Regular and large bags of coffee beans are independently filled at a factory.
The weights of coffee in regular bags, RRR, are normally distributed with mean 250 g and standard deviation 6 g.
The weights of coffee in large bags, LLL, are normally distributed with mean 480 g and standard deviation 10 g.
The random variable W W\,W represents the total weight of coffee in 2 randomly selected regular bags minus the weight of coffee in 1 randomly selected large bag.
W∼N(a,b)W \sim \text{N}(a, b)W∼N(a,b) where a a\,a and b b\,b are positive constants.
Find the value of a a\,a and the value of bbb.
Find the probability that a randomly chosen large bag contains more than 1.9 times the amount of coffee in a randomly chosen regular bag.
A random sample of 3 regular bags is taken.
Find the probability that the weight of the first regular bag in the sample is at least 4 g more than the mean weight of all 3 regular bags in the sample.