Case 7 Quality Associates

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Quality Associates, Inc. is a consulting firm who advises its clients about statistical and sampling methods that can be used to control their manufacturing procedures. In this particular case we consider a production line designed to fill bottles of a shampoo with a mean weight of 12 ounces of shampoo per bottle. Quality Associates, Inc. made a quality control testing of the manufacturing machine of this client (Alibaba Machinery), to determine if the process is operating properly or if, perhaps, a machine malfunction has caused the process to begin underfilling or overfilling the bottles. The client picked a sample of 800 bottles taken during a time when the machine was operating satisfactorily. The sample standard deviation for these data was 0.21; and thus (with so much data) the population standard deviation was assumed to be 0.21. Then Quality Associates recommended that random samples of size 30 should be taken periodically to monitor the machine operation on an ongoing basis. By analyzing the new samples, Alibaba Machinery could quickly learn whether the machine is operating satisfactorily. If the machine is not operating satisfactorily, corrective action can be taken to eliminate the problem. The design specification indicated that the mean weight of the bottles of shampoo should be 12 fluid ounces (fl.oz.).

The manufacturing company who created the machine claimed that it has been set such that the mean weight of the bottles of shampoo is 12 fl.oz. Furthermore, the production process has been designed so that each bottle of shampoo is filled independently. The hypothesis test suggested by Quality Associates is:

Ho: µ = 12
Ha: µ≠12
Corrective action will be taken anytime Ho is rejected.

EXECUTIVE SUMMARY (main findings/issues/assumptions):

The quality control design calls for taking a sample of 30 bottles at hourly intervals for a day operation. Four samples are collected at the following times: (i) morning (9am – 10am);
(ii) mid-morning (11am – 12pm);
(iii) mid-afternoon (1pm – 2pm); and
(iv) afternoon (3pm – 4pm).
Sampling method are the probability sampling techniques known as Cluster Sampling (p.295) and Systematic Sampling. The working day is divided into ‘clusters’ depending on the time of the day in which the production process is occurring. During each of the chosen clusters (which are purposefully chosen to be at hourly intervals), a sample of only 30 are selected, using systematic sampling method. Target population is (p.275): the population we want to make inferences about. Sampled population is: the population from which the sample is actually taken. After each of the four samples have been analysed individually, some analysis is also undertaken for the overall 120 observations as a whole. This is for comparative purposes, since it is expected that as the size of the sample increases, the sample mean should be closer to the population mean. ASSUMPTIONS:

Some variation in weights of the bottles of shampoo is expected. This expected variation is due to a number of factors, which may include the temperature of the machine, variations in the thickness of the shampoo, …???..... It is assumed that the population standard deviation is 0.21 fl.oz. Furthermore, it is assumed that the sampling distribution of the sample mean can be approximated by a normal distribution because the sample size of each sample is size 30. This assumption is based on the Central Limit Theorem, which states that in selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large.

Moreover, the assumption is made that lunch time at Alibaba Machinery is from 12 – 1pm, during which time the machine is turned off....
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