A company manufactures dog food packaged in bags labeled "30 pounds." The amount of dog food that is put into a bag is a random variable that can be described by a normal distribution that has a standard deviation theta = 1 pound. the mean amount of dog food going into the bag is controlled by operators and the target value is μ = 32 pounds, to assure that most customers get at least 30 pounds in their bag. It is impossible for the dog food filling machine to go "out of control" so that the mean amount being put into the bag is more than or less than 32 pounds. In order to protect against such shifts in the mean, each hour five bags are weighed and x̄, the sample mean bag is computed. Let x̄ denote the random variable sample mean before the five bags are chosen. Assume that the bag-filling machine is operating correctly so that μ = 32 pounds.
Is the mean of the machine shifts to μ = 29 pounds what proportion of bags will contain less than 30 pounds of dog food? Report both the z-score and the corresponding probability.