1. St. Edward’s University
Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University

2. Statistical Methods for Quality Control
Statistical Process Control
Acceptance Sampling
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UCL CL
LCL

3. Quality Terminology
Quality is “the totality of features and characteristics of a product or service that bears on its ability to satisfy given needs.”

4. Quality Terminology
Quality assurance refers to the entire system of policies, procedures, and guidelines established by an organization to achieve and maintain quality.
The objective of quality engineering is to include quality in the design of products and processes and to identify potential quality problems prior to production.
Quality control consists of making a series of inspections and measurements to determine whether quality standards are being met.

5. Statistical Process Control (SPC)
The goal of SPC is to determine whether the process can be continued or whether it should be adjusted to achieve a desired quality level.
If the variation in the quality of the production output is due to assignable causes (operator error, worn-out tooling, bad raw material, ) the process should be adjusted or corrected as soon as possible.
If the variation in output is due to common causes (variation in materials, humidity, temperature, ) which the manager cannot control, the process does not need to be adjusted.

6. SPC Hypotheses
SPC procedures are based on hypothesis-testing methodology.
The null hypothesis H0 is formulated in terms of the production process being in control.
The alternative hypothesis Ha is formulated in terms of the process being out of control.
As with other hypothesis-testing procedures, both a Type I error (adjusting an in-control process) and a Type II error (allowing an out-of-control process to continue) are possible.

7. Decisions and State of the Process
Type I and Type II Errors
State of Production Process
Decision
H0 True
In Control
Ha True
Out of Control
Continue
Process
Correct
Decision
Type II Error
Allow out-of-control
process to continue
Adjust
Process
Type I Error
Adjust in-control
process
Correct
Decision

8. Control Charts
SPC uses graphical displays known as control charts to monitor a production process.
Control charts provide a basis for deciding whether the variation in the output is due to common causes (in control) or assignable causes (out of control).

9. Control Charts
Two important lines on a control chart are the upper control limit (UCL) and lower control limit (LCL).
These lines are chosen so that when the process is in control, there will be a high probability that the sample finding will be between the two lines.
Values outside of the control limits provide strong evidence that the process is out of control.

10. Types of Control Charts
An x chart is used if the quality of the output is measured in terms of a variable such as length, weight, temperature, and so on.
x represents the mean value found in a sample of the output.
An R chart is used to monitor the range of the measurements in the sample.
A p chart is used to monitor the proportion defective in the sample.
An np chart is used to monitor the number of defective items in the sample.

11. x Chart Structure x UCL Center Line Process Mean When in Control LCL
Time

12. Control Limits for an x Chart
Process Mean and Standard Deviation Known

13. Example: Granite Rock Co.
Control Limits for an x Chart: Process Mean
and Standard Deviation Known
The weight of bags of cement filled by Granite’s packaging process is normally distributed with a mean of 50 pounds and a standard deviation of 1.5 pounds.
What should be the control limits for samples of 9 bags?

14. Example: Granite Rock Co.
Control Limits for an x Chart: Process Mean
and Standard Deviation Known
 = 50,  = 1.5, n = 9
UCL = (.5) =
LCL = (.5) =

15. Control Limits for an x Chart
Process Mean and Standard Deviation Unknown
where:
x = overall sample mean
R = average range
A2 = a constant that depends on n; taken from
“Factors for Control Charts” table = _

16. Factors for x and R Control Charts
Factors Table (Partial)

17. Control Limits for an R Chart
UCL = RD4
LCL = RD3
where:
R = average range
D3, D4 = constants that depend on n; found in “Factors for Control Charts” table _ _ _

18. Factors for x and R Control Charts
Factors Table (Partial)

19. Example: Granite Rock Co.
Control Limits for x and R Charts: Process Mean
and Standard Deviation Unknown
Suppose Granite does not know the true mean and standard deviation for its bag filling process. It wants to develop x and R charts based on twenty samples of 5 bags each.
The twenty samples resulted in an overall sample mean of pounds and an average range of .322 pounds.

20. Example: Granite Rock Co.
Control Limits for R Chart: Process Mean
and Standard Deviation Unknown
x = 50.01, R = .322, n = 5
UCL = RD4 = .322(2.114) = .681
LCL = RD3 = .322(0) = 0 _ = _ _

21. Example: Granite Rock Co.
R Chart

22. Example: Granite Rock Co.
Control Limits for x Chart: Process Mean
and Standard Deviation Unknown
x = 50.01, R = .322, n = 5
UCL = x + A2R = (.322) =
LCL = x - A2R = (.322) = = = =

23. Example: Granite Rock Co.
x Chart

24. Control Limits for a p Chart
where:
assuming:
np > 5
n(1-p) > 5
Note: If computed LCL is negative, set LCL = 0

25. Example: Norwest Bank
Every check cashed or deposited at Norwest Bank must be encoded with the amount of the check before it can begin the Federal Reserve clearing process. The accuracy of the check encoding process is of utmost importance. If there is any discrepancy between the amount a check is made out for and the encoded amount, the check is defective.

26. Example: Norwest Bank
Twenty samples, each consisting of 250 checks, were selected and examined when the encoding process was known to be operating correctly. The number of defective checks found in the samples follow.

27. Example: Norwest Bank Control Limits for a p Chart
Suppose Norwest does not know the proportion of defective checks, p, for the encoding process when it is in control.
We will treat the data (20 samples) collected as one large sample and compute the average number of defective checks for all the data. That value can then be used to estimate p.

28. Example: Norwest Bank Control Limits for a p Chart
Estimated p = 80/((20)(250)) = 80/5000 = .016

29. Example: Norwest Bank
p Chart

30. Control Limits for an np Chart
assuming:
np > 5
n(1-p) > 5
Note: If computed LCL is negative, set LCL = 0

31. Interpretation of Control Charts
The location and pattern of points in a control chart enable us to determine, with a small probability of error, whether a process is in statistical control.
A primary indication that a process may be out of control is a data point outside the control limits.
Certain patterns of points within the control limits can be warning signals of quality problems:
Large number of points on one side of center line.
Six or seven points in a row that indicate either an increasing or decreasing trend.
. . . and other patterns.

32. Acceptance Sampling
Acceptance sampling is a statistical method that enables us to base the accept-reject decision on the inspection of a sample of items from the lot.
Acceptance sampling has advantages over 100% inspection including: less expensive, less product damage, fewer people involved, and more.

33. Acceptance Sampling Procedure
Lot received
Sample selected
Sampled items
inspected for quality
Results compared with
specified quality characteristics
Quality is not
satisfactory
Quality is
satisfactory
Accept the lot
Reject the lot
Send to production
or customer
Decide on disposition
of the lot

34. Acceptance Sampling
Acceptance sampling is based on hypothesis-testing methodology.
The hypothesis are:
H0: Good-quality lot
Ha: Poor-quality lot

35. The Outcomes of Acceptance Sampling
Type I and Type II Errors
State of the Lot
Decision
H0 True
Good-Quality Lot
Ha True
Poor-Quality Lot
Accept H0
Accept the Lot
Correct
Decision
Type II Error
Consumer’s Risk
Reject H0
Reject the Lot
Type I Error
Producer’s Risk
Correct
Decision

36. Probability of Accepting a Lot
Binomial Probability Function for Acceptance Sampling
where:
n = sample size
p = proportion of defective items in lot
x = number of defective items in sample
f(x) = probability of x defective items in sample

37. Example: Acceptance Sampling
An inspector takes a sample of 20 items from a lot.
Her policy is to accept a lot if no more than 2 defective
items are found in the sample.
Assuming that 5 percent of a lot is defective, what is
the probability that she will accept a lot? Reject a lot?
n = 20, c = 2, and p = .05
P(Accept Lot) = f(0) + f(1) + f(2) = =
P(Reject Lot) = =

38. Example: Acceptance Sampling
Using the Tables of Binomial Probabilities

39. Selecting an Acceptance Sampling Plan
In formulating a plan, managers must specify two values for the fraction defective in the lot.
a = the probability that a lot with p0 defectives will be rejected.
b = the probability that a lot with p1 defectives will be accepted.
Then, the values of n and c are selected that result in an acceptance sampling plan that comes closest to meeting both the a and b requirements specified.

40. Operating Characteristic Curve
.10
.20
.30
.40
.50
.60
.70
.80
.90
Probability of Accepting the Lot
1.00
Percent Defective in the Lot p0 p1 b
(1 - a) a
n = 15, c = 0
p0 = .03, p1 = .15
a = .3667, b = .0874

41. Multiple Sampling Plans
A multiple sampling plan uses two or more stages of sampling.
At each stage the decision possibilities are:
stop sampling and accept the lot,
stop sampling and reject the lot, or
continue sampling.
Multiple sampling plans often result in a smaller total sample size than single-sample plans with the same Type I error and Type II error probabilities.

42. A Two-Stage Acceptance Sampling Plan
Inspect n1 items
Find x1 defective items in this sample
Yes
x1 < c1 ?
Accept
the lot No
Reject
the lot
Yes
x1 > c2 ? No
Inspect n2 additional items
Find x2 defective items in this sample No
x1 + x2 < c3 ?
Yes

43. End of Chapter