# Using the (strong) law of large numbers problem

this is a practice problem I am doing. It says "Suppose that in a community the distributions of heights of men and women in centimeters are N(173, 40) and N(160, 20) respectively. Calculate P that the average height of 10 randomly selected men is at least 5 centimeters larger than the average height of six randomly selected women." I am tempted to say that I should just look at the probability that height of 1 randomly selected man is at least 5 cm greater than height of 1 randomly selected woman, but I am struggling to make this rigorous. Help appreciated! P.S. Central Limit Theorem might be used here too, this is a chapter review question so not sure what exactly can be useful.

Consider the fact that from a normal distribution $N(\mu, \sigma)$, the distribution of means from samples of size $n$ is $N(\mu, \sigma/\sqrt{n})$. We can use this to make new distributions for the mean height of 10 random men, and the mean height of 6 random women; then consider the fact that the distribution for a random sample from $N(\mu_1, \sigma_1)$ minus a random sample from $N(\mu_2, \sigma_2)$ is $N(\mu_1-\mu_2, \sqrt{\sigma_1^2 + \sigma_2^2})$.
• I got to that last step, from there, do I just find the probability that $x \geq 5?$ Using normal distribution z tables etc. – prboq Nov 18 '11 at 6:15
• Precisely!  – smackcrane Nov 18 '11 at 6:16