When I was doing sampling/hypothesis testing questions back then, I was told the mean (rather the sample mean/unbiased mean) is as follows $$\overline{x} = \dfrac{\sum x}{n}$$
But when I am presented with known given equations such as $\sum (x-200) = -66$ and $n =30$ I am expected to remember to use the following formulation to find the sample mean: $$\overline{x} = \dfrac{\sum(x-200)}{30} +200$$
May I know why is this so, any proof to back it up?
On
I think you could probably prove it in this simple manner: choose $y = x-200$. You have that $$\bar{y} = \frac{\sum y}{n}$$ Then $$\overline{x-200} = \frac{\sum(x-200)}{n}$$ By the linearity of the mean $\overline{Ax-B} = A\bar{x}-B$ you get that $$\bar{x}-200 = \frac{\sum(x-200)}{n}$$ and so the result you where searching for $$\bar{x} = \frac{\sum(x-200)}{n}+200$$
$$\sum_{n=1}^{30}(f(n)+b)=\left( \sum_{n=1}^{30} f(n)\right) +30b$$
Because of commutativity of addition.