a) You find P[X > 320] = P[Z > (320-330) / (80/sqrt(40))] = 1 - Phi(-0.79) = Phi(0.79) = .7852
If you don't know how that Z and Phi got involved, you're in trouble with this problem.
b) Find P[ 320 < X < 350 ]
c) Find P[X > 350]
b,c and d) I'm not sure if that sample stdev=2.44 is supposed to play a part in the calculations...I'm not sure how to reconcile it with the sample mean=6.05. Note that the stdev of a Poisson distribution is equal to the square root of the mean (and 2.44 is nearly the sqrt(6.05)).
If I had to write an answer down, I'd say sample mean = 6.05, sample stdev = sqrt(6.05) = 2.46, then use the formula for confidence interval:
(sample mean) +/- z[alpha] * (sample stdev) / sqrt(40)