I've part of a solution I need help grasping. First, here's the question:
"Let the average lifetime of certain electronic components be 6 months, and the standard deviation of the lifetime equal to 2 months. Suppose now that the components are independent, but that the distribution of their lifetime is unknown, and we only known the mean and standard deviation of the lifetime. We consider a machine made up of n components of this type placed in parallel. Furthermore, only one component is active at a time (standby redundancy). Use the central limit theorem to find the smallest value of n for which the probability that the machine functions during at least 15 years is greater than 90%."
Now the solution we were given is:
"Let S(n)= T1+T2+...+Tn, where Ti denotes a lifetime. If n is large enough, by the central limit theorem, we can write that S(n) ~~ N(n*6,n*4). We seek n(min) such that P[S(n)>= 180]>0.90..." (By the way ~~ "means approximately equal to")
So, my question
Why are both our mean and standard deviation multiplied by n in N(6*n,4*n)? I know, of course, why 6 and 4 are there (as they represent mu and sigma squared respectively). Is it something to do with the fact that we take n samples?