You have a factory with 40 production machines that are essentially identical, each producing at a mean daily rate of 100 products with a standard deviation of 15. You may assume that they produce independently of one another. Consider the average daily production per machine tomorrow, which is a random variable.
a. Find the mean of this random variable. Compare it to the mean for a single machine.
b. Find the standard deviation of this random variable. Compare it to the standard deviation for a single machine.
c. What is the approximate probability distribution of this random variable? How do you know?
d. Find the (approximate) probability that your average daily production per machine will be more than 102 products tomorrow.
e. Find the (approximate) probability that your average daily production per machine will be between 97 and 103 products tomorrow.