Veronica Jules owns and operates the Dancer Bottling Company in Springfield Massachusetts. The company bottles soft drinks and beer and…

Veronica Jules owns and operates the Dancer Bottling Company in Springfield Massachusetts. The company bottles soft drinks and beer and distributes the products in the surrounding communities..
The company has four bottling machines, which can be adjusted to fill bottles at any mean level between 2 ounces and 72 ounces. The machines exhibit some variation in actual fill from the mean setting. For example, if the mean setting is 16 ounces, the actual fill may be slightly more or less than that amount.
Three of the four filling machines are relatively new, and their fill variation is not as great as that of the older machine. Veronica has observed that the standard deviation in fill for the three new machines is about 1% 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 settings exceeding 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 slightly in 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 two brands of soft drinks using two machines, and it can use the other two machines to bottle beer. Although each filling machine has its own warehouse and the products are loaded from the warehouse directly on a truck, products from two or more machines can be loaded on the same truck. However, an individual customer almost always receives bottles on a particular day from just one machine.
On Saturday morning Veronica received a call at home from the M.L.H. Grocery store manager. The manager was upset because the shipment of 16-ounce bottles of beer she received yesterday contained several bottles that were not adequately filled. The manager wanted Veronica to replace the entire shipment at once.
Veronica rushed through breakfast and was about to head down to the store to investigate the problem. She started to think about how she could find out which filling machine was responsible for the problem. If she could at least determine whether it was the old machine or one of the newer machines, she could save her maintenance people a lot of time and effort checking all the machines.
Her plan was to select a sample of 64 bottles of beer from the store and measure the contents. Veronica figures that she might be able to determine, on the basis of average contents, whether it is the 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 Veronica 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

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