Approximate distributions

Let \(\displaystyle X_1, ..., X_{35}\) be independent Poisson random variables having mean and variance 2.


Let \(\displaystyle Y_1, ..., Y_{15}\) be independent Normal random variables having mean 1 and variance 2.


(a.) Specify the (approximate) distributions of \(\displaystyle \bar{X}\).
(b.) Find the probability \(\displaystyle P(1.8 \leq \bar{X} \leq 2.3)\)
(c.) Specify the (approximate) distributions of \(\displaystyle \bar{Y}\).
(d.) Specify the (approximate distributions of \(\displaystyle \bar{X} + \bar{Y}\).
(e.) Find the probability \(\displaystyle P(Y_1 + ... + Y_{15} \leq 20)\).


My attempt,
(a.) Since \(\displaystyle n > 30\). I can use the central limit theorem to state \(\displaystyle \bar{X} \sim N(2, \frac{2}{35})\)


(b.) Apparently this one is incorrect and I'm unsure where my error is.
\(\displaystyle \sigma = \sqrt{\frac{2}{35}} = 0.24\)
\(\displaystyle P(-0.2/0.24 \leq Z \leq 0.3/0.24) = P(-0.84 \leq Z \leq 1.25)\)

(c.) \(\displaystyle \bar{Y} \sim N(1, \frac{2}{15})\)


(d.) \(\displaystyle \bar{X} + \bar{Y} ~N(35*2 + 15*1, 35*\frac{2}{35} + 15*\frac{2}{15}) = \sim N(50, 4)\)


(e.) \(\displaystyle P(Z \leq \frac{5}{\sqrt{30}}) = P(Z \leq 0.91)\)




Any insight on any part is greatly appreciated. Thanks.