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QUESTION 1
PART A
pieces of chalks in the sample that are considered defective

Z-Score= deviations from the true mean/standard deviation
(1.5)/(0.75)=2

From normal distribution tables probabilty equivalent to Z-Score value of 2= 0.4772

using the probability value(0.4772) we can obtain number of chalk considered as defective as follows
(sample averege – expected everage)/standard deviation = Probability value(0.4772)

(3.6 -µ )/0.75 =0.4772
making µ the subject= 3.2421 approximately 3 defective pieces of chalks

PART B

Defects per Million Opportunity (DPMO) = Defect Per Opportunity (DPO) * 1000000
DPO=(Number of defects)/(sample size*number of opportunities per unit in sample)
DPO= (3)/(3.6*100) 0.008333333

DPMO = (0.00833 * 1000000) 8330

PART C
According to Six-Sigma conversion Table the quality level corresponding to DPMO of 8330 is level 3

QUESTION 2

PART A
A run Chart and computing the average value line.
Sample defects
1 3
2 8
3 6
4 9
5 7
6 1
7 9
8 3
9 6
10 7
11 11
12 14
13 7
14 8
15 4
16 5
17 2
18 4
19 6
20 1

Average value/ centre line
(6+7+8+11+13)/4 11.25

PART B
Six-Sigma quality level.

PART C
Their could be special cause present because of increase variations in the run chart.
Flactuation from one sample to the other is higher this makes the probability of availability of special cause possible.

QUESTION 3

PART A
construct a Pareto diagram for the data.

Errors frequency cumulative frequency %
Long delays 5793 50.10%
shipping errors 3628 81.48%
billing errors 960 89.79%
delivery errors 651 95.42%
electronic charge errors 530 100%

TOTAL 11562

PART B
Conclusions reached.

From the cummulative frequency table the company should focus more efforts in in reducing long delays errors.
By starting with it the company would have covered 50.10% of all complaints presented.

QUESTION 4

NUMBER CONVEYOR SPEED (cm/sc) CUT LENGTH (mm)
1 8.1 1046
2 7.9 1030
3 8 1039
4 7.4 1027
5 7.5 1028
6 7.4 1025
7 8 1035
8 7.2 1015
9 8 1038
10 7.9 1036
11 7.6 1026
12 8 1041
13 7.5 1029
14 7.2 1010
15 7.3 1020
16 7.4 1024
17 8 1034
18 8 1036
19 7.4 1023
20 7.1 1011
21 7.9 1030
22 7.3 1014
23 7.8 1030
24 7.2 1016
25 7.3 1020
26 8 1040
27 5.5 1013
28 6.9 1025
29 7 1020
30 7.5 1022
31 6.7 1020
32 8.1 1035
33 9 1052
34 7.1 1021
35 7.6 1024
36 8.5 1029
37 7.5 1015
38 8 1030
39 5.2 1010
40 6.5 1025
41 8 1031
42 6.9 1030
43 7.6 1034
44 6.5 1034
45 5.5 1020
46 6 1025
47 5.5 1023
48 7.6 1028
49 8.6 1020
50 6.3 1026

PART A
A run chart and draw a conclusion.

the trend observed in the run chart is slightly stationary for conveyor speed. Similarly, cut length shows an increasing trend as the number of tubes increases.
PART B
Quality level.
The quality level corresponding to 1026mm is level 3

PART C
Scatter diagram

PART D
Recomedations to improve the process.

production should be done in blocks of 20 tubes. There is less flactuations in cut length between 1-20.