I am trying to see if my solution is accurate. The question goes (I am aware that the question may have been asked before on this site, but nowhere have I seen a reason why the CLT can/cannot be used):
An engineer believes that $W$, the amount of weight (in units of $1000$ pounds) that a certain span of a bridge can withstand without structural damage, is normally distributed with mean $400$ and variance $40^2$. Suppose that the weight (again, in units of $1000$ pounds) of a car is a random variable with mean $3$ and standard deviation $0.3$. Approximately how many cars would have to be on the bridge span for the probability of structural damage to exceed $0.1$?
The reason I am asking if I can use the central limit theorum is because we do not know the distribution of the weight of each car. However, here is my work:
I started by defining $C_n$= total weight of $n$ cars. We need to find n such that: $$P(C_n\geq W)>.1\to P(C_n-W\geq0)>.1$$ Can I use the central limit theorem and say: $$P(C_n-W\geq0)>.1=P\left(\frac{(C_n-W)-E(C_n-W)}{\sqrt{Var(C_n-W)}}\geq\frac{0-E(C_n-W)}{\sqrt{Var(C_n-W)}}\right)>.1$$ $$\approx1-\phi\left(\frac{-E(C_n-W)}{\sqrt{Var(C_n-W)}}\right)>.1\to\phi\left(\frac{-E(C_n-W)}{\sqrt{Var(C_n-W)}}\right)\leq.9$$
I found:
$E(C_n-W)=E(C_n)-E(W)=nE(C_i)-400=3n-400$
Since all $C_i$ are independent, and $C_n$ and $W$ are independent: $Var(C_n-W)=Var(C_n)+Var(W)=nVar(C_i)+Var(W)=.3^2n+1600=.09n+1600$
After simplifying, I got $n>116.2$, therefore the $n$ at which the probability exceeds $.1$ is $117$.
Is this a valid proof, or did I assume too much by using the central limit theorum?