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. ). PROBLEM/ISSUE: 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): (SUMMARY & HIGHLIGHTS OF OUR FINDING) – FOR US TO WORK ON Data/Facts/Issues? METHODOLOGY 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. 1 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.
If Ho is rejected, further investigation will be required. The acceptable weight limit is within 11. 90fl. oz. and 12. 10fl. oz. LIMITATIONS The quality control inspection in this case study does not consider the quality of the shampoo which is being poured into the bottles; this is assumed to be constant. Moreover, the quality of the bottle (i. e. no defect) and proper labeling are not considered in this study, but also assumed to be constant. Another limitation may be the systematic sampling method used to collect each sample.
One of the conditions that should be satisfied when a random sample is selected from an infinite population is that each element selected comes from the same population. To ensure that this condition is satisfied, the bottles must be selected at approximately the same point in time (p. 270). This is so that the inspector avoids the possibility of selecting some bottles when the machine is operating properly, and other bottles when the machine is not operating properly and is underfilling or overfilling the bottles.
In the methodology assumed to be used for this case study, the 30 bottles selected for each sample are selected within a period of one hour. PRELIMINARY FINDINGS: (OUR DESCRIPTIVE STATISTICS —– TABLES & GRAPHS Analysis & Interpretation of table & graphs in relation to solving the issue raised at the start. Box plots (p. 112): (i) Sample 4 has the highest bottle weights. The lowest bottle weights are in Sample 3. (ii) Sample 4 has the highest median weight, followed by Sample 2, and then Samples 1 and 3 show the lower median weights. iii) No outliers??? (iv) Samples 1 and 2 appear to have the most variation, while Samples 3 and 4 have less variation. (v) For Sample 3, the median is higher than the mean. For all of the other three samples, the mean and median are equivalent (to 2 decimal places). The mean is a type of average which is affected by every observation in the sample, whereas the median is not. Thus the relatively low mean is an indication that there are low-lying weights which have affected the mean.
Observing the data shows that sample 3 has the lowest weight out of all the samples, which is 11. 36 fl. oz. STATISTICAL FINDINGS: (OUR HYPOTHESIS TESTING… SUPPORTED WITH TABLES & GRAPHS For this hypothesis test a two-tailed test is required, about a population mean, with the population standard deviation known (p. 361). This is because Alibaba Machinery does not want to either underfill or overfill its shampoo. Underfilling may result in a complaint from customers or a government agency, and overfilling would be an unnecessary cost to the company.
According to the said hypothesis test, the null hypothesis is to be rejected if p-value =a. The test was calculated for each sample at the 0. 01 level of significance, as follows: Insert tables Step by Step, up to Decision Analysis | Sample 1| Sample 2| Sample 3| Sample 4| Total| Result| p-value = a| p-value = a| p-value = a| p-value = a| p-value =a| Analysis| Accept H0| Accept H0| Reject H0| Accept H0| Accept H0| It is noted that the p-value is less than the level of significance for all of the samples except for sample 3.
As such, further investigation should be made by the company to determine the reason why the weights were significantly less than 12 fl. oz. , and corrective action should be taken to eliminate the problem. The problem may not be due to malfunctioning of the machine, since the machine operated satisfactorily again in Sample 4. There are various possible reasons why the weights of the bottles of shampoo in Sample 3 were significantly less than 12 fl. oz. One reason may be the following: * the shampoo itself may have been a thinner mixture.
This may be because a new container of shampoo was started to be used for filling the bottles, at 1pm, and it was not given a shake/stir before the bottle-filling started. * The machine was turned off over lunch time (12pm – 1pm) and when it was turned on, for some reason it is operating slower so it underfills the bottles during part of the first hour (i. e. from 1pm – 2pm). * MAJOR FINDINGS Highlight of our findings MANAGERIAL IMPLICATIONS/RECOMMENDATIONS Highlights/brief the importance of quality control to the success of the business, support with statistical examples Our recommendation: