The relative frequency distribution of the weights of all passengers using an elevator is mound-shaped, with mean $150$pounds and standard deviation $35$pounds. Suppose there are n passengers in the elevator, describe and explain the probability distribution of the total weight of n passengers in the elevator.
So, here's how I describe the distribution:
I let $X$ be the total weight of $n$ passengers.
I think that if $n < 30$, then $X$ follows T-Distribution with $n-1$ degrees of freedom.
If $n \geq 30$, by CLT, then $X$ follows a Normal Distribution.
I am also making an assumption that the weights of all passengers are normally distributed.
However, in the later parts of the same question, it asks to show the probability that the total weight of $12$ passengers in the elevator not more than $2000$ is $0.0495$. The only way to get the probability of $0.0495$ apparently is through a normal distribution. If I follow the way I described the distribution above, since $n = 12 < 30$, it should follow a T-Distribution, which yields the probability of $0.064$. But this is not a number the question wanted me to show.
So, I suspect that the way I am thinking about the distribution is wrong. What is wrong with the way I think about the probability distribution of this scenario?