A bottling company bottles soda and beer and distributes the products in the surrounding area.

The company has four bottling machines, which can be adjusted to fill bottles at any mean fill level between 2 ounces and 72 ounces. The machines exhibit some variation in actual fill from the mean setting. For instance, if the mean setting is 16 ounces, the actual fill may be slightly more or less than that amount.

Three of the four filling machine are relatively new, and their fill variation is not as great as that of the older machine. Don has observed that the standard deviation in fill for the three new machines about I% of the mean fill level when the mean fill is set at 16 ounces or less, and it is 0.5% of the mean at settingsexceeding 16 ounces. The older machine has a standard deviation of about 1.5% of the mean setting regardless of the mean fill

setting. However, the older machine tends to underfill bottles more than overfill, so the older machine is set at a mean fill slightlin excess of the desired mean to compensate for the propensity to underfill. For example. when 16 ounce bottles are to be filled, the machine is set at a mean fill level of 16.05 ounces.

The company can simultaneously fill bottles with two brands of soda using two machines. and it can use the other two machine to bottle beer. Although each filling machine has its own warehouse and the products are loaded from the warehouse directly onto a truck, products from two or more filling machines may be loaded on the same truck. However, an individual store almost always receives bottles on a particular day from just one machine.

On Saturday morning the owner of the bottling company received a call at home from the grocery store manager. She was very upset because the shipment of 16 ounce bottles of beer received yesterdaycontained several bottles that were not adequately filled. The manager wanted the owner, Don, to replace the entire shipment at once.

Don gulped down his coffee and prepared to head to the store to chock out the problem. He started thinking how he could determine which machine was responsible for the problem. If he could at least determine whether it was the old machine or one of the new

ones, he could save his maintenance people a lot of time and effort checking all the machines.

His plan was to select a sample of 64 bottles of beer from the store and measure the content. Don figured that he might be able to determine, on the basis of the average content, whether it was more likely that the beer was bottled by a new machine or by the old one.

The results of the sampling showed an average of 15.993 ounces. Now Don needs some help in determining whether a sample mean of 15.993 ounces or less is more likely to come from the new machines or the older machine.

i'm confused as to how to calculate the probability that each machine filled the beer to 15.993 oz or below. i need the formulas worked out so i can compare my findings. i calculated the standard deviation of the old machines as 1.5% of mean (16.05)=.24075. Is that the correct sd, or is it .015?