# Statistics

In a population, heights of males are normally distributed with u=180 cm and sigma^2=16 cm^2, while the heights of females are normally distributed with u=170 cm and sigma^2= 25 cm^2.
a) One random male and one random female are selected from the population
i)What is the probability that the male is more than 5 cm taller than the female?
ii)What is the distribution of the average of the two heights?

b)Now suppose that a random sample of 16 males and 16 females are selected from the
population.
i) What is the probability that the average height of the males is less than 178cm?
ii) What is the probability that the average height of the males is more than 12cm greater than the average height of the females?

So i said that X= height of one random male and Y= height of one random female. For this question I think I should find the z-score for both random variables and multiply the two probabilities since both variables are independent of each other. But I don't know how to get the probability of X+5>Y for a normal distribution and the other parts of the question, any help would be greatly appreciated, thanks.

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