You are recruiting 25 space-travellers to populate a new planet many light years away. The travellers will need to be cryogenically frozen and to fit in the cryogenic pods they will need to be between 165cm and 198cm tall. (Show all working)

1. Amongst the species to which you belong height can be approximated by a normal distribution with mean 180cm and standard deviation 10cm. What is the probability that a randomly selected potential traveller will fall within the permitted height range?
2. What is the probability that a randomly selected group of 25 travellers will all be within the permitted height range?
3. The maximum total weight of the 25 travellers must not exceed 2000kg meaning that their average weight should not be above 80kg. If individuals' weight is described by a normal distribution with mean 78kg and standard deviation 4kg what is the distribution of the mean weight of a sample of 25?
4. What is the probability that a randomly chosen group of 25 travellers is underneath the weight limit?
5. A colleague of yours suggests that to get the overall probability of a group of 25 travellers meeting both the height and weight constraints you should multiply the probability in (2) with the probability in (4). Do you agree? Why or why not?