STATISTICS 9

Standarddeviation, 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

Median:the ounces for the 30 bottles are as follows when arranged

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 |

Themedian = (14.8 + 14.8) /2

=14.8

Variance= 8.783 /(30-1)

=0.302862

Standarddeviation = 0.302862

=0.55

95%confidence interval

Fromthe statistics table, the normal distribution value for the 95%confidence interval is 1.96 this value is important in determiningthe upper and the lower limits of an interval. The lower limit willbe determined by subtracting 1.96 from the soda bottle ounces.Alternatively, the upper limit of the interval will be given byadding 1.96 to ounces of the bottle.

95%confidence intervals 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 |

Hypothesistesting

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 < -1.96, or > 1.96, the null hypothesis shouldbe rejected.

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. Therefore, a bottle contains less than 16 ounces.

Fromthe conclusion, in a bottle of soda, there are less than 16 ounces.One of the likely causes for having a bottle of soda with less than16 ounces may be due to faulty measuring devices. The individualsworking in the plant may not be aware of the inaccuracy of themeasuring devices leading to the inaccurate results. However, theplant can avoid the deficit in the soda bottles through ensuring thatall the measuring devices used in the plant are checked regularly tosee if they are faulty.

Anotherreason for having bottles of soda with less than 16 ounces isleakage. When preparing and packaging the soda bottles, there may beleakages emanating from improper handling of the soda bottles. Thisproblem can be corrected to avoid a deficit in the future through theplant ensuring that there are proper handling of soda bottles duringpreparation and packaging. This will help in mitigating leakages inthe processes. Besides, a deficit in the soda bottles may be as aresult of unqualified employees in the plant unqualified employeesmay not have the skills of measuring the right quantities leading tothe deficits. This problem can be corrected through the plantensuring that it has qualified individuals that are keen on the issueof accurate measurement this will ensure that the deficit is avoidedin the future.