A Soda Bottling Glitch

Due to complaints that bottles of soda labeled 16 ounces did not include the full 16 ounces, a statistical analysis was completed. Bottling staff randomly chose 30 filled bottles of soda and then measured the amount of soda in each bottle. The results are as follows:

The Mean, or average, amount of soda in the Sample of 30 bottles was 14.87 ounces. The Median amount of soda in the Sample was 14.8 ounces. The Standard Deviation of the Sample was .566 ounces.

In order to determine how likely it is that these measurements reflect the population - all the bottles of soda - a confidence interval was calculated. When we use a 95% confidence level, we are saying that we have calculated the correct interval within which the true mean lies, in 95% of all possible samples. In this case, with a Standard Deviation of .566, a 95% confidence level would range from a lower value of 13.76 and upper value of 15.98 (Mean + (1.96) (.566), as 1.96 represents the outer limits of the 95% confidence interval (Brase & Brase, 2012).

Management is now concerned that the true mean of soda in the bottles is lower than 16 ounces, and they decide to do a lower tailed hypothesis test. A lower tailed hypothesis test is appropriate where the alternative hypothesis is that the population mean is lower than the hypothetical mean (Wilcox, 2009).

The null hypothesis is Ho: Mean > 16 and the alternative hypothesis is H1: Mean < 16

Based on the research variables, the test statistic is computed using the following formula:

Z = Mean – Ho 1.13s/Square root of n .566/5.47 with n = 30, s = .566

Z, therefore, equals -10.97, significantly lower than even the .0001, or 99.999 confidence interval Z score of 3.719 (Hypothesis Testing, 2013).

It is, therefore, appropriate to reject the null hypothesis that the Mean amount of liquid in the population of soda bottles at the plant is > 16 ounces.

With such an extreme discrepancy between the amounts of soda each bottle should hold, the amounts of soda they hold, and extrapolating from the above statistical testing, management must now set about figuring out the reason for the shortfall. One such cause may be a lack of rigor or accuracy in measuring the soda in each of the 30 bottles that were part of the sample. If the conclusion reflects that it is highly likely that the mean of 95% of any other samples would be less than 16 ounces, colloquially, that would translate to a very high proportion of the bottles that are short soda. However, if the sample statistics were unreliable the conclusion would be called into question (Wilcox, 2009). The samples would need to be rigorously measured for accuracy for the data collected to also be accurate.

Secondly, the bottles may not have been randomly selected. They could have been selected from one specific section of the plant or have been handled by specific workers who may not have been careful with the amounts that they filled the bottles with. The sample would therefore not be random, and the results would be skewed and therefore unreliable.

If the bottles are mechanically filled, there would probably be one or more machine settings that determine how much soda each bottle is filled with. This might need to be re-calibrated or reset if it is not currently correct. A random sampling of soda output from the machines can be used to determine if they are calibrated properly. For example, 1 in every 10 bottles can be sampled for their volume.

Lastly, the bottles may be purposely filled less than the printed amount. While it may not appear to a significant difference of volume between 14.87 ounces and 16 ounces, in time-study experiments, that would add up to a great deal of soda saved, and subsequently, a great deal of savings for the company.

Assuming that management is not purposefully shorting their customers, and since the anecdotal evidence – complaints from multiple customers – is in line with the statistical evidence, the company should immediately take the following steps. First, since the results of the first statistical study were significantly skewed, it would best to do another random sample, with the bottles chosen by different workers, with a supervisor assuring the randomness of the sample. If the results are like the first analysis, mechanical settings should be reviewed and adjusted, if necessary. If such adjustments are done, a third random sample should be chosen, and another statistical analysis completed. If the third analysis is like the first two, indicating that most bottles are being shorted, it is likely that more complicated mechanical problems must be investigated.

Assuming, however, that changes in settings or mechanical repairs do remediate the problem, certain plans should be put in place, to prevent such problems from occurring in the future. These plans should include a schedule of calibration checks and mechanical checks, with such checks recorded in a logbook so there is evidence that they have been done. The company should also institute a regular schedule of random sampling of filled bottles to ensure that the shorting does not re-occur, or to catch it immediately if it does.

Lastly, in terms of public relations, given that there had been customer complaints, the company may want to create a public relations campaign around the issue. For example, the company could publicize an 800 number, and explain that there was a problem, apologize, and assure the customer that the company is working to resolve the mistake. Then the company could suggest that if any customer purchases a bottle that he or she believes is short of 16 ounces of soda, they should immediately call the 800 number and the issue will be addressed.

References

Brase, C.H. & Brase, C.P. (2012). Understanding basic statistics. Boston: Brooks/Cole.

Hypothesis Testing (2013). Boston: Boston University School of Public Health. Accessed 18 December 2013. http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BSBS704_HypothesisTest-Means- Proportions/BS704_HypothesisTest-Means-Proportions3.html

Wilcox, R. (2009). Basis statistics: Understanding conventional methods and modern Insights. New York: Oxford University Press.