α β + 4 β - α^{2} - 4 - 4 α = 1$
$\Rightarrow α^{2} - β^{2} + 2 α β + 4 β - 4 α = 0$
$\Rightarrow (α - β) (α - β + 4) = 0$
$\Rightarrow α - β = - 4$
Hence
$∴ | α - β | = 4, (α \neq β)$

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Let X1,X2,...X18 be eighteen observation such that 18∑i=1(Xi−α)=36 and 18∑i=1(Xi−β)2=90, where α and β are distinct real numbers. If the standard deviation of these observations is 1, then the value of |α−β| is

Let $X_{1}, X_{2}, . . . X_{18}$ be eighteen observation such that $18 \sum i = 1 (X_{i} - α) = 36$ and $18 \sum i = 1 (X_{i} - β)^{2} = 90$ , where $α$ and $β$ are distinct real numbers. If the standard deviation of these observations is $1$ , then the value of $| α - β |$ is