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# Performance: Control Chart for Process Mean

13-14
Boston-based Legal Seafoods prides itself on having instituted an advanced quality control system that includes the control of both food quality and service quality. The following are successive service times at one of the chain's restaurants on a Saturday night in May 2007 (time is stated in minutes from customer entry to appearance of waitperson)
5, 6, 5,5,5,7, 4, 12, 4, 4.5, 2, 5, 5, 5, 6, 6, 13, 2, 5, 4, 4.5, 6.5, 4, 1,
2 3, 5, 5, 4, 4, 8, 12 3, 4.5, 6.5, 6, 7, 10, 6, 6.5, 5, 3, 6.5 7,
Aggregate the data into groups of four, and construct a control chart for the process mean. Is the waiting time at the restaurant under control?

13-17
The following data are tensile strengths, in pounds, for a sample of string for industrial use made at a plant. Construct a control chart for the mean, using groups of 5 observations each. Test for statistical control of the process mean.
5, 6, 4, 6, 5, 7, 7, 7, 6, 5, 3, 5, 5, 5, 6, 5, 5, 6, 7, 7, 7, 7, 6, 7, 5,
5, 5, 6, 7, 7, 7, 7, 7, 5, 5, 6, 4 6, 6, 6, 7, 6, 6, 6, 6, 6, 7, 5, 7, 6,

13-23
Create R and s charts for problem 13-17. Is the process in control?

13-25

The capacity of the fuel tank of the 2007 Volvo S40 is designed to be 12.625 gallons. The actual capacity of tanks produced is controlled using a control chart. The data of 9 random samples of size 5 each collected on 9 different days are tabulated below. Draw X-bar, R, and s charts. Is the process in control? If it is not, remove the sample that is out of control, and redraw the charts.
1 2 3 4 5 6 7
12.667 12.600 12.599 12.607 12.738 12.557 12.646
12.598 12.711 12.583 12.524 12.605 12.745 12.647
12.685 12.653 12.515 12.718 12.640 12.626 12.651
12.700 12.703 12.653 12.615 12.653 12.694 12.607
12.722 12.579 12.599 12.554 12.507 12.574 12.589
8 9
12.710 12.529
12.627 12.725
12.605 12.306
12.648 12.551
12.545 12.600

13-29
BASF Inc. makes CDs for use in computers. A quality control engineer at the plant tests batches of 50 disks at a time and plots the proportions of defective disks on a control chart. The first 10 batches used to create the chart had the following numbers of defective disks: 8, 7, 6, 7, 8, 4, 3, 5, 5, 8. Construct the chart and interpret the results.

13-31

The following are the numbers of imperfections per yard of yarn produced in a mill in Pennsylvania in May 2007: 5, 3, 4, 8, 2, 3, 1, 2, 5, 9, 2, 2, 2, 3, 4, 2, 1. Is there evidence that the process is out of control?

13-32

The following are the numbers of blemishes in the coat of pain of new auto mobile made by Ford in June 2007: 12, 25, 13, 20, 5, 22, 8, 17, 31, 40, 9, 62, 14, 16, 9, 28. Is there evidence tat the painting process is out of control?

13-33

The following are the numbers of imperfection in rolls of wallpaper made by Laura Ashley: 5, 6, 3, 4, 5, 2, 7, 4, 5, 3, 5, 5, 3, 2, 0, 5, 5, 6, 7, 6, 9, 3, 3, 4, 2, 6. Construct a c chart for the process, and determine whether there is evidence that the process is out of control.

#### Solution Preview

Solution is attached.

Solution to problem (13-25):

Day sample1 sample2 sample3 sample4 sample5 x-bar s R
1 12.667 12.598 12.685 12.700 12.722 12.674 0.047237 0.124
2 12.600 12.711 12.653 12.703 12.579 12.649 0.059323 0.132
3 12.599 12.583 12.515 12.653 12.599 12.590 0.04951 0.138
4 12.607 12.524 12.718 12.615 12.554 12.604 0.074204 0.194
5 12.738 12.605 12.640 12.653 12.507 12.629 0.083722 0.231
6 12.557 12.745 12.626 12.694 12.574 12.639 0.079654 0.188
7 12.646 12.647 12.651 12.607 12.589 12.628 0.028178 0.062
8 12.710 12.627 12.605 12.648 12.545 12.627 0.060287 0.165
9 12.529 12.725 12.306 12.551 12.600 12.542 0.152314 0.419
Sum = 113.653 113.765 113.399 113.824 113.269 113.582 0.634428 1.653

Capacity is designed to be = 12.625 gallons
Therefore,  = 12.625
The x-bar for day 1, (12.674)is calculated as below.

In the similar way the x-bars of all the 9 days for 5 samples are calculated.

s- is the standard deviation of the five samples for each day, which are calculated using the standard deviation formula.

R is the Range, which is the difference of the highest and lowest value in each row of the five samples for each day.

From the table of constants for control limits, we have A2 = 0.5768 for n = 5 , k = 9
From the above table, we have

Therefore,
Lower control limit for designed mean =

Upper control limit for the designed mean =

Therefore, for the process mean to be in control, the sample means (that is, x-bars) should be between the limits of 12.519 and 12.731.

From the above table, we see that all ...

#### Solution Summary

A control chart for process mean performance is examined.

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