I came across a problem today in which it indicated the weight of a baseball $X$ has mean 5 and variance $4\over25$. It then indicated a carton contains 144 baseballs, and we are to obtain $T$ the total weight of the baseballs in one carton, assuming the weights of all the baseballs in the carton are independent. I thought of two approaches to this - one is "correct" (the Correct Method below) and the other is not (Incorrect Method). I'm trying to where the fault is in my reasoning. Why is it not the cases that both answers should be right?
Correct Method
Let $T=\sum_{i=1}^{144}X_i$, where $X_i$ is the weight of the $i$-th baseball, for $i=1, 2, ..., 144$. Then:
$Var(T) = Var[\sum_{i=1}^{144}X_i]=\sum_{i=1}^{144}Var(X_i)=144(\frac{4}{25})=23.04$
Incorrect Method
Let $T = 144X_1$ since each $X_i$ are iid. Then:
$Var(T)=Var(144X_1)=144^2Var(X_1)=144^2\left(\frac{4}{25}\right)=829.44$
I realize the incorrect method is obviously not correct, but why aren't these two quantities the same? What am I missing?