# Statistics

STATISTICS 9

Calculationof standard deviation, median and mode

 Bottle No. Ounces (x-14.87) (x-14.87)^2 1 14.5 -0.37 0.1369 2 14.6 -0.27 0.0729 3 14.7 -0.17 0.0289 4 14.8 -0.07 0.0049 5 14.9 0.03 0.0009 6 15.3 0.43 0.1849 7 14.9 0.03 0.0009 8 15.5 0.63 0.3969 9 14.8 -0.07 0.0049 10 15.2 0.33 0.1089 11 15 0.13 0.0169 12 15.1 0.23 0.0529 13 15 0.13 0.0169 14 14.4 -0.47 0.2209 15 15.8 0.93 0.8649 16 14 -0.87 0.7569 17 16 1.13 1.2769 18 16.1 1.23 1.5129 19 15.8 0.93 0.8649 20 14.5 -0.37 0.1369 21 14.1 -0.77 0.5929 22 14.2 -0.67 0.4489 23 14 -0.87 0.7569 24 14.9 0.03 0.0009 25 14.7 -0.17 0.0289 26 14.5 -0.37 0.1369 27 14.6 -0.27 0.0729 28 14.8 -0.07 0.0049 29 14.8 -0.07 0.0049 30 14.6 -0.27 0.0729 Total 446.1 8.783

Mean= 446.1 /30

=14.87

Medianwhen arranged, the ounces for the 30 bottles are as follows

 14 14 14.1 14.2 14.4 14.5 14.5 14.5 14.6 14.6 14.6 14.7 14.7 14.8 14.8 14.8 14.8 14.9 14.9 14.9 15 15 15.1 15.2 15.3 15.5 15.8 15.8 16 16.1

Therefore,the median = (14.8 + 14.8) /2

=14.8

Variance= 8.783 /(30-1)

=0.302862

Standarddeviation = 0.302862

=0.55

95%confidence interval

Thenormal distribution value for the 95% confidence interval from thestatistics table is 1.96 this value is vital in determining theupper and the lower limits of a given interval (Moore, 2008). Thelower limit will be given by subtracting 1.96 from the soda bottleounces. On the other hand, the upper limit of the interval will begiven by adding 1.96 to ounces of the bottle. 95% confidenceintervals for the bottles table

 Bottle No. Ounces Lower Limit Upper Limit Interval 1 14.5 12.54 16.46 12.54 to 16.46 2 14.6 12.64 16.56 12.64 to 16.56 3 14.7 12.74 16.66 12.74 to 16.66 4 14.8 12.84 16.76 12.84 to 16.76 5 14.9 12.94 16.86 12.94 to 16.86 6 15.3 13.34 17.26 13.34 to 17.26 7 14.9 12.94 16.86 12.94 to 16.86 8 15.5 13.54 17.46 13.54 to 17.46 9 14.8 12.84 16.76 12.84 to 16.76 10 15.2 13.24 17.16 13.24 to 17.16 11 15 13.04 16.96 13.04 to 16.96 12 15.1 13.14 17.06 13.14 to 17.06 13 15 13.04 16.96 13.04 to 16.96 14 14.4 12.44 16.36 12.44 to 16.36 15 15.8 13.84 17.76 13.84 to 17.76 16 14 12.04 15.96 12.04 to 15.96 17 16 14.04 17.96 14.04 to 17.96 18 16.1 14.14 18.06 14.14 to 18.06 19 15.8 13.84 17.76 13.84 to 17.76 20 14.5 12.54 16.46 12.54 to 16.46 21 14.1 12.14 16.06 12.14 to 16.06 22 14.2 12.24 16.16 12.24 to 16.16 23 14 12.04 15.96 12.04 to 15.96 24 14.9 12.94 16.86 12.94 to 16.86 25 14.7 12.74 16.66 12.74 to 16.66 26 14.5 12.54 16.46 12.54 to 16.46 27 14.6 12.64 16.56 12.64 to 16.56 28 14.8 12.84 16.76 12.84 to 16.76 29 14.8 12.84 16.76 12.84 to 16.76 30 14.6 12.64 16.56 12.64 to 16.56

Hypothesistest

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

Nullhypothesis: a bottle contains 16 ounces

Alternativehypothesis: a bottle contains less than 16 ounces

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

Calculationof z-statistic z = (14.87 – 16) / 0.55

=-1.13 / 0.55

Z= -2.055

Conclusionz (-2.055) is greater than -1.96, which implies that the nullhypothesis should be rejected. Therefore, the alternative hypothesisis accepted. Hence, a bottle contains less than 16 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 be due to faulty measuring devices.The measuring devices may be faulty such that, when measuring theprecise content, the devices deviate from giving the rightmeasurement. This makes the soda bottles have less than 16 ounces.The individuals working in the plant may not be aware of theinaccuracy of the measuring devices leading to the inaccurateresults. However, the plant can avoid the deficit in the soda bottlesthrough ensuring that all the measuring devices used in the plant arechecked regularly to see if they are faulty.

Anotherprobable reason for having bottles of soda with less than 16 ouncesmay be due to leakage. When preparing and packaging the soda bottles,there may be leakages emanating from improper handling of the sodabottles. This problem can be corrected to avoid a deficit in thefuture through the plant ensuring that there are proper handling ofsoda bottles during preparation and packages. This will aid inmitigating leakages in the processes. Besides, a deficit in the sodabottles may be as a result of unqualified employees in the plantunqualified employees may not have the skills of measuring the rightquantities leading to the deficits. This problem can be correctedthrough the plant ensuring that it has qualified individuals that arekeen on the issue of accurate measurement this will ensure that thedeficit is avoided in the future.

References

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

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