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1 1 Slide © 2003 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edwards University

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2 2 Slide © 2003 Thomson/South-Western | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | UCL CL LCL Statistical Methods for Quality Control n Statistical Process Control n Acceptance Sampling

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3 3 Slide © 2003 Thomson/South-Western Quality Terminology n Quality is the totality of features and characteristics of a product or service that bears on its ability to satisfy given needs.

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4 4 Slide © 2003 Thomson/South-Western Quality Terminology n Quality assurance refers to the entire system of policies, procedures, and guidelines established by an organization to achieve and maintain quality. n 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. n Quality control consists of making a series of inspections and measurements to determine whether quality standards are being met.

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5 5 Slide © 2003 Thomson/South-Western Statistical Process Control (SPC) n 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. n 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. n 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.

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6 6 Slide © 2003 Thomson/South-Western SPC Hypotheses n SPC procedures are based on hypothesis-testing methodology. n The null hypothesis H 0 is formulated in terms of the production process being in control. n The alternative hypothesis H a is formulated in terms of the process being out of control. n 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.

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7 7 Slide © 2003 Thomson/South-Western Decisions and State of the Process n Type I and Type II Errors State of Production Process State of Production Process Decision Decision CorrectDecision Type II Error Allow out-of-control process to continue CorrectDecision Type I Error Adjust in-control process AdjustProcess ContinueProcess H 0 True In Control H a True Out of Control

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8 8 Slide © 2003 Thomson/South-Western Control Charts n SPC uses graphical displays known as control charts to monitor a production process. n 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).

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9 9 Slide © 2003 Thomson/South-Western Control Charts n Two important lines on a control chart are the upper control limit (UCL) and lower control limit (LCL). n 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. n Values outside of the control limits provide strong evidence that the process is out of control.

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10 Slide © 2003 Thomson/South-Western Types of Control Charts n 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. n x represents the mean value found in a sample of the output. n An R chart is used to monitor the range of the measurements in the sample. n A p chart is used to monitor the proportion defective in the sample. n An np chart is used to monitor the number of defective items in the sample.

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11 Slide © 2003 Thomson/South-Western x Chart Structure UCL LCL Process Mean When in Control Center Line Time x

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12 Slide © 2003 Thomson/South-Western Control Limits for an x Chart n Process Mean and Standard Deviation Known

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13 Slide © 2003 Thomson/South-Western Example: Granite Rock Co. n Control Limits for an x Chart: Process Mean and Standard Deviation Known The weight of bags of cement filled by Granites 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?

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14 Slide © 2003 Thomson/South-Western Example: Granite Rock Co. n Control Limits for an x Chart: Process Mean and Standard Deviation Known = 50, = 1.5, n = 9 = 50, = 1.5, n = 9 UCL = (.5) = 51.5 UCL = (.5) = 51.5 LCL = (.5) = 48.5 LCL = (.5) = 48.5

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15 Slide © 2003 Thomson/South-Western Control Limits for an x Chart n Process Mean and Standard Deviation Unknown where: x = overall sample mean R = average range R = average range A 2 = a constant that depends on n ; taken from A 2 = a constant that depends on n ; taken from Factors for Control Charts table Factors for Control Charts table = _

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16 Slide © 2003 Thomson/South-Western Factors for x and R Control Charts n Factors Table (Partial)

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17 Slide © 2003 Thomson/South-Western UCL = RD 4 UCL = RD 4 LCL = RD 3 LCL = RD 3where: R = average range R = average range D 3, D 4 = constants that depend on n ; found in Factors for Control Charts table D 3, D 4 = constants that depend on n ; found in Factors for Control Charts table Control Limits for an R Chart _ _ _

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18 Slide © 2003 Thomson/South-Western Factors for x and R Control Charts n Factors Table (Partial)

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19 Slide © 2003 Thomson/South-Western Example: Granite Rock Co. n 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.

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20 Slide © 2003 Thomson/South-Western n Control Limits for R Chart: Process Mean and Standard Deviation Unknown x = 50.01, R =.322, n = 5 x = 50.01, R =.322, n = 5 UCL = RD 4 =.322(2.114) =.681 UCL = RD 4 =.322(2.114) =.681 LCL = RD 3 =.322(0) = 0 LCL = RD 3 =.322(0) = 0 Example: Granite Rock Co. _ = _ _

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21 Slide © 2003 Thomson/South-Western Example: Granite Rock Co. n R Chart

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22 Slide © 2003 Thomson/South-Western n Control Limits for x Chart: Process Mean and Standard Deviation Unknown x = 50.01, R =.322, n = 5 x = 50.01, R =.322, n = 5 UCL = x + A 2 R = (.322) = LCL = x - A 2 R = (.322) = Example: Granite Rock Co. = = =

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23 Slide © 2003 Thomson/South-Western Example: Granite Rock Co. n x Chart

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24 Slide © 2003 Thomson/South-Western Control Limits for a p Chart where:assuming: np > 5 np > 5 n (1- p ) > 5 n (1- p ) > 5 Note: If computed LCL is negative, set LCL = 0 Note: If computed LCL is negative, set LCL = 0

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25 Slide © 2003 Thomson/South-Western 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.

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26 Slide © 2003 Thomson/South-Western 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.

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27 Slide © 2003 Thomson/South-Western Example: Norwest Bank n 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.

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28 Slide © 2003 Thomson/South-Western Example: Norwest Bank n Control Limits for a p Chart Estimated p = 80/((20)(250)) = 80/5000 =.016

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29 Slide © 2003 Thomson/South-Western Example: Norwest Bank n p Chart

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30 Slide © 2003 Thomson/South-Western Control Limits for an np Chart assuming: np > 5 np > 5 n (1- p ) > 5 n (1- p ) > 5 Note: If computed LCL is negative, set LCL = 0

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31 Slide © 2003 Thomson/South-Western Interpretation of Control Charts n 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. n A primary indication that a process may be out of control is a data point outside the control limits. n 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. 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. Six or seven points in a row that indicate either an increasing or decreasing trend.... and other patterns.... and other patterns.

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32 Slide © 2003 Thomson/South-Western Acceptance Sampling n 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. n Acceptance sampling has advantages over 100% inspection including: less expensive, less product damage, fewer people involved,... and more.

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33 Slide © 2003 Thomson/South-Western Acceptance Sampling Procedure Lot received Sample selected Sampled items inspected for quality Results compared with specified quality characteristics Accept the lot Reject the lot Send to production or customer Decide on disposition of the lot Quality is not satisfactory satisfactory Quality is Quality issatisfactory

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34 Slide © 2003 Thomson/South-Western Acceptance Sampling n Acceptance sampling is based on hypothesis-testing methodology. n The hypothesis are: H 0 : Good-quality lot H 0 : Good-quality lot H a : Poor-quality lot H a : Poor-quality lot

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35 Slide © 2003 Thomson/South-Western The Outcomes of Acceptance Sampling n Type I and Type II Errors State of the Lot State of the Lot Decision Decision CorrectDecision Type II Error Consumers Risk CorrectDecision Type I Error Producers Risk Reject H 0 Reject the Lot Accept H 0 Accept the Lot H 0 True Good-Quality Lot H a True Poor-Quality Lot

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36 Slide © 2003 Thomson/South-Western n 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 f ( x ) = probability of x defective items in sample Probability of Accepting a Lot

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37 Slide © 2003 Thomson/South-Western 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 n = 20, c = 2, and p =.05 P (Accept Lot) = f (0) + f (1) + f (2) P (Accept Lot) = f (0) + f (1) + f (2) = = =.9246 =.9246 P (Reject Lot) = P (Reject Lot) = =.0754 =.0754

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38 Slide © 2003 Thomson/South-Western Example: Acceptance Sampling n Using the Tables of Binomial Probabilities

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39 Slide © 2003 Thomson/South-Western Selecting an Acceptance Sampling Plan n In formulating a plan, managers must specify two values for the fraction defective in the lot. a = the probability that a lot with p 0 defectives will be rejected. a = the probability that a lot with p 0 defectives will be rejected. b = the probability that a lot with p 1 defectives will be accepted. b = the probability that a lot with p 1 defectives will be accepted. n 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.

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40 Slide © 2003 Thomson/South-Western Operating Characteristic Curve Probability of Accepting the Lot Percent Defective in the Lot p0p0p0p0 p1p1p1p1 (1 - ) n = 15, c = 0 p 0 =.03, p 1 =.15 =.3667, =.0874 =.3667, =.0874

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41 Slide © 2003 Thomson/South-Western Multiple Sampling Plans n A multiple sampling plan uses two or more stages of sampling. n At each stage the decision possibilities are: stop sampling and accept the lot, stop sampling and accept the lot, stop sampling and reject the lot, or stop sampling and reject the lot, or continue sampling. continue sampling. n 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.

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42 Slide © 2003 Thomson/South-Western A Two-Stage Acceptance Sampling Plan Inspect n 1 items Find x 1 defective items in this sample x1 < c1 ?x1 < c1 ?x1 < c1 ?x1 < c1 ? x1 > c2 ?x1 > c2 ?x1 > c2 ?x1 > c2 ? Inspect n 2 additional items Accept the lot Reject x 1 + x 2 < c 3 ? Find x 2 defective items in this sample Yes Yes No No No Yes

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43 Slide © 2003 Thomson/South-Western End of Chapter

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