Suppose some people are asked to each randomly pick 30 apples and put into a bag.
In this case, the weights of all the bags will surely be different because different apples weigh differently. So for one of the bag, if let $X_1$ be the weight of the first apple, $X_2$ be the second apple, and all the way to $X_{30}$ the weight of the $30th$ apple, then $X$, which is the weight of the bag is $X=X_1+X_2+\cdots+X_{30}$.
Now, if I want to approximate the sampling distribution of $\bar{X}$, I presume that I can simply just say $\bar{X}=\frac{1}{30}\sum _{ i=1 }^{ 30 }{ X_{ i } } $. Is this right?
In the problem, I need to continue to find the sampling distribution for the variation of the weights of the bags from the approximated sampling distribution of $\bar{X}$ that I have found. I am stuck at this part. I don't see how $\bar{X}$ links with the variance of the bag weights.