Don Carpita owns and operates Carpita Bottling Company in Lakeland, Wisconsin. The company bottles soda and beer and distributes the products in the counties surrounding Lakeland. 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 machines 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 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 set-ting. 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 bottles with two brands of soda using two machines, and it can use the other two machines to bottle beer. Although each filling machine has its own ware-house 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, Don received a call at home from the J. R. Summers grocery store manager. She was very upset because the shipment of 16-ounce bottles of beer received yesterday contained several bottles that were not adequately filled. The manager wanted Don to replace the entire shipment at once. Don gulped down his coffee and prepared to head to the store to check out the problem. He started thinking how he could deter-mine 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 contents. Don figured that he might be able to determine, on the basis of the average contents, 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
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Don Carpita owns and operates Carpita Bottling Company in Lakeland, Wisconsin. The company bottles soda and beer and distributes the products in the counties surrounding Lakeland.
The company has four bottling machines, which can be adjusted to fill bottles at any
Three of the four filling machines 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 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 set-ting. 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 bottles with two brands of soda using two machines, and it can use the other two machines to bottle beer. Although each filling machine has its own ware-house 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, Don received a call at home from the J. R. Summers grocery store manager. She was very upset because the shipment of 16-ounce bottles of beer received yesterday contained several bottles that were not adequately filled. The manager wanted Don to replace the entire shipment at once.
Don gulped down his coffee and prepared to head to the store to check out the problem. He started thinking how he could deter-mine 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 contents. Don figured that he might be able to determine, on the basis of the average contents, 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.
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
