Quality assurance is an mportant part of the production and a guard of company reputation. Quality control engineers work all year round, yet their role is of particular significance when the claims of inappropriate quality appear.
The current problem is associated with lack of soda in soda bottles. The company advertises 16 ounces; however, there are accusations that there are less soda in the bottles. The quality engineer has to run an investigation. Therefore, the plan of the research is as follow. It is definitely impossible to test all the produced bottles. The statistical methods address the issue. The bottles that are produced are population; the characteristics of population cannot be determined exactly. However, statistics offers reliable methods for assessing the parameters. For this, the sample is chosen from the population. Sample consist of the population members (soda bottles) that are randomly chosen from the population. The statistical science states that if the sample was chosen appropriately (random choice and the sufficient number of sample members), the characteristics of the sample are close to population’s characteristics (Triola, 2015). Therefore, 30 bottles were randomly chosen. This is the sample with n = 30 members.
1.
The characteristics of the sample are assessed. The characteristics necessary for the following analysis are mean and standard deviation. The mean is the average value of the bottles. Let us denote X as a real volume of soda, X is the mean value, and xi is measurement of each member of the sample. Then, the mean value calculates:
X=i=1nxin.
The standard deviation is a measure of dispersion. It measures how much the data are scattered around the mean and calculates (Triola, 2015):
s=i=1n(xi-x)2n-1.
For the studied case:
X=15.13+15.76+16.31++16.4930=15.812 ounces.
s=(15.13-15.81)2(15.13-15.81)2+(16.31-15.81)2+(16.49-15.81)220-1=0.512 ounces.
2.
The confidence interval assesses the range of dispersion of the mean. The confidence interval provides a range in which the mean value is contained with 95% assurance. The confidence interval is:
∆x=t(0.95;n-1)∙sn.
where t(0.95;29) is the Student’s criteria found in tables, t(0.95;29) = 2.04. (Triola, 2015)
∆x=2.04∙0.51230=0.191 ounces.
Therefore, the real value of the volume of soda in the bottle is between X±∆x. This is 15.812 ± 0.19. The volume of soda in the bottles changes from 15.622 to 16.003 ounces, or 95% of all the population values are within interval 15.622 and 16.003 ounces.
Another measure of central tendency is the median, which is obtained when all the values are arranged in an ascending order, and the central value is taken. In the studied case, the values are arranged and the mean value of 15th and 16th measurements are taken. The median is 15.755 ounces.
3.
- null hypothesis H0: the volume of soda equals 16 ounces, x=16;
- alternate hypothesis H1: the volume of soda does not equal 16 ounces, x<16.
The hypothesis is tested by calculation of the t-statistics:
t=x-Xsn=15.812-16.0000.512/30=2.004.
The hypothesis is accepted basing on the critical t-value, obtained from table for 0.05 significance level and 29 (n – 1, 30 – 1 = 29) degree of freedom (Triola, 2015). The tabulated critical value for one-tailed test is t(0.95;29) = 1.699. The one-tailed test is applied since the engineer is interested if the mean value is less than the advertised 16 ounces volume.
The null hypothesis is accepted if t < tcritical. Otherwise (when t > tcritical), the null hypothesis is rejected and the alternate is accepted. The calculated t is 2.004 > tcritical, then the engineer has to reject the null hypothesis and accept the alternate.
Alternatively, the p-value can be obtained from the statistical software by automatic running the t-test. Mini tab software provides the p-value = 0.027. If p-value > 0.05, the null hypothesis is accepted, if p < 0.05, the alternate hypothesis is accepted. Since 0.027 < 0.05, the null hypothesis is rejected, and the alternate is accepted. The result confirms the analysis based on critical t-value.
This means that the claim about the fact that the volume of soda in the bottle is confirmed. There is enough statistical evidence that the volume is less than the advertised value.
4.
The statistical analysis indicated that there is lack of soda in the bottles. This is the evidence for the quality engineer to run production analysis and search for technological (or any other) reasons for volume deficit. Here is the list of the possible reasons.
1. The technical tools responsible for soda volume have to tested. Possibly, the devise adds smaller amount of soda. It is either programmed in a wrong way, or programming settings have busted.
2. If the bottles are filled to the certain line, then the manager has to test the bottles volume. Possibly, the bottles can contain less soda than 16 ounces. The bottles supplier has to be changed.
3. In addition, it is possible that more sugar was added to soda, and the density increased. If the device measures weight, the volume will be less. The technological parameters have to be tested.
The technical investigation of the production has to be run: dosage settings adjusted, density tested, recipes corrected, physical volume of bottles tested. In the future, the periodical volume control (once a month) should be performed.
The managerial advice to the company is to advertise the volume of soda as 16 ounces ± 3%. Therefore, the minor random fluctuations of volume will not raise the lack volume claims.
References
Triola, M. F. (2015). Essentials of statistics.
