Suppose that a particular production process fills detergent in boxes of a given size. Specifically, this process fills the boxes with an amount of detergent (in ounces) that is normally distributed with mean 50 and standard deviation 0.5.

a. Simulate this production process for the filling of 500 boxes of detergent. Find the mean and standard deviation of your simulated sample weights. How do your sample statistics compare to the theoretical population parameters in this case? How well do the empirical rules apply in describing the variation in the weights in your simulated detergent boxes?

b. A box of detergent is rejected by quality control personnel if it is found to contain less than 49 ounces or more than 51 ounces of detergent. Given these quality standards, what proportion of all boxes are rejected? What step(s) could the supervisor of this production process take to reduce this proportion to 1%?