Here is the problem I'm trying to solve:
The central limit theorem can be used to analyze round-off error. Suppose that the round-off error is represented as a uniform random variable on [−1,1].
(i) Suppose 100 numbers are added. Use the CLT to approximate the probability that the total round-off error exceeds 7 in absolute value (that is, greater than 7 or less than -7).
(ii) Suppose 100 numbers are added. Use Chebyshev’s inequality to obtain an upper boundon the probability that the total round-off error exceeds 7 in absolute value.
(iii) Which is closer to the true probability, the approximation in part (i) or the upperbound in part (ii)? Explain.
So far part 1 I let the round off error sum of 100 numbers be represented by random variable Y. I then got $E(Y) = \sum_{i=1}^{100}E(X_i) = 0$ since $E(X_i) = 0$ for all $X_i$ in the interval. Likewise I got, $Var(Y) = \sum_{i=1}^{100}Var(X_i) = 100(\frac{1}{3}) = \frac{100}{3}$ since $Var(X_i) = \frac{1}{3}$ for all $X_i$ in the interval. I then approximated the probability that the total round-off error exceeds 7 in absolute value using the CLT. $P(Y > 7) = 1 - P(Y \leq 7) = 1 - pnorm(\frac{7-0}{\sqrt{\frac{100}{3}}}) = .1127$ in R.
I don't really understand part II. Can someone give me advice? Thanks in advance!