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

Statistical Methods for Quality Control Statistical Process Control Acceptance Sampling | | | | | | | | | | | | | | | | | | | | | | | | | UCL CL LCL

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

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.

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.

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.

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

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).

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.

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.

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

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

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?

Example: Granite Rock Co. Control Limits for an x Chart: Process Mean and Standard Deviation Known  = 50,  = 1.5, n = 9 UCL = 50 + 3(.5) = 51.5 LCL = 50 - 3(.5) = 48.5

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 = _

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

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 _ _ _

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

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 50.01 pounds and an average range of .322 pounds.

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 _ = _ _

Example: Granite Rock Co. R Chart

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 = 50.01 + .577(.322) = 50.196 LCL = x - A2R = 50.01 - .577(.322) = 49.824 = = =

Example: Granite Rock Co. x Chart

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

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.

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.

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.

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

Example: Norwest Bank p Chart

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

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.

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.

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

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

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

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

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) = .3585 + .3774 + .1887 = .9246 P(Reject Lot) = 1 - .9246 = .0754

Example: Acceptance Sampling Using the Tables of Binomial Probabilities

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.

Operating Characteristic Curve .10 .20 .30 .40 .50 .60 .70 .80 .90 Probability of Accepting the Lot 0 5 10 15 20 25 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

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.

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

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