HYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of


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1 HYPOTHESIS TESTING FOR THE PROESS APABILITY RATIO A thesis resented to the faculty of the Russ ollege of Engineering and Technology of Ohio University In artial fulfillment of the requirement for the degree Master of Science Satyajit Datar November 2002
2 This thesis entitled HYPOTHESIS TESTING FOR THE PROESS APABILITY RATIO BY SATYAJIT DATAR has been aroved for the Deartment of Industrial and Manufacturing Systems Engineering and the Russ ollege of Engineering and Technology Dale T. Masel Assistant Professor of Industrial and Manufacturing Systems Engineering Dennis Irwin Dean, Russ ollege of Engineering and Technology
3 DATAR, SATYAJIT M.S. November Industrial and Manufacturing Systems Engineering Hyothesis Testing for the Process aability Ratio (70.) Director of Thesis: Dale T. Masel Process aability Indices are used in the industry as decision making tools. The value of the indices are estimated from a samle and should be qualified. Qualification of an estimate is rovided in the form of a confidence interval for the estimate or from a hyothesis test. in industry. and A hyothesis test for thesis is to develo a hyothesis test for k are two of the most widely used rocess caability indices has been develoed by Kane (1986). The aim of this k. The hyothesis test for k is develoed from the aroximate two sided confidence interval by Nagata and Nagahata (1992 & 1994). An Excel alication has been develoed which allows the user to comute cutoff and samle sizes for userdefined samling lans. The alication also comutes k and their confidence intervals from rocess data. and Aroved: Dale T. Masel Assistant Professor of Industrial and Manufacturing Systems Engineering
4 Acknowledgements I would like to take this oortunity to thank a number of eole associated with this thesis. I would like to thank Dr. Gerth for the various courses in statistics that heled me gain a better understanding of various statistical concets. I would like to thank him for his hel and guidance during the course of this thesis. I would like to exress my gratitude to Dr. Masel for roviding valuable suggestions and inuts during the course of this thesis and also during the writing of my thesis. This thesis would not have been comlete without his cooeration and constant hel and suort. I wish to thank Mr. Seki of JUSE for roviding me with the articles from JUSE ublications. I would like to thank Dr. Hale and Dr. Mehta from my thesis committee for their suort. Finally, I would like to thank my family, my colleagues and friends at Ohio University for their hel and encouragement in comleting this thesis.
5 5 Table of ontents Acknowledgements... 4 Table of ontents... 5 List of Tables... 8 List of Figures... 9 hater 1: Introduction Process caability Analysis Research Objective Research Develo the hyothesis test comutations for k oding the O urve for the rocess caability ratio oding the user interface Testing using Test data Outline of the Thesis hater 2: Literature Review Literature Review Process aability Indices Statistical Inference... 20
6 Statistical Hyothesis Testing onfidence Intervals for Process aability Ratios Oerating haracteristic urve The Process aability Analysis rocess hater 3: A Hyothesis Test for k Hyothesis Testing Develoment of a Hyothesis test for k Need for a rogram The User Interface The Use Interface: Hyothesis Test sheet The User Interface: PRs and onfidence Intervals hater 4: Results and Validation Results for the Process aability Ratio Results for the Process aability Ratio k Validation Validation of Results for and k A aability Analysis Study Validation of data for aability Analysis Study Hyothesis Test and onfidence Interval for Hyothesis Test and onfidence Interval for k... 60
7 4.4.4 Results from the rogram hater 5: onclusions onclusions Future Scoe References Aendix A: Data for Process aability Study
8 List of Tables 8 Table 2.1: Three Processes with their and k values Table 3.1: Samle size and cutoff Value Determination for Testing Table 4.1: Table of Samling Plans for Process aability Ratio Table 4.2: Table of Samling Plans for Process aability Ratio [3] k Table 4.3: Table of Samling Plan Table 4.4: Samle size and cutoff value table by Kane [3] Table 4.5: orrected Samle size and cutoff value table Table 4.6: Summary of results for Table 4.7: Summary of results for Data using Statgrahics k Data using Statgrahics... 58
9 9 Figure 2.1: Three rocesses with List of Figures Figure 2.2: Oerating characteristic curve for k =1 and their other aability Indices Figure 2.3: The rocess caability Analysis Process Figure 3.1: alculation of cutoff and samle size value for Figure 3.2: alculation of cutoff and samle size value for k k Figure 3.3: Hyothesis Test Sheet Figure 3.4: Data inut for samling lan Figure 3.5: Dislay of samle size and cutoff value Figure 3.6: O curve for Figure 3.7: PRs and onfidence Interval Sheet Figure 3.8: Data sheet with rocess data Figure 3.9: X chart for rocess data Figure 3.10: R chart for rocess data Figure 3.11: Process aability Ratio and onfidence Interval Figure 4.1: O curve for Figure 4.2: O curve for Figure 4.3: O curve for Figure 4.4: O curve for for samling lan for samling lan for samling lan for samling lan
10 Figure 4.5: O curve for Figure 4.6: Histogram for Figure 4.7: Histogram for Figure 4.8: Xbar chart for Figure 4.9: Range chart for Figure 4.10: Xbar chart for Figure 4.11: Range chart for Figure 4.12: alculation of Figure 4.13: alculation of 10 omarison hart Data using Statgrahics k Data using Statgrahics Data using Statgrahics Data using Statgrahics k Data using Statgrahics k Data using Statgrahics and confidence interval k and confidence interval... 62
11 11 hater 1: Introduction 1.1 Process caability Analysis Every manufacturing rocess has variation associated with it. Since rocess variation can never be totally eliminated, the variability in a rocess should be minimized to imrove roduct quality. Process caability analysis deals with the techniques used to understand the variability of a rocess and its effect on the roduct erformance. Process caability analysis is an imortant engineering decisionmaking tool and has found alication in a number of areas: as a criterion for vendor selection, reducing variability in a manufacturing rocess, secifying rocess requirements for new equiment, redicting how well the rocess will hold tolerances, assisting roduct designers in selecting or modifying a rocess and formulating quality imrovement rograms. Process caability analysis techniques have heled manufacturers control the quality of goods roduced. The result of a statistical rocess caability analysis is the rocess caability index. Process caability indices were first introduced by Juran [1]. Process caability indices are used to determine how well the outut of a rocess meets the secification requirements set by the customer. The new indices like index and the k index were the first indices develoed. Although m, mk, ( u, v) have been develoed to rovide additional information about the rocess, the majority of the organizations using rocess caability indices still use and k.
12 Process caability analysis has found widesread accetance in industry. 12 Xerox, AT&T Bell Laboratories and Motorola, Inc. are some of the cororations that are using rocess caability indices to monitor and imrove the quality of their roducts. By 1991, all of the big three US automakers were using statistical control and rocess caability indices to monitor and imrove roduct quality [1]. They also required their suliers to rovide roof of quality via rocess caability indices. Manufacturers now require suliers to rovide rocess caability index values with sulied goods as art of the contract. This is also true of machine qualification and rocess caability studies. Therefore it is necessary to show that the rocess caability ratio meets or exceeds target values [2]. One can test this requirement using a hyothesis test, where the value of the index can be comared to a critical value on which a decision of rejection or accetance can be made. Usually when conducting the hyothesis test, the oulation mean and standard deviation are unknown and must be estimated from the data. The values of the indices are estimated by estimators like Ĉ and Ĉ k. The ability of Ĉ and Ĉ k to reflect the true value of the index deends on the inherent variability of the rocess and the samling lan. A samling lan consists of a samle size and cutoff criteria by which a decision on the caability of the rocess can be made. High Ĉ and Ĉ k values are required to demonstrate rocess caability with small samle sizes. Often samle sizes are fixed without giving due consideration to the samling variability of the data. Kane [3] develoed a method by which the comuted samle sizes account for the abovementioned variability. hou, et al. [4] rovided tables for selecting samle sizes.
13 From a study of the literature on rocess caability indices, Kane [3] has 13 develoed a hyothesis test for size and cutoff criteria for, as well as rocedures for determining the samle. hou, et al. [4] rovided tables for selecting samle sizes. But, little work has been done on the hyothesis test for because k. This may be Ĉ k follows the joint distribution of two comlex noncentral tdistributions. Develoing a hyothesis test for k will aid in the decisionmaking rocess. With the hyothesis test it will also be ossible to develo samling lans for will develo a hyothesis test for critical value for k. k. This research k and a method to comute the samle size and 1.2 Research Objective The aim of this thesis was to develo a hyothesis test and samling lan for the rocess caability ratio k, which will enable suliers to test rocess caability. This will enable a statistically based method of choosing the right samle sizes and critical values for demonstrating rocess caability. A comuter rogram using Visual Basic for Alication (VBA) was develoed to calculate the samle size and cutoff values for any Tye I and Tye II error robabilities.
14 1.3 Research The develoment of the hyothesis test for k followed the method used by 14 Kane to develo the hyothesis test for. The aroximate confidence intervals for k develoed by Nagata and Nagahata [5] (see equation 2.12) were used to arrive at the formula for a hyothesis test. In the Oerating haracteristic (O) curve method develoed by Kane, [3] the Tye I and Tye II robabilities must be fixed and coordinated with the samle size in order to establish a critical value that can be used to judge the caability of the rocess. In this thesis, the method was comuterized so that the samle size and critical value for both and robabilities. k could be generated for any given Tye I and Tye II error Develo the hyothesis test comutations for k. This ste involved the actual mathematical develoment of the hyothesis test comutations for k. The formulae are derived based on the twosided confidence interval for k develoed by Nagata and Nagahata [4] (see equation 2.12). After the formulae had been develoed inuts and oututs were identified for coding uroses oding the O urve for the rocess caability ratio. The mathematical formulation for the comutations of the O curve has been develoed by Kane [3]. This ste involves coding the O curve comutations for.
15 1.3.3 oding the user interface. 15 The required inuts were identified and coded to rovide the oututs, samle size and cutoff value Testing using Test data. Testing and validation were conducted with data where the required results were known from the literature or were calculated. The outut from the code was then validated against the known results. 1.4 Outline of the Thesis hater 2 of the thesis deals with the literature review in the field of rocess caability indices and k. In hater 3 the develoment of the hyothesis test for k is discussed. This chater also discusses the rogram architecture. hater 4 includes the results and validation and hater 5 deals with the conclusions and recommendations based on the results of this research.
16 16 hater 2: Literature Review 2.1 Literature Review In statistics, estimates are made about arameters of a oulation by taking samles from the oulation. The arameter estimate is a random variable and is called a statistic. Every arameter estimate has associated with it a articular distribution. The value of the estimate deends on several variables including samle size and samling techniques. This chater is divided into four sections. The first section deals with rocess caability indices. The second section deals with statistical inference, which involves hyothesis testing and confidence intervals for rocess caability ratios. The third section deals with oerating characteristic curves for rocess caability indices. The fourth section briefly outlines the entire caability analysis rocess. 2.2 Process aability Indices The science of rocess caability analysis, first introduced by Juran [1], began as a comarison of the rocess outut distribution with the roduct tolerances. Frequency histograms, log lots and control charts were used to comare rocess data to roduct tolerances. Process caability indices were born out of the need for an index that could relate information from the various lots into a single value. Pearn, Kotz and Johnson [6] discussed the distributional roerties of the three basic indices,, k and, and their estimators. A new index m was roosed, which was more mk
17 sensitive to the dearture of the rocess mean from the target value and thus able to distinguish between offtarget and ontarget rocesses. The index is the simlest caability index and is defined as follows [1]: 17 USL LSL = 6σ (2.1) where USL = Uer secification limit, LSL = Lower secification limit, and σ = rocess standard deviation. An estimate of is Ĉ, which is given by: ˆ USL LSL = (2.2) 6σˆ where σˆ = samle standard deviation. Note that the only random variable in equation (2.2) is σˆ. The samle standard deviation σˆ, follows a chisquared distribution. Therefore, Kane [3] concluded that the samling distribution for Ĉ is also related to the chisquared distribution. From the definition of the rocess caability ratio, it is aarent that measures otential caability as defined by the actual rocess sread and does not consider the rocess mean [1], so it gives no indication of actual rocess erformance.
18 measures the sread of the secifications relative to the 6 σ sread in the rocess. k was created to comensate for this weakness. [1]: The rocess caability ratio 18 k is defined by a three ste rocedure as follows µ = USL u (2.3) 3σ k = min( l µ LSL l = 3σ, u min ) = ( µ LSL, USL µ ) 3σ (2.4) (2.5) where µ = rocess mean. An estimate of k is Ĉ k, which is given by: ( ˆ µ LSL, USL ˆ µ ) min ˆ k = (2.6) 3 ˆ σ where µˆ = rocess samle mean. Note that function, the distribution of Ĉ k is a function of both σˆ and µˆ. Also, because of the min Ĉ k involves the joint distribution of two noncentral t distributed random variables [1]. This distribution is very comlex and makes determining the exact confidence interval on Ĉ k very difficult.
19 The formula for k takes into account the rocess mean. Deviation from the rocess mean is reflected in the value of the index. The caability of onesided secification limits can be determined by using the aroriate u or l. k by itself does not adequately measure rocess caability. It is an inadequate measure of rocess centering. The three rocesses in Figure 2.1 [6] have their tabulated in Table 2.1. and 19 k values B A LSL µ USL Figure 2.1: Three rocesses with k =1 and their other aability Indices. Table 2.1: Three Processes with their and k values. Process µ σ k A B
20 and To characterize rocess centering satisfactorily, 20 k must be comared to. k together give a good indication of rocess caability with regard to both rocess sread and rocess location. Processes are designed to achieve certain target values. Deviation from a target value can result in loss of quality and cost escalation [7]. and k do not account for deartures from target values. Since their develoment, several new indices, such as m, mk, ( u, v) have been develoed which address the shortcomings of and k. The focus of this thesis, however, is on and used indices in industry [1]. k as they are the most widely 2.3 Statistical Inference In statistical inference, samles of data from a oulation are analyzed to draw conclusions about the oulation. onclusions and inferences about oulation arameters are solely based on random samles. Since the samles result in statistics, which themselves are random variables, hyothesis testing is used to make statistical inferences on the samle statistic Statistical Hyothesis Testing Statistical hyothesis testing involves a null hyothesis, H 0, and an alternate hyothesis, H 1. The rejection of H 0 leads to the accetance of the alternate hyothesis, H 1. A null hyothesis concerning a oulation arameter is stated to secify an exact value of the arameter, whereas the alternative hyothesis allows for
21 the ossibility of a range of values. When a null hyothesis is acceted it means that there is not enough evidence to reject it. where Two errors are ossible in hyothesis testing [8]: Tye I error ( α ): Rejecting H 0 when it is true. Tye II error ( β ) : Acceting H 0 when it is false. α and β denote the robability of occurrence of the Tye I and Tye II errors resectively [7]. A useful concet for evaluating the erformance of a test is called the ower of the test. The ower of the test is its ability to correctly reject the null hyothesis. The ower of the test, Power (θ ), is defined as: where θ = testing arameter or statistic ( θ ) = 1 β ( θ ) β ( θ ) = robability of a tye II error for θ. 21 Power (2.7) β ( θ ) should be minimized to increase the ower of the test. Minimizing its value by increasing α to its largest accetable value, or by increasing the samle size. In the ideal condition a dearture from H 0 will be detected with certainty and the ower of the test will be equal to 1. But for a fixed samle size, both α and β cannot simultaneously be minimized. Therefore, a small value of α is selected, and a rejection region, which minimizes β for the samle size n is calculated.
22 The urose of using a rocess caability index is to judge whether a rocess is caable or not caable. This is a decisionmaking rocedure. In rocess caability analysis the following hyothesis is tested. versus the alternative hyothesis For the rocess caability ratio H 0 : The rocess is not caable, H 1 : The rocess is caable., the hyotheses are as follows [3] 22 H 0 : c 0 where H 1 : c 0 is the standard minimal criterion for > c 0 (2.8). The value c 0 is a lower bound rocess caability value and is usually decided by the organization deending uon the critical nature of the rocess. Kane [3] develoed a hyothesis test for the rocess caability ratio. Kane investigated this test and rovides a table to assist calculation of samle sizes and critical values c, to test rocess caabilities. heng [9] develoed the hyothesis test for m using the O curve aroach develoed by Kane [3]. The imetus behind this develoment was to develo a decisionmaking rocedure that was scientifically based. han, heng and Siring [7] also used the O curve aroach to analyze Ĉ when µ = T and a more generalized m case where ( USL T ) ( T LSL).
23 2.3.2 onfidence Intervals for Process aability Ratios 23 Statistical estimation deals with arameter estimates of oulation arameters. A oint estimate of a oulation arameter is a single value of the arameter, which is calculated from a random samle. It is unlikely for a oint estimate to equal the oulation arameter. It is referable to rovide an interval estimate within which the value of the arameter will robably be found. onsiderable work has been done in the area of develoing confidence intervals for rocess caability ratios. Several researchers have develoed confidence intervals for the rocess caability estimators ˆ. Ĉ and Ĉ k. Kane [3] rovided the statistical base for develoing the confidence interval for A 100 ( 1 α )% confidence interval for is given by [1] 2 2 χ α χ / 2 α ˆ 1 / 2 P < < ˆ = 1 α with ( n 1) d.o.f. (2.9) n 1 n 1 The construction of confidence intervals for k is difficult because the distribution of k involves the joint distribution of two noncentral tdistributed random variables. Several authors have constructed aroximate confidence intervals by making various assumtions about the distribution of the oint estimator hou, Owen and Borrego [4] derived the lower confidence limit for with the exact distribution of Ĉ k. Ĉ k. k by working
24 Kusher and Hurley [10] suggested a formula for the lower 100(1α )% limit by using the normal aroximation for the samling distribution of Ĉ k. 24 ˆ k 1 z 1 α 1/ 2 ( 2n 2) (2.10) given by: The 100 ( 1 α )% twosided confidence interval suggested by Heavlin [11] is 1/ 2 ( ) ( ) ˆ n ± + ˆ k z1 α / 2 k 1 + (2.11) 9n n 3 2 n 3 n 1 Nagata and Nagahata [5], [12] develoed an aroximate two sided confidence interval for k which is given by ˆ k ˆ 2 ˆ 2 k 1 z + ˆ k 1 + z + α / 2, k α / 2 (2.12) 2 f 9n 2 f 9n with f = n 1. The aroximate formula develoed by Nagata and Nagahata (see equation 2.12 above) was used to develo the hyothesis test for k. 2.4 Oerating haracteristic urve The β error is the oerating characteristic of the hyothesis test and is, in general, a function of the test arameter, θ [8]. The oerating characteristic (O) curve lots β ( θ ) versus the arameter θ. The O curve is a grahical reresentation of the robability of committing a Tye II error.
25 Kane [3] develoed the O curve for the case where the test arameter θ is the rocess caability ratio. The samle variance follows a chisquared distribution. Kane used this distribution to study the samling variation in 25 Ĉ. The ower equation develoed by Kane is: Power ( ) P[ c] = ˆ (2.13) Power 2 2 = P χ n 1 2 (2.14) c ( ) The β error is calculated using the equation ( ) = 1 Power( ) β (2.15) The ower is a critical comonent in comuting the samle size and critical value in a samling lan. It is the tradeoff between tye I and tye II errors that rovide the two ieces of information necessary to determine the samle size and cutoff criteria. Using equation (2.15) the β error is comuted at different the different β error values versus the corresonding values. Plotting values gives the oerating characteristic curve. By selecting α and β error values and the accetable and rejectable quality levels for, Kane [3] determined the required samle size and critical value for a hyothesis test. In other words, different samle sizes and critical values result in different O curves. For examle, by stating a minimum accetable caability level, ( low), that will be acceted with robability α and a maximum
26 rejectable caability level, ( high) that will be rejected with robability β, one can restate equations (2.9) and (2.14) to obtain: 26 ( high) ( low) = 2 χ ( n 1) 1 2 χ ( β ) ( α ) ( n 1) (2.16) c = n χ 1 ( 2 ( low) ( ) ) n 1 α (2.17) It is now ossible to determine the samle size and critical value from equations (2.16) and (2.17). The oerating characteristic curve (O) for is shown below in Figure O( ) 0.0 Figure 2.2: Oerating characteristic curve for. In industry, samle sizes between 30 and 50 are commonly used for machine qualification studies [2]. The samle sizes are chosen arbitrarily. The O curve method can be used to comare different testing schemes.
27 27 Focussed rocess definition Data collection Sto No Define the tye of study (machine or rocess caability) hoose the weaons In Statistical ontrol? No Systematic assignable cause? Get to know the rocess Evaluate measurement system Yes Estimate rocess caability Yes Adjust for systematic cause Develo samling lan Qualify the results Document results Figure 2.3: The rocess caability Analysis Process. 2.5 The Process aability Analysis rocess. The rocess caability analysis rocess involves the following stes as develoed by Kotz and Lovelace [1]. Focussed rocess definition: A roduct is usually manufactured in more than one ste and often is also defined by more than one critical characteristic. The analyst therefore has to choose from more than one manufacturing rocess. A single rocess and arameter, which will
28 indicate the overall outgoing quality of the roduct, is usually chosen after a careful study of the roduct requirements. 28 Define tye of aability Study Machine or Process: Machine caability studies concern the machines along with the fixtures, tools and gauges that are used in the manufacturing rocess. Machine caability studies are essential for qualifying new equiment, studying the effects of modifications on a machine and comaring machine caability. A machine caability study reflects a machine s reeatability on a shortterm basis. On the other hand, the urose of a rocess caability study is to determine the feasibility of the entire rocess to roduce quality roducts. It involves normal oerating conditions, oerators who tyically run the equiment and regular roduction raw material. Here the rocess reroducibility is measured over a longer eriod of time. Because additional sources of variability are resent, rocess caability is usually lower than machine caability. hoose the Indices: This rocess involves selecting the indices by which to estimate caability. There are many indices to choose from but Often and, k and m are the most oular. k are used together. Using the two indices together gives a better ersective on caability versus erformance versus centering. A choice has to be made between univariate and multivariate indices deending uon the nature of the
29 roduct and the rocess. The choice of index is usually governed by the customer s 29 demand. Practitionerrocess bonding: get to know the rocess: This is a very imortant ste in the caability analysis rocess. Understanding the rocess hels to identify which dimension of the rocess outut truly reflects the rocess behavior. Evaluate the measuring system: The data collected for analysis should be free of measurement error. Measurement error affects the erformance of rocess caability indices. Measurement error is of two tyes. Bias results when the measured variable is constantly offset by a fixed amount. Measurement errors occur when the measurement system adds variability to the measurement rocess and incorrectly measures the variable. When bias is encountered the average of the rocess is affected. This significantly affects and m but is not affected. The resence of stochastic measurement error always results in a decrease in the estimates of, k and m. k Develo the samling lan: It is critical to develo a samling lan because the accuracy of the rocess caability estimates is deendent on the samling size and the method used for samling. Inadequate samle sizes result in oor estimates of the indices, which results
30 in incorrect interretations about the rocess. The rocess caability standard 30 develoed by the ANSI Z1 ommittee on Quality Assurance (1996) outlines suggestions for develoing samling lans. The ability of Ĉ and Ĉ k to reflect the true value of the index deends on the samle size and the samling lan. High Ĉ k values are required to demonstrate rocess caability with small samle sizes. Larger samles result in more recision in estimation and consequently a shorter confidence interval. Ĉ and Data collection: The rocedure for the data collection rocess has to be documented correctly. Nonnormal data, cyclical atterns and correlation in the data should be noted as well as the conditions under which the rocess was run. Oerating conditions include the oerator shift, etc. Verify statistical control: ontrol charts are used to verify the stability of a rocess. The X and R charts are commonly used in caability studies as they include both timetotime variability and random error of the rocess. Process caability and the effectiveness of the control charts are related to each other, which further rovides a relation between rocess caability and control chart sensitivity. In cases where subgrous larger than one are not ossible, Moving Range, usum or EWMA charts are used to determine statistical control. Process caability should not be estimated when the rocess is not in control.
31 Estimate the rocess caability: 31 The rocess caability index can be calculated once the data has been collected and characterized and checked for statistical control. Qualify the results: A oint estimate of a rocess caability index must be qualified by either confidence intervals or a hyothesis test. This ensures that the results are not misinterreted. Also, the managers using the results should be aware of the samling conditions under which the estimates were calculated. Document results: The estimate calculations as well as the analysis should be documented. Documentation of the test conditions, deviations from standard assumtions and samling should be included in the reort. The documentation should also include the confidence intervals and hyothesis test results along with the rocess caability index. Documentation is required for future studies and investigations.
32 32 hater 3: A Hyothesis Test for k 3.1 Hyothesis Testing Hyothesis testing and confidence interval techniques of statistical inference are closely related to each other. onfidence interval estimation involves calculating confidence limits within which the arameter in question lies with a certain robability. 2 For a oulation with mean µ and known σ, the hyothesis test and confidence interval estimation is based on the random variable Z X µ =. σ / n The testing of the hyothesis H 0 : µ = µ 0 versus the hyothesis H 1 : µ µ 0 at a significance level α is equivalent to comuting a 100 ( 1 α )% confidence interval on µ and rejecting H 0 if µ 0 is not inside the confidence interval. If µ 0 is inside the confidence interval then the hyothesis is not rejected. 3.2 Develoment of a Hyothesis test for k Develoing a confidence interval for the rocess caability ratio difficult, because the samling distribution of k was k follows a noncentral t distribution. Several aroximate confidence intervals have been develoed for k and were mentioned in the revious chater. For the develoment of the hyothesis test for the confidence interval formulated by Nagata and Nagahata [5] was used. Nagata and Nagahata develoed the confidence interval based on the normal distribution, given by: k
33 33 ˆ k ˆ 2 ˆ 2 k 1 k 1 z k ˆ α / 2 + k + zα / 2 + (3.1) 2 f 9n 2 f 9n The hyothesis for by Kane [3]. k is develoed similarly to the hyothesis develoed for H 0 : k c ko (3.2) H 1 : k > c ko (3.3) The samle size and the cutoff were calculated by selecting α and β error values and the accetable and rejectable quality levels for k 3.2 illustrate the calculations for samle size and cutoff values.. Figure 3.1 and Figure H 0 H1 α / 2 β α / 2 1 β (low) k c (high) k k Figure 3.1: alculation of cutoff and samle size value for k. For calculating c k we use the Uer Secification Limit for k (low) c k ˆ 2 k ( low) 1 = ˆ k ( low) + zα / 2 + (3.4) 2 f 9n
34 From Figure c k ˆ 2 k ( high) 1 = ˆ k ( high) zβ + (3.5) 2 f 9n ck ˆ k ( high) zβ = (3.6) ˆ 2 k ( high) f 9n Substituting equation (3.4) in equation (3.6) results in the following equation: ˆ 2 k ( low) ˆ 1 k ( low) + zα / f 9n zβ = (3.7) ˆ 2 k ( high) f 9n In equation (3.7) n is the only unknown variable. For a given α, β, AQL and RQL the value of the samle size n can be calculated. Once the value of n is known, substitution in equation (3.4) yields the cutoff value. H 0 H1 α / 2 β α / 2 k (low) (high) k Figure 3.2: alculation of cutoff and samle size value for c k k.
35 The cutoff value c k can only be a single value. The sread of the distribution is a function of the samle size n. By lacing c k with α and β robabilities on either side we can calculate the cutoff value. From Figure c k ˆ 2 k ( low) 1 = ˆ k ( low) + zα / 2 + (3.8) 2 f 9n If the cutoffs for the intervals coincide then c k ˆ 2 k ( high) 1 = ˆ k ( high) zβ + (3.9) 2 f 9n Therefore from equation (3.8) and (3.9) ˆ k This results in ˆ k ( low) ˆ 2 low ˆ 2 k ( ) 1 k high z ˆ ( ) 1 α / 2 + = k ( high) z + (3.10) 2 f 9n 2 f 9n + β ˆ 2 ˆ 2 k ( low) 1 ˆ k ( high) 1 ( low) + zα / 2 + k ( high) zβ + = 0 (3.11) 2 f 9n 2 f 9n Again n is the only unknown quantity and can be calculated from equation (3.11). Once n has been calculated the cutoff value is determined from either equation (3.8) or equation (3.9).
36 3.2 Need for a rogram 36 As mentioned earlier Kane [3] has develoed a hyothesis test for the rocess caability ratio. Kane investigated this test and develoed a table (Table 3.1) for calculating samle sizes and critical values [3]. Table 3.1: Samle size and cutoff Value Determination for Testing [3]. Samle Size ( high) ( low) α = β = 0.10 α = β = c (low) ( high) ( low) c (low) The rogram eliminates the need for such a table. It calculates the samle size and critical values for for and k for user defined values and also lots the O curve. The rogram also calculates the rocess caability indices and k confidence intervals from user entered data. and
37 3.3 The User Interface 37 The interface is an Excel sreadsheet. The user interface is illustrated in the following Figures (Refer Figures 3.3 through 3.11). The Excel object model has been rogrammed using Visual Basic for Alications (VBA) to rovide added functionality. The required data is entered in the designated cells and the results of the calculations are dislayed on the sheet. The user interface consists of 2 Excel sreadsheets. The first sheet titled Hyothesis Test comutes the samle size and cutoff from the user defined samling lans. The second sheet titled PRs and onfidence Intervals allows the user to inut data from a rocess. X and R charts are then lotted to determine whether the rocess is in control or not. Once it is determined that the rocess is in control, the indices confidence intervals. and k can be comuted along with their
38 The Use Interface: Hyothesis Test sheet The Hyothesis Test Sheet is as shown in Figure 3.3 below. Figure 3.3: Hyothesis Test Sheet. By clicking on the otion button or otion button k the user is able to select the rocess caability ratio for which the samle size and cutoff value has to be comuted. omuting the cutoff value for a caability test requires secification of the accetable quality level ( (high) or (high) k ), rejectable quality level
39 39 ( (low) or (low) k ), and the alha and beta error robabilities. The command button ENTER VALUES romts the user to inut the required data as shown in Figure 3.4 below. Figure 3.4: Data inut for samling lan. There are a number of calculations that need to be erformed before the solution is obtained. The command button ALULATE checks for valid data and then roceeds with the calculations. It runs the Excel object Goalseek within Excel to arrive at the samle size. The cutoff value is then calculated from this samle size.
40 The calculations are erformed in the background hidden from the user and the results are dislayed on the sheet as shown in Figure 3.5 below. 40 Figure 3.5: Dislay of samle size and cutoff value. The command button PLOT uses the samle size and cutoff values to generate a table to lot the O curve for for the user defined samling lan. The table is hidden from the viewer. After the table has been generated the O curve is lot. The O curve for the articular samling lan is dislayed on a searate sheet as shown in Figure 3.6. The command button RESET set all cells to initial state. The
41 command button ALULATE PRs AND.I. brings u the PRs and onfidence Interval sheet. 41 Figure 3.6: O curve for The User Interface: PRs and onfidence Intervals The PRs and onfidence Intervals sheet is as shown below in Figure 3.7. The command button DATA TABLE romts the user for samle size and sub grou size. A data table is created for the user defined samle size and sub grou size. After the table is oulated with data from an actual rocess, the samle mean and samle
42 range values are comuted by the command button AL MEANS as shown in Figure Figure 3.7: PRs and onfidence Interval Sheet.
43 43 Figure 3.8: Data sheet with rocess data. Before comuting the desired rocess caability ratio for a rocess, the data has to be checked for normality and the rocess should exhibit control. Once the normality is verified, the X and R charts are lotted. Plotting of the charts involved comuting the uer control limit and the lower control limit for both the charts. The command button XBAR PLOT comutes the control limits for the X and lots the chart on a searate sheet titled XBar chart. The user visually insects the chart to verify rocess control. Only after verifying rocess control can the rocess caability indices be comuted.
44 44 The X chart is as shown in Figure 3.9 below. Figure 3.9: X chart for rocess data. Similarly, the command button RBAR PLOT comutes the uer and lower confidence limits for the R chart.
45 3.10 below. The R chart is lotted as a searate sheet titled RBar chart as shown in Figure 45 Figure 3.10: R chart for rocess data. The command buttons and k comutes the rocess caability ratio and calculates the confidence interval. The command buttons and k romt the user for the uer secification limit (USL) and lower secification limit (LSL) values for the articular rocess and then comute the rocess caability index and the uer and lower confidence intervals. The results are dislayed on the sreadsheet as shown
46 in Figure The command button RESET sets the cells to their initial state and deletes the grahs. 46 Figure 3.11: Process aability Ratio and onfidence Interval. Thus, the alication allows the user to conduct rocess caability studies for user secified samling lans.
47 47 hater 4: Results and Validation 4.1 Results for the Process aability Ratio The rogram was tested for several combinations to roduce samling lans for both the indices. The results from the rogram are listed below. Samling lans for and k were develoed for similar secifications. The samle size and cutoff values for the samling lans for 4.1 below. are tabulated in Table Table 4.1: Table of Samling Plans for Process aability Ratio. Samling Plans Sr. No. α β (low) (high) Samle Size, n utoff, c The rogram also lots the oerating characteristic curve (O urve) for the rocess caability ratio. The O curve resents a grahical reresentation of the samling lans and is useful for comaring the behavior of different lans. The following Figures (Figure 4.1Figure 4.4) illustrate the O curves for the different samling lans shown in Table 4.1.
48 48 Oerating haracteristic curve for O() Figure 4.1: O curve for for samling lan 1. Oerating haracteristic curve for O() Figure 4.2: O curve for for samling lan 2.
49 49 Oerating haracteristic curve for O() Figure 4.3: O curve for for samling lan 3. Oerating haracteristic curve for O() Figure 4.4: O curve for for samling lan 4. Figure 4.5 is a comarison of the O curves roduced by the different samling lans. The O curve allows users to evaluate and comare different testing schemes. The O curve for samling lan1 shows that at 1.41, O( ) aroximately equals 0.40 which imlies that there is a 40% chance of judging the rocess not caable
50 (acceting H 0 ). The true rocess caability must be aroximately 1.6 before there is only a 5% chance of judging a rocess not caable using a cutoff value of 1.41.Similarly the O curves for the other 3 samling lans can be analyzed to evaluate the testing schemes. 50 O urve omarison hart O() Plan1 Plan2 Plan3 Plan4 Figure 4.5: O curve for omarison hart. 4.2 Results for the Process aability Ratio k Samling lans for the rocess caability ratio secifications as. k were develoed for the same
51 The samle size and cutoff values for the samling lans for in Table 4.2 below. 51 k are tabulated Table 4.2: Table of Samling Plans for Process aability Ratio k. Samling Plans Sr. No. α β k (low) k (high) Samle Size, n utoff, c Validation for This section deals with the validation of the rogram outut results. The results are comared to the results from Kane s aer [3]. A rocess caability analysis study is conducted for both the indices Validation of Results for and k Kane [3] resents an examle on how to use his tables to obtain an aroriate samle size and critical value for. In his examle: α = β = 0.05, ( high) = 1. 6, and ( low) = Using his table (see Table 3.3) he obtains: samle size = 70, and critical value = Using a critical value of 1.37 and a samle size of 70, there is a 5% risk of judging a rocess with above 1.6 not caable (i.e., Acceting the null hyothesis
52 H 0 ) and a 5% risk of judging a rocess with the null hyothesis H 0 ). below 1.2 as caable (i.e., Rejecting Results from the rogram: samle size = 68 and critical value = 1.41 The slight difference in the results from the rogram and Kane s tables is illustrated in the Table 4.3 below. Table 4.3: Table of Samling Plan. For Samle size utoff Kane's Tables Program The difference may be attributed to the Goalseek function in Excel. To calculate the samle size the following equation is set to zero: ( high) ( low) χ 2 ( n 1) χ ( 1 β ) ( α ) 2 ( n 1) = 0 For this articular examle Goalseek finds a solution by setting the equation to 0.000l, which is not exactly zero. This roduces the slight difference in the samle size. The difference in the cutoff value is significant and this can be attributed to an inaccuracy in Kanes table [3]. The values in the c / ( low) column are incorrectly comuted. The correct equation for comuting c / ( low) is given by hou et al [4]. The error in the values decreases with increase in the samle size. The table created by Kane (Table 4.4) and the corrected table (Table 4.5) are as shown below.
53 53 Table 4.4: Samle size and cutoff value table by Kane [3]. Samle Size ( high) ( low) α = β = 0.10 α = β = c (low) ( high) ( low) c (low) Table 4.5: orrected Samle size and cutoff value table. Samle Size ( high) ( low) α = β = 0.10 α = β = c (low) ( high) ( low) c (low)
54 4.4 A aability Analysis Study 54 A caability analysis study was erformed to validate the hyothesis test develoed for the rocess caability ratio both the indices. This allowed for comarison of the results for k. The secifications were similar for the and k. The following samling lan was set to erform a caability analysis study: For : α = β = 0.1, ( high) = 1. 7, and ( low) = 1. 2 These secifications roduced a samle size of 29 and a cutoff value of For k : α = 0.1, β = 0.1 k ( high) = 1. 7, and k ( low) = 1. 2 These secifications roduced a samle size of 31 and a cutoff value of The data for a caability analysis study is included in the aendix. The test data were obtained from Montgomery [2] (ages 187,192). The data reresent the inside diameter measurement (mm) on iston rings for an automotive engine roduced by a forging rocess. The secification limits on the iston ring are ± 0.05mm Validation of data for aability Analysis Study Before a caability study is erformed the data has to be checked for normality and the rocess has to exhibit control. The statistical software, Statgrahics was used to lot the frequency histogram The frequency histogram is often used to examine the data. The histogram dislays variation and centering of the data. It also allows for checking normality of the data. The following two figures (Figure 4.6 and Figure 4.7) show the histograms for the data for and k analysis.
55 55 Histogram for Data frequency Figure 4.6: Histogram for Data Data using Statgrahics Histogram for kdata frequency Xbark Figure 4.7: Histogram for k Data using Statgrahics The histograms show that the rocess is aroximately normal, centered on nominal and the sread is within the secifications. It can be concluded that the data is normally distributed. The X and R charts are the most commonly recommended control charts for caability uroses [1], because they include both timetotime variability and random
56 56 variability. The X and R charts and the results are as shown below (Figure 4.8Figure 4.11 and Table 4.6Table 4.7). As both the charts exhibit control it can be concluded that the rocess is in control at the secified levels. Xbar hart for Data UL = Xbar TR = LL = Subgrou Figure 4.8: Xbar chart for Data using Statgrahics. Range hart for Data UL = TR = Range LL = Subgrou Figure 4.9: Range chart for Data using Statgrahics.
57 57 Table 4.6: Summary of results for Data using Statgrahics. Number of Subgrous: 29 Subgrou Size: 5 0 Subgrous excluded UL: +3.0 sigma LL: 3.0 sigma enterline Xbar hart Rbar hart Estimates Process Mean Process Sigma Mean Range Xbar hart for kdata Xbar UL = TR = LL = Subgrou Figure 4.10: Xbar chart for k Data using Statgrahics.
58 58 Range hart for kdata Range UL = TR = LL = Subgrou Figure 4.11: Range chart for k Data using Statgrahics. Table 4.7: Summary of results for k Data using Statgrahics. Number of Subgrous: 31 Subgrou Size: 5 0 Subgrous excluded UL: +3.0 sigma LL: 3.0 sigma enterline Xbar hart Rbar hart Estimates Process Mean Process Sigma Mean Range
59 The following samling lan was used to erform a caability analysis study: 59 For : α = β = 0.1, ( high) = 1. 7, and ( low) = 1. 2 For k : α = 0.1, β = 0.1 k ( high) = 1. 7, and k ( low) = Hyothesis Test and onfidence Interval for The hyothesis test for To demonstrate caability is as follows H 0 : ˆ c H : ˆ > c 1 Ĉ must exceed This value is comuted from the samling lan for the rocess caability analysis study for. ˆ USL LSL = 6σˆ onfidence interval for R ˆ σ = = /2.326 = d 2 ˆ = = ( ) is comuted using equation (2.9)
60 Hyothesis Test and onfidence Interval for k Hyothesis test for k To demonstrate caability H 0 : ˆ k c k H ˆ 1 : k > c k Ĉ k must exceed This value is comuted from the samling lan for the rocess caability analysis study for k. ˆ k = min( ˆ l, ˆ u ) = min ( ˆ µ LSL, USL ˆ µ ) 3 ˆ σ R ˆ σ = = /2.326 = d 2 ˆ k = min( ˆ l, ˆ u ) = min ( ˆ µ LSL, USL ˆ µ ) 3 ˆ σ onfidence interval for Results from the rogram Ĉ = 1.63 k k is comuted using equation (2.12) k 2.04 The following figures (Figure 4.12 and Figure 4.13) illustrate the rocess caability ratio and confidence interval values from the rogram
61 61 Figure 4.12: alculation of and confidence interval.
62 62 Figure 4.13: alculation of k and confidence interval.
63 63 Both Ĉ and Ĉ k have values greater than the cutoff values for their resective samling lans. Thus, the rocess exhibits control at the required secifications. The confidence intervals on both and k are extremely wide. This is because of the moderately small samle size involved. In this case the confidence interval reveals little about the caability of the rocess. A hyothesis test becomes a useful tool to qualify the results.
64 64 hater 5: onclusions 5.1 onclusions This thesis has develoed a hyothesis test for the rocess caability ratio The roer utilization of estimation qualification is imortant in the case of k. k as it is widely used in managerial decisions. Qualification of an estimate is rovided in the form of confidence intervals or hyothesis testing. The hyothesis test develoed in this thesis will rovide ractitioners with an additional method for estimate qualification. as well as The rogram rovides an interface for users to develo samling lans for. This eliminates the need for tables as develoed by Kane. The rogram can calculate samle sizes and cutoff values for userdefined data. The rogram also allows the user to erform rocess caability analysis studies. It allows the user to k check for rocess control using the _ X and _ R charts and then comutes the required index and confidence interval. The samling lans develoed for both the indices can be utilized for caability analysis studies as shown in the validation examle. Due to the nature of develoment of caability indices several recautions need to be taken when using the indices. The data from the rocess should be normal and the rocess should be in control. and k are not designed to handle nonnormal data. There are secific indices that address nonnormality in the data. The hyothesis test for k has been develoed from an aroximate confidence interval. This fact should
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