Let the observations x_i (1 i10) satisfy the equations _i=1^10(x_i-5)=10 and _i=1^10(x_i-5…

Question

Let the observations $x_i (1\le i\le10)$ satisfy the equations $\displaystyle\sum_{i=1}^{10}(x_i-5)=10$ and $\displaystyle\sum_{i=1}^{10}(x_i-5)^2=40$. If  and $\lambda$ are the mean and the variance of the observations $x_1-3, x_2-3,\ldots,x_{10}-3$, then the ordered pair $(\mu,\lambda)$ is equal to:

Accepted Answer

Mean of $(x_i-5) = \dfrac{10}{10}=1 \implies$ mean of $x_i=6$; mean of $x_i-3 = 6-3=3=\mu$

Variance of $(x_i-5)$: $\dfrac{40}{10}-1^2=3$. Since variance is invariant to translation, variance of $x_i-3$ also $=3=\lambda$.

$$(\mu,\lambda)=(3,3)$$

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