BOTTLING COMPANY CASE STUDY 8

BottlingCompany Case Study

Standarddeviation, median and mode

Bottle No. | Ounces | Deviations (x-15.85) | (x-15.85)^2 |

1 | 14.23 | -1.62 | 2.6244 |

2 | 14.32 | -1.53 | 2.3409 |

3 | 14.98 | -0.87 | 0.0289 |

4 | 15 | -0.85 | 0.7225 |

5 | 15.11 | -0.74 | 0.5476 |

6 | 15.21 | -0.64 | 0.4096 |

7 | 15.42 | -0.43 | 0.1849 |

8 | 15.47 | -0.38 | 0.1444 |

9 | 15.65 | -0.20 | 0.0049 |

10 | 15.74 | -0.11 | 0.0121 |

11 | 15.77 | -0.08 | 0.0064 |

12 | 15.80 | -0.05 | 0.0025 |

13 | 15.82 | -0.03 | 0.0009 |

14 | 15.87 | 0.02 | 0.0004 |

15 | 15.98 | 0.13 | 0.0169 |

16 | 16 | 0.15 | 0.0225 |

17 | 16.02 | 0.17 | 0.0289 |

18 | 16.05 | 0.20 | 0.04 |

19 | 16.21 | 0.36 | 0.1296 |

20 | 16.21 | 0.36 | 0.1296 |

21 | 16.23 | 0.38 | 0.1444 |

22 | 16.25 | 0.40 | 0.16 |

23 | 16.31 | 0.46 | 0.2116 |

24 | 16.32 | 0.47 | 0.2209 |

25 | 16.34 | 0.49 | 0.2401 |

26 | 16.46 | 0.61 | 0.3721 |

27 | 16.47 | 0.62 | 0.3844 |

28 | 16.51 | 0.66 | 0.4356 |

29 | 16.91 | 1.06 | 1.1236 |

30 | 16.96 | 1.11 | 1.2321 |

Total | 475.62 | 11.9227 |

Mean= 446.1 /30

=15.85

Medianwhen arranged in an ascending order, the ounces for the 30 bottlesare as follows

14.23 |

14.32 |

14.98 |

15 |

15.11 |

15.21 |

15.42 |

15.47 |

15.65 |

15.74 |

15.77 |

15.8 |

15.82 |

15.87 |

15.98 |

16 |

16.02 |

16.05 |

16.21 |

16.21 |

16.23 |

16.25 |

16.31 |

16.32 |

16.34 |

16.46 |

16.47 |

16.51 |

16.91 |

16.96 |

Therefore,the median = (15.98 + 16) /2

=15.99

Variance= 11.9227 /(30-1)

=0.4111

Standarddeviation = 0.4111

=0.64

95%confidence interval

Thenormal distribution value for the 95% confidence interval from thestatistics table is 1.96 this value is critical for thedetermination of the upper and the lower limits of a given interval(Moore, 2008). The lower limit will be given by subtracting 1.96 fromthe ounces of the soda bottles. On the other hand, the upper limit ofthe interval will be obtained by adding 1.96 to the ounces of thebottles. The 95% confidence intervals for the bottles is as providedin the following table

Bottle No. | Ounces | Lower Limit | Upper Limit |

1 | 14.23 | 12.27 | 16.19 |

2 | 14.32 | 12.36 | 16.28 |

3 | 14.98 | 13.02 | 16.94 |

4 | 15 | 13.04 | 16.96 |

5 | 15.11 | 13.15 | 17.07 |

6 | 15.21 | 13.25 | 17.17 |

7 | 15.42 | 13.46 | 17.38 |

8 | 15.47 | 13.51 | 17.43 |

9 | 15.65 | 13.69 | 17.61 |

10 | 15.74 | 13.78 | 17.7 |

11 | 15.77 | 13.81 | 17.73 |

12 | 15.80 | 13.84 | 17.76 |

13 | 15.82 | 13.86 | 17.78 |

14 | 15.87 | 13.91 | 17.83 |

15 | 15.98 | 14.02 | 17.94 |

16 | 16 | 14.04 | 17.96 |

17 | 16.02 | 14.06 | 17.98 |

18 | 16.05 | 14.09 | 18.01 |

19 | 16.21 | 14.25 | 18.17 |

20 | 16.21 | 14.25 | 18.17 |

21 | 16.23 | 14.27 | 18.19 |

22 | 16.25 | 14.29 | 18.21 |

23 | 16.31 | 14.35 | 18.27 |

24 | 16.32 | 14.36 | 18.28 |

25 | 16.34 | 14.38 | 18.30 |

26 | 16.46 | 14.50 | 18.42 |

27 | 16.47 | 14.51 | 18.43 |

28 | 16.51 | 14.55 | 18.47 |

29 | 16.91 | 14.95 | 18.87 |

30 | 16.96 | 15 | 18.92 |

Hypothesistest

Theclaim being tested is whether a bottle contains less than 16 ounces.Therefore, the null and alternative hypothesis would be stated asfollows

NullHypothesis: a bottle contains less than 16 ounces

AlternativeHypothesis: a bottle contains 16 ounces

Letα = 0.05 therefore, from the z-table, z critical value is 1.96.Hence, in case z < -1.96, or > 1.96, the null hypothesis shouldbe rejected (Anderson et al., 2012).

Calculationof z-statistic z = (15.85 – 16) / 0.64

=-0.15 / 0.64

Z= 0.2344

Conclusionz (0.2344) is less than 1.96, which implies that the null hypothesisshould be accepted. Therefore, the alternative hypothesis isrejected. Hence, it can be concluded that a bottle contains less than16 ounces.

Fromthe conclusion, it is apparent that, in a bottle of soda, there areless than 16 ounces. One of the likely causes for having a bottle ofsoda with less than 16 ounces may emanate from faulty measuringdevices. The measuring devices may be faulty such that, whenmeasuring the exact content, the devices deviate from providing theright measurement. The measuring devices may be faulty, in this case,resulting in less than the required measurements. The plant can avoidthe deficit in the soda bottles through ensuring that all themeasuring devices used in the plant are regularly inspected to checkif they are faulty. Also, old measuring materials should be replacedwith new ones to avoid the problem of inaccuracy in the future.

Anotherprobable reason for having bottles of soda having less than 16 ouncesmay be due to leakage. When preparing and packaging the soda bottles,there may be leakages originating from improper handling of the sodabottles by the employees. This problem can be rectified to avoid adeficit in the future by the plant ensuring that there is properhandling of soda bottles during preparation and packaging. This willaid in mitigating leakages in the processes. Furthermore, a deficitin the soda bottles may be because of incompetent employees in theplant unqualified employees may not have the skills of measuring theright quantities, which may lead to the deficits. This problem can becorrected through the plant ensuring that it has qualifiedindividuals that are keen on the issue of accurate measurement. Also,employees dealing with issues of measurements should regularly betrained. This will ensure that the deficit is avoided in the future.

References

Anderson,D. R., Sweeney, D. J., & Williams, T. A. (2012). *Statisticsfor business and economics*.Mason, Ohio: South-Western Cengage Learning.

Moore,D. S. (2008). *Thebasic practice of statistics*.New York: W.H. Freeman and Co.