Statistical Process Controls: Please solve and submit the following questions (50

points).

1. A major record-of-the-month club collected data on the reasons for returned

shipments during a quarter. Results are: wrong selection, 50,000; refused, 195,000;
wrong address, 68,000; order canceled, 5,000; and other, 15,000. Construct a Pareto
diagram and recommend what are the major problems that demand investment.

Pareto Chart of Reason

350000
100
300000
80
250000

Percent
200000 60
totals

150000
40
100000
20
50000

0 0
reason Refused Wr. Add Wr. Sel. Other Ord Can
Count 195000 68000 50000 15000 5000
Percent 58.6 20.4 15.0 4.5 1.5
Cum % 58.6 79.0 94.0 98.5 100.0

It does appear that communication is the largest issue in the organization. The
customers are refusing their orders as 60% of our total losses because we are
apparently not getting them proper service. We also have the wrong addresses
for 20% of our customers with issues and 50,000 we have the wrong address for.
Communication with the customer is apparently the largest flaw in need of most
investment.

2. Calculate the average, median, mode, range, and standard deviation for each
group of numbers.
 (a) 50, 45, 55, 55, 45, 50, 55, 45, 55
N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
C1 9 0 50.56 1.55 4.64 45.00 45.00 50.00 55.00 55.00

Variable Range
C1 10.00 MODE = 45

(b) 89,87,88,83,86,82,84
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
C1 7 0 85.571 0.997 2.637 82.000 83.000 86.000 88.000

Variable Maximum Range Mode = 83

C1 89.000 7.000

(c) 11,17,14,12,12,14,14,15,17,17
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
C1 10 0 14.300 0.700 2.214 11.000 12.000 14.000 17.000

Variable Maximum Range Mode

C1 17.000 6.000 13

(d) 16,25,18,17,16,21,14
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
C1 7 0 18.14 1.40 3.72 14.00 16.00 17.00 21.00 25.00

Variable Range mode

C1 11.00 17

(e) 45,39,42,42,43
Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3
C1 5 0 42.200 0.970 2.168 39.000 40.500 42.000 44.000

Variable Maximum Range Mode

C1 45.000 6.000 42

3. Control charts for X and R are to be established on a certain dimension part,

measured in millimeters. Data were collected in subgroup sizes of 6 and are given
below. Calculate the trial central line and control limits. Assume assignable causes
and revise the central line and limits.
-----------------------------------------------------------------------------------------------------------
Subgroup Subgroup
Number X R Number X R
------------------------------------------------------------------------------------------------------------
1 20.35 0.34 14 20.41 0.36
2 20.40 0.36 15 20.45 0.34
3 20.36 0.32 16 20.34 0.36
4 20.65 0.36 17 20.36 0.37
5 20.20 0.36 18 20.42 0.73
6 20.40 0.35 19 20.50 0.38
7 20.43 0.31 20 20.31 0.35
8 20.37 0.34 21 20.39 0.38
9 20.48 0.30 22 20.39 0.33
10 20.42 0.37 23 20.40 0.32
11 20.39 0.29 24 20.41 0.34
12 20.38 0.30 25 20.40 0.30
13 20.40 0.33
-----------------------------------------------------------------------------------------------------------

Xbar-R Chart of X
U C L=20.6556
20.6
Sample M ean

20.5
_
_
20.4 X=20.4004

20.3

20.2
LC L=20.1452
1 2 3 4 5
Sample

1
U C L=0.4320
0.4
Sample Range

0.3
_
0.2 R=0.2156

0.1

0.0 LC L=0
1 2 3 4 5
Sample

4. The following table gives the average and range in kilograms for tensile tests on
an improved plastic cord. The subgroup size is 4. Determine the trial central line
 and control limits. If any points are out of control, assume assignable causes, and
determine the revised limits and central line.

Xbar-R Chart of X
520 U C L=519.75
Sample Mean

500
_
_
480 X=482.52

460

LC L=445.29
440
1 2 3 4 5 6 7
Sample

60 U C L=58.29

45
Sample Range

30 _
R=25.55

15

0 LC L=0
1 2 3 4 5 6 7
Sample

5. The Get-Well Hospital has completed a quality improvement project on the time
to admit a patient using: XC- and R charts. They now wish to monitor the activity
using median and range charts. Determine the central line and control limits with
the latest data in minutes, as given here.
 Xbar-R Chart of XBar
6.8 U C L=6.759

6.4
Sample Mean

_
_
X=6.064
6.0

5.6
LC L=5.369
5.2
1 2 3 4 5 6 7 8
Sample

2.0
U C L=1.748
1.5
Sample Range

1.0
_
R=0.679
0.5

0.0 LC L=0
1 2 3 4 5 6 7 8
Sample

6. The viscosity of a liquid is checked every half hour during one three-shift day.
What does the run chart indicate? Data are 39, 42,38,37,41,40,36,35,37,36,
39,34,38,36,32,37,35,34,33,35,32,38,34,37,35,35, 34, 31, 33, 35, 32,
 36,31,29,33,32,31,30,32, and 29.

Run Chart of C1
42

40

38

36
C1

34

32

30

1 5 10 15 20 25 30 35 40
Observation
Number of runs about median: 14 Number of runs up or down: 26
Expected number of runs: 20.20000 Expected number of runs: 26.33333
Longest run about median: 7 Longest run up or down: 4
Approx P-Value for C lustering: 0.01917 A pprox P-Value for Trends: 0.44910
Approx P-Value for Mixtures: 0.98083 A pprox P-Value for Oscillation: 0.55090

The viscosity of the liquid appears to be dropping over time from the average. If this
deterioration trend continues, then the liquid will have to be replaced to prevent
damage to equipment.

7. Determine the trial central line and control limits for a p chart using the following
data, which are for the payment of dental insurance claims. Plot the values on
graph paper and determine if the process is stable. If there are any out -of-control
 points, assume an assignable cause and determine the revised central line and
control limits.

P Chart of No. N.C

0.07 1

0.06

0.05
Proportion

0.04 UCL=0.04016

0.03

0.02 _
P=0.01747

0.01

0.00 LCL=0

1 3 5 7 9 11 13 15 17 19 21 23 25
Sample
 P Chart of No. N.C
0.04
UCL=0.03676

0.03
Proportion

0.02
_
P=0.01542

0.01

0.00 LCL=0

1 3 5 7 9 11 13 15 17 19 21 23
Sample

8. Determine the trial limits and revised control limits for a u chart using the data in
the table for the surface finish of rolls of white paper. Assume any out -of-control
points have assignable causes.
 U Chart of Total N.C
6
1
1
5 UCL=5.043
Sample Count Per Unit

4
_
U=3.315
3

2
LCL=1.588
1
1

1
0
1 4 7 10 13 16 19 22 25 28
Sample
Tests performed with unequal sample sizes

U Chart of Total N.C

5 UCL=5.097
Sample Count Per Unit

4
_
U=3.358
3

2
LCL=1.620

1
1 3 5 7 9 11 13 15 17 19 21 23
Sample
Tests performed with unequal sample sizes
9. A quality technician has collected data on the count of rivet nonconformities in
four meters travel trailers. After 30 trailers, the total count of nonconformities is
316. Trial control limits have been determined and a comparison with the data
shows no out-of-control points. What is the recommendation for the central line
and the revised control limits for a count of nonconformities chart?

If the data has been compared to the original data done for testing and all of the
data indicates there are no out of spec process controls, then the revised control
limits will have to be moved to fit the newly gathered data for the next run. If the
nonconformities occurred on the low side of the spectrum, then move the revised
control limits lower. If the phenomenon occurred on the high side, then move the
control limits up.
10. By means of a scatter diagram, determine if a relationship, exists between
product temperatures and percent foam for a soft drink.

Scatterplot of temp vs foam

50.0

47.5

45.0
temp

42.5

40.0

37.5

35.0
15 20 25 30 35 40 45
foam

From the look of the graph of Product Temperature VS Foam, the data indicates
that there is a correlation between the temperature and the percentage of foam in
the product. My suggestion if foam is a deterrent to a quality end product, is to add
equipment that would eliminate air entrapment to the liquid in the beginning or add
coolers to the process to control the temperature of the product.