Statistics

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 &lt -1.96, or &gt 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.