Probability/Statistics question URGENT!

The question says:
In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. Furthermore, there is a weight limit of 2500 lb. Assume that the average weight of students, faculty, and staff on campus is 154 lb, that the standard deviation is 29 lb, and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken:

I am stuck on this part
(d) What is the chance that a random sample of 16 persons on the elevator will exceed the weight limit?

Thanks a lot

You need to use the property that the sum of n normal distributed variables has :
1) also a normal distribution
2) as mean the sum of the means
3) has as variance the sum of the variances
so we take 16 times N(154,29)
=> N(154*16, sqrt(16*29²))
= N( 2464, 116)
we have a limit of 2500 lb
=> P(sum > 2500) ?
we go to normalised normal distribution to calculate the z value
z = (2500-2464)/116 = 36/116 = 9/29
we look up this value in a table for z values
z = 9/29 = 0.310345
P[z < 0.310345] = 0.6217
=> P[sum < 2500] = 0.6217
=> P[sum > 2500] = 1-0.6217=0.3783
37.83 %