Chapter 5 Exercise Solutions

Transcription

1 -bar R Chapter Eercise Solutions Notes:. Several eercises in this chapter differ from those in the th edition. An * indicates that the description has changed. A second eercise number in parentheses indicates that the eercise number has changed. New eercises are denoted with an.. The MINITAB convention for determining whether a point is out of control is: () if a plot point is within the control limits, it is in control, or () if a plot point is on or beyond the limits, it is out of control.. MINITAB uses pooled standard deviation to estimate standard deviation for control chart limits and capability estimates. This can be changed in dialog boes or under Tools>Options>Control Charts and Quality Tools>Estimating Standard Deviation.. MINITAB defines some sensitizing rules for control charts differently than the standard rules. In particular, a run of n consecutive points on one side of the center line is defined as 9 points, not. This can be changed under Tools > Options > Control Charts and Quality Tools > Define Tests. -. (a) for n =, A =.77, D =., D = m.... m R R Rm R.7 m UCL A R..77(.7).7 R CL. LCL A R..77(.7).9 UCL DR.(.7) 9.9 R CL R.7 LCL DR(.7). R X-bar Chart for Bearing ID (all samples in calculations) R chart for Bearing ID (all samples in calculations). 9. UCL = UCL =.7. CL =.. CL =.7. LCL = No No. LCL = -

2 -bar R Chapter Eercise Solutions - (a) continued The process is not in statistical control; is beyond the upper control limit for both No. and No.. Assuming an assignable cause is found for these two out-of-control points, the two samples can be ecluded from the control limit calculations. The new process parameter estimates are:.; R.; ˆ R / d./..9 UCL.;CL.; LCL. UCL 9.;CL.; LCL. R R R -bar Chart for Bearing ID (samples, ecluded) R chart for Bearing ID (samples, ecluded). 9. UCL = UCL =... CL =. CL =.. LCL = No No. LCL = pˆ Pr{ LSL} Pr{ USL} Pr{ } Pr{ } Pr{ } Pr{ } ( 7.7) (.9)

3 Range Mean Chapter Eercise Solutions -. (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of E-V. UC L=... X=. 7.. LC L=.77 UC L=. R=. LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. n,., R., ˆ X R / d./.9.. Actual specs are V. With i = (observed voltage on unit i ) : USL T = +, LSL T = ˆ USL LSL ( ) C P.9, so the process is capable. ˆ (.) MTB > Stat > Quality Tools > Capability Analysis > Normal Process Capability Analysis of E-V Process Data LSL -. Target * USL. Mean. N StDev(Within). StDev(Overall). LSL USL Within Overall Potential (Within) Capability Cp.9 CPL. CPU. Cpk. CCpk.9 Overall Capability Pp. PPL. PPU. Ppk. Cpm * Observed Performance PPM < LSL. PPM > USL. PPM Total. Ep. Within Performance PPM < LSL. PPM > USL. PPM Total. Ep. Overall Performance PPM < LSL. PPM > USL. PPM Total. -

4 Range Mean Percent Chapter Eercise Solutions - continued (c) MTB > Stat > Basic Statistics > Normality Test Probability Plot of E-V Normal Mean. StDev. N AD.7 P-Value.. E-V A normal probability plot of the transformed output voltage shows the distribution is close to normal. -. (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of E-Dia UC L=7. X=.9 - LC L=-.7 UC L=. R=. LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. -

5 Chapter Eercise Solutions - continued ˆ R/ d./. 7. (c) USL = +, LSL = ˆ USL LSL ( ) CP., so the process is capable. ˆ (7.) MTB > Stat > Quality Tools > Capability Analysis > Normal Process Capability Analysis of E-Dia Process Data LSL -. Target * USL. Mean.9 N StDev(Within) 7.9 StDev(Overall).9 LSL USL Within Overall Potential (Within) Capability Cp. CPL. CPU.9 Cpk.9 CCpk. Overall Capability Pp. PPL. PPU.7 Ppk.7 Cpm * Observed Performance PPM < LSL. PPM > USL. PPM Total. Ep. Within Performance PPM < LSL. PPM > USL 9.79 PPM Total 7.9 Ep. Overall Performance PPM < LSL. PPM > USL.7 PPM Total 9. -

6 Range Mean Chapter Eercise Solutions -. (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of Thickness (E-Th). UC L=.9.. X=.9.. LC L=.. UC L=.... R=.9. LC L= Test Results for Xbar Chart of E-Th TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points: Test Results for R Chart of E-Th TEST. One point more than. standard deviations from center line. Test Failed at points: * WARNING * If graph is updated with new data, the results above may no * longer be correct. -

7 Range Mean Chapter Eercise Solutions - continued The process is out-of-control, failing tests on both the and the R charts. Assuming assignable causes are found, remove the out-of-control points (samples, ) and recalculate control limits. With the revised limits, sample is also out-of-control on the chart. Removing all three samples from calculation, the new control limits are: Xbar-R Chart of Thickness (E-Th) (s,, removed from control limits calculations). UCL=.77.. X=.9.. LCL=.. UCL=.... R=.. LCL= ˆ R/ d./.9. (c) Natural tolerance limits are: ˆ.9 (.) [.9,.] -7

8 Chapter Eercise Solutions - continued (d) Assuming that printed circuit board thickness is normally distributed, and ecluding samples,, and from the process capability estimation: ˆ USL LSL. (.) CP. ˆ (.) MTB > Stat > Quality Tools > Capability Analysis > Normal Process Capability Analysis of Thickness (E-Thw/o) (Estimated without s,, ) Process Data LSL. Target * USL. Mean.9 N StDev(Within).9 StDev(Overall). LSL USL Within Overall Potential (Within) Capability Cp. CPL.99 CPU.7 Cpk.99 CCpk. Overall Capability Pp.9 PPL.9 PPU.97 Ppk.9 Cpm * Observed Performance PPM < LSL. PPM > USL. PPM Total. Ep. Within Performance PPM < LSL 7. PPM > USL 9.7 PPM Total 7. Ep. Overall Performance PPM < LSL 9. PPM > USL. PPM Total. -

9 Range Mean StDev Mean Chapter Eercise Solutions -. (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-S (E-Vol) Under Options, Estimate select Sbar as method to estimate standard deviation. Xbar-S Chart of Fill Volume (E-Vol). UC L=.7.. X= LC L= UC L=... S=.. LC L=. 7 9 The process is in statistical control, with no out-of-control signals, runs, trends, or cycles. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R (E-Vol) Xbar-R Chart of Fill Volume (E-Vol). UC L=.9.. X= LC L= UC L=... R=.. LC L= The process is in statistical control, with no out-of-control signals, runs, trends, or cycles. There is no difference in interpretation from the s chart. -9

10 s^ (Variance) Chapter Eercise Solutions - continued (c) Let =.. n =, s =.. CL s.. UCL s ( n). ( ). ( ).. /, n./, LCL s ( n). ( ). ( ).7. ( / ), n (./ ), MINITAB s control chart options do not include an s or variance chart. To construct an s control chart, first calculate the sample standard deviations and then create a time series plot. To obtain sample standard deviations: Stat > Basic Statistics > Store Descriptive Statistics. Variables is column with sample data (E-Vol), and By Variables is the sample ID column (E-). In Statistics select Variance. Results are displayed in the session window. Copy results from the session window by holding down the keyboard Alt key, selecting only the variance column, and then copying & pasting to an empty worksheet column (results in E-Variance). Graph > Time Series Plot > Simple Control limits can be added using: Time/Scale > Reference Lines > Y positions Control Chart for E-Variance. UCL =.... CL =.. LCL = signals out of control below the lower control limit. Otherwise there are no runs, trends, or cycles. If the limits had been calculated using =.7 (not tabulated in tetbook), sample would be within the limits, and there would be no difference in interpretation from either the s or the R chart. -

11 Range Mean Chapter Eercise Solutions -. (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of Net Weight (E-Wt). UC L=... X=.. LC L=.99. UC L=..7. R=.7.. LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. n ;.; R.7; ˆ R / d.7/.. -

12 Percent Frequency Chapter Eercise Solutions - continued (c) MTB > Graph > Histogram > Single (E-Wt) Histogram of Net Weight (E-Wt)... E-Wt.. MTB > Graph > Probability Plot > Single (E-Wt) Probability Plot of Net Weight (E-Wt) Normal - 9% CI Mean.7 StDev. N AD.7 P-Value < E-Wt Visual eamination indicates that fill weights approimate a normal distribution - the histogram has one mode, and is approimately symmetrical with a bell shape. Points on the normal probability plot generally fall along a straight line. -

13 Chapter Eercise Solutions - continued (d) ˆ USL LSL. (.) CP., so the process is not capable of meeting ˆ (.) specifications. MTB > Stat > Quality Tools > Capability Analysis > Normal Under Estimate select Rbar as method to estimate standard deviation. Process Capability Analysis of Net Weight (E-Wt) Process Data LSL.7 Target * USL.7 Mean. N StDev(Within). StDev(Overall).9 LSL USL Within Overall Potential (Within) Capability Cp. CPL.9 CPU.7 Cpk.7 CCpk. Overall Capability Pp. PPL.9 PPU.7 Ppk.7 Cpm *..... Observed Performance PPM < LSL. PPM > USL. PPM Total. Ep. Within Performance PPM < LSL 7. PPM > USL 79. PPM Total 99. Ep. Overall Performance PPM < LSL. PPM > USL.7 PPM Total 7.9 (e).7. pˆ lower Pr{ LSL} Pr{.7} (.7).7. The MINITAB process capability analysis also reports Ep. "Overall" Performance PPM < LSL. PPM > USL.7 PPM Total 7.9 -

14 StDev Mean StDev Mean Chapter Eercise Solutions -7. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-S (E-Vl) Xbar-S Chart of Output Voltage (E-V). UC L=.7.. X=. 7.. LC L=.9. UC L=... S=.7.. LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. -. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-S (E-Dia) Xbar-S Chart of Deviations from Nominal Diameter (E-Dia) UC L=.9 X=.9 - LC L=-. UC L=.7 S=. LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. -

15 Range Mean Chapter Eercise Solutions -9. (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R (E-9ID) 7. Xbar-R Chart of Inner Diameter (E-9ID) UC L=7. 7. X= LC L= UC L=.9.. R=... LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. The control limits on the charts in Eample - were calculated using S to estimate, in this eercise R was used to estimate. They will not always be the same, and in general, the control limits based on S will be slightly different than limits based on R. -

16 Chapter Eercise Solutions -9 continued (c) ˆ R/ d. /..999 ˆ USL LSL , so the process is not capable of meeting CP. ˆ (.999) specifications. MTB > Stat > Quality Tools > Capability Analysis > Normal Under Estimate select Rbar as method to estimate standard deviation. Process Capability Analysis of Inner Diameter (E-9ID) Process Data LSL 7.9 Target * USL 7. Mean 7. N StDev(Within).999 StDev(Overall). LSL USL Within Overall Potential (Within) Capability Cp.7 CPL.7 CPU. Cpk. CCpk.7 Overall Capability Pp. PPL.7 PPU.9 Ppk.9 Cpm * Observed Performance PPM < LSL. PPM > USL. PPM Total. Ep. Within Performance PPM < LSL. PPM > USL. PPM Total. Ep. Overall Performance PPM < LSL. PPM > USL.9 PPM Total. pˆ Pr{ LSL} Pr{ USL} Pr{ 7.9} Pr{ 7.} Pr{ 7.9} Pr{ 7.} (.) (.) -

17 Range Mean Chapter Eercise Solutions -. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R (E-ID) Xbar-R Chart of Inner Diameter (E-ID) UC L=7. 7. X= LC L= UC L=.9.. R=... LC L= Test Results for Xbar Chart of E-ID TEST. One point more than. standard deviations from center line. Test Failed at points: 7,, 9 TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:, 7,, 9, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 9, The control charts indicate that the process is in control, until the -value from the 7 th sample is plotted. Since this point and the three subsequent points plot above the upper control limit, an assignable cause has likely occurred, increasing the process mean. -7

18 Chapter Eercise Solutions - (-9). n ; in-lb; in-lb; and A.99; B.9; B.7 centerline UCL A.99() 9.9 LCL A.99() 7. centerline c.977() 9.77 S UCL B.9().9 S LCL B.7().7 S -* (-). n items/sample; ; R ; m samples (a) i i i i Ri i i ; R m m UCL A R.().9 LCL A R.(). UCL DR.(). R LCL DR() R natural tolerance limits: ˆ R d i / (/.) [.,.7] (c) ˆ USL - LSL. (.) CP., so the process is not capable. ˆ (.79) (d) pˆ scrap Pr{ LSL} Pr{ } (.).7, or.7% pˆ rework Pr{ USL} Pr{ USL} (.) or.%. (e) First, center the process at, not, to reduce scrap and rework costs. Second, reduce variability such that the natural process tolerance limits are closer to, say, ˆ.. -

19 Chapter Eercise Solutions -* (-). n items/subgroup; ; S 7; m subgroups (a) i i i i i Si i 7 S. m UCL A S.(.). LCL A S.(.) 7. UCL BS.(.). S LCL BS(.) S m i natural process tolerance limits: S. ˆ [.,.7] c.9 (c) ˆ USL - LSL. (.) CP., so the process is not capable. ˆ (./.9) (d) pˆ rework Pr{ USL} Pr{ USL} (.99).97.7./.9 or.7%. pˆ scrap Pr{ LSL} (.99).9, or.9%./.9 Total =.% +.9% =.99% (e) 9 pˆ rework (.).9977., or.%./.9 9 pˆ scrap (.)., or.%./.9 Total =.% +.% =.% Centering the process would reduce rework, but increase scrap. A cost analysis is needed to make the final decision. An alternative would be to work to improve the process by reducing variability. -9

20 Range Mean Range Mean Chapter Eercise Solutions - (-). (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of Critical Dimension (E-a,..., E-a) UC L=. X=. LC L=7. UC L=. R=. LC L= The process is in statistical control with no out-of-control signals, runs, trends, or cycles. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Under Options, Estimate use subgroups : to calculate control limits. Xbar-R Chart of Critical Dimension (E-b,..., E-b) UC L=. X=. 9 7 LC L=7. UC L=. R=. LC L= 9 7 Starting at #, the process average has shifted to above the UCL =.. -

21 Range Mean Chapter Eercise Solutions - continued (c) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Under Options, Estimate use subgroups : to calculate control limits. Xbar-R Chart of Critical Dimension (E-c,..., E-c) UC L=. X=. LC L=7. UC L=. R=. LC L= The adjustment overcompensated for the upward shift. The process average is now between and the LCL, with a run of ten points below the centerline, and one sample (#) below the LCL. -

22 Range Mean Chapter Eercise Solutions -* (-). (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of Strength Test (E-aSt). UC L=... X= LC L=7.9 UC L=.9 R=.7 LC L= Yes, the process is in control though we should watch for a possible cyclic pattern in the averages. -

23 Range Mean Chapter Eercise Solutions - continued MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Under Options, Estimate use subgroups : to calculate control limits. Xbar-R Chart of Strength Test (E-bSt). UC L=... X= LC L= UC L=.9 R=.7 LC L= 9 7 Test Results for R Chart of E-bSt TEST. One point more than. standard deviations from center line. Test Failed at points:,, 7,,,, TEST. 9 points in a row on same side of center line. Test Failed at points:,,, A strongly cyclic pattern in the averages is now evident, but more importantly, there are several out-of-control points on the range chart. -

24 StDev Mean StDev Mean Chapter Eercise Solutions - (-). (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-S Xbar-S Chart of Strength Test (E-aSt) Original Data. UCL=... X= LCL=7. UCL=7. S=.7 LCL= Under Options, Estimate use subgroups : to calculate control limits. Xbar-S Chart of Strength Test (E-bSt) Original plus New Data. UCL=... X= LCL= UCL=7.7.. S=.7. LCL= 9 Test Results for Xbar Chart of E-bSt TEST. One point more than. standard deviations from center line. Test Failed at points:,, Test Results for S Chart of E-bSt TEST. One point more than. standard deviations from center line. Test Failed at points:,,, 7,,,, 7 -

25 Chapter Eercise Solutions - continued Yes, the s chart detects the change in process variability more quickly than the R chart did, at sample # versus sample #. -7 (-). n ;.; R.7 old old old (a) for n new = d (new).9 UCL old A(new) Rold. (.7) 7. d (old). d (new).9 LCL old A(new) Rold. (.7). d (old). d (new).9 UCLR D(new) Rold.7 d (.7). (old). d (new).9 Rold d (old). CL R Rnew (.7). d (new).9 (new) Rold d (old). LCLR D (.7) The control limits for n = are tighter (.9,.7) than those for n = (., 7.). This means a shift in the mean would be detected more quickly with a sample size of n =. -

26 Chapter Eercise Solutions -7 continued (c) for n = d (new).7 UCL old A(new) Rold.7 (.7). d (old). d (new).7 LCL old A(new) Rold.7 (.7). d (old). d (new).7 UCLR D(new) Rold. d (.7).7 (old). d (new).7 Rold d (old). CL R Rnew (.7).7 d (new).7 (new) Rold d (old). LCLR D. (.7).7 (d) The control limits for n = are even "tighter" (.,.), increasing the ability of the chart to quickly detect the shift in process mean. -. n old =, old = 7., R old =., n new = d (new).9 UCL old A(new) Rold 7.. (.) 7. d (old). d (new).9 LCL old A(new) Rold 7.. (.) 7.9 d (old). UCL R D (new) d (new).9 Rold.7 (.). d (old). d (new).9 Rold d (old). CL R Rnew (.).7 d (new).9 (new) Rold d (old). LCLR D (.) -

27 Chapter Eercise Solutions -9 (-). n 7; 7; R ; m samples (a) i i i i i 7 Ri i R.9 m UCL A R.9(.9) 7.7 LCL A R.9(.9). UCL DR.9(.9).97 R LCL DR.7(.9). R m ˆ ; ˆ R / d.9/.7. i (c) ˆ USL LSL ( ) CP.9, the process is not capable of meeting ˆ (.) specifications. pˆ Pr{ USL} Pr{ LSL} Pr{ USL} Pr{ LSL} Pr{ } Pr{ } (.) (.) (d) The process mean should be located at the nominal dimension,, to minimize nonconforming units. pˆ (.7) (.7)

28 Chapter Eercise Solutions - (-7). n ;.; R 9.; m samples (a) i i i i i.. Ri i 9. R. m UCL A R..77(.).7 LCL A R..77(.).9 UCL DR.(.).7 R LCL DR(.) R m i ˆ R/ d./.. pˆ Pr{ USL} Pr{ LSL} Pr{ USL} Pr{ LSL} (.) (.7) (c) pˆ (.) (.)

29 Chapter Eercise Solutions - (-). n ;.; S.; m samples (a) ˆ S / c./.9. UCL A S..7(.). LCL A S..7(.) 7. UCL BS.9(.). S LCL BS(.) S (c) Pr{in control} Pr{LCL UCL} Pr{ UCL} Pr{ LCL}. 7. (.) (.79) (-9). Pr{detect} Pr{not detect} [Pr{LCL UCL}] [Pr{ UCL} Pr{ LCL}] UCL LCL new new 9 9 n n (7) ().. - (-). X ~ N; n ; ; R 9.; USL=; LSL=9 ˆ / ˆ R d 9./..99 and (.99).99 is larger than the width of the tolerance band, () =. So, even if the mean is located at the nominal dimension,, not all of the output will meet specification. ˆ USL LSL ( ) CP. ˆ (.99) -9

30 Chapter Eercise Solutions -* (-). n ; ;.. These are standard values. (a) centerline UCL A.(.). LCL A.(.).7 centerline d.(.). R UCL D.(.) 9. R LCL D (.) R (c) centerline c.7979(.).99 S UCL B.(.). S LCL B (.) S -

31 Chapter Eercise Solutions - (-). n ; ; R.; m samples (a) UCL A R.77(.). LCL A R.77(.) 7.7 UCL DR.(.) 9. R LCL DR(.) R ˆ R/ d./..9 (c) ˆ USL LSL ( ) CP., so the process is not capable of meeting ˆ (.9) specifications. (d) Pr{not detect} Pr{LCL UCL} Pr{ UCL} Pr{ LCL} UCL new LCL new. 7.7 ˆ ˆ n n.9.9 (.) ( 7.).9.9 -

32 Range Mean Range Mean Chapter Eercise Solutions -. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of TiW Thickness (E-Th) UC L=. X=.9 LC L=. UC L=7.9 R=. Test Results for Xbar Chart of E-Th TEST. One point more than. standard deviations from center line. Test Failed at points: LC L= The process is out of control on the chart at subgroup. Ecluding subgroup from control limits calculations: Xbar-R Chart of TiW Thickness (E-Th) Ecluding subgroup from calculations UCL=. X=9. LCL=7.9 UCL=. R=.7 LCL= Test Results for Xbar Chart of E-Th TEST. One point more than. standard deviations from center line. Test Failed at points: No additional subgroups are beyond the control limits, so these limits can be used for future production. -

33 Percent Chapter Eercise Solutions - continued Ecluding subgroup : 9. ˆ R/ d.7/.9. (c) MTB > Stat > Basic Statistics > Normality Test Probability Plot of TiW Thickness (E-Th) Normal Mean.7 StDev 9. N AD.9 P-Value.7. E-Th 7 A normal probability plot of the TiW thickness measurements shows the distribution is close to normal. -

34 Chapter Eercise Solutions - continued (d) USL = +, LSL = ˆ USL LSL ( ) CP., so the process is capable. ˆ (.) MTB > Stat > Quality Tools > Capability Analysis > Normal Process Capability Analysis of TiW Thickness (E-Th) Process Data LSL. Target * USL. Mean.7 N StDev(Within). StDev(Overall) 9.9 LSL USL Within Overall Potential (Within) Capability Cp. CPL. CPU.9 Cpk. CCpk. Overall Capability Pp.9 PPL. PPU. Ppk. Cpm * 7 Observed Performance PPM < LSL. PPM > USL. PPM Total. Ep. Within Performance PPM < LSL 9. PPM > USL.9 PPM Total. Ep. Overall Performance PPM < LSL. PPM > USL.7 PPM Total. The Potential (Within) Capability, Cp =., is estimated from the within-subgroup variation, or in other words, is estimated using R. This is the same result as the manual calculation. -

35 Range Mean Chapter Eercise Solutions -7. MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of TiW Thickness (E-7Th) Using previous limits with new subgroups UCL=. X=9. LCL= UCL=. R=.7 LCL= 9 7 Test Results for Xbar Chart of E-7Th TEST. One point more than. standard deviations from center line. Test Failed at points: The process continues to be in a state of statistical control. -

36 Range Mean Chapter Eercise Solutions -. n ; 9.; R.7; n old old old new d (new). UCL old A(new) Rold 9.. (.7).9 d (old).9 d (new). LCL old A(new) Rold 9.. (.7) d (old).9. d (new). (new) Rold d (old).9 UCLR D.7 (.7) 9.9 d (new). Rold d (old).9 CL R Rnew (.7) 9.7 d (new). (new) Rold d (old) LCLR D.9 ˆ R d new new (new) (.7) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Select Xbar-R options, Parameters, and enter new parameter values. Xbar-R Chart of TiW Thickness (E-Th) New subgroups with N=, Limits derived from N= subgroups 7 UCL=.9 X= LCL=. UCL=9.9 R=9.7 LCL= 7 9 The process remains in statistical control. -

37 StDev Mean StDev Mean Chapter Eercise Solutions -9. The process is out of control on the chart at subgroup. After finding assignable cause, eclude subgroup from control limits calculations: MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-S Xbar-S Chart of Thickness (E-Th) Ecluding subgroup from calculations UCL=. X=9. LCL=7. UCL=7. S=7.7 LCL= Xbar-S Chart of E-Th Test Results for Xbar Chart of E-Th TEST. One point more than. standard deviations from center line. Test Failed at points: No additional subgroups are beyond the control limits, so these limits can be used for future production. Xbar-S Chart of Thickness (E-7Th) subgroups of new data, with prior limits UCL=. X=9. LCL= UCL=7. S=7.7 LCL= 9 7 The process remains in statistical control. -7

38 Chapter Eercise Solutions - (-). n ; ; R ; m samples (a) i i i i i Ri i R m UCL A R.(). LCL A R.() 97.9 UCL DR.(). R LCL DR() R m ˆ R/ d /..97 i ˆ USL LSL ( ) Cp. ˆ (.97) The process is not capable of meeting specification. Even though the process is centered at nominal, the variation is large relative to the tolerance. (c) risk Pr{not detect} (.) (.7)

39 Chapter Eercise Solutions - (-). ; L ; n ; ; 9 k 9. Pr{detecting shift on st sample} Pr{not detecting shift on st sample} L k n L k n (.) (.) (.) (.) (-). (a).; R.9 UCL A R..77(.9).9 LCL A R..77(.9).77 UCL DR.(.9). R LCL DR(.9) R # is out of control on the Range chart. So, ecluding # and recalculating: ; R.79 UCL A R.77(.79). LCL A R.77(.79).9 UCL DR.(.79) 7. R LCL DR(.79) R Without sample #, ˆ R/ d.79/..9 (c) UNTL ˆ (.9). LNTL ˆ (.9)

40 Chapter Eercise Solutions - continued (d) 7 99 pˆ (.9) (.) (e) To reduce the fraction nonconforming, first center the process at nominal pˆ (.) (.) Net work on reducing the variability; if ˆ.7, then almost % of parts will be within specification pˆ (.997) (.997) (-). n ; 7.; R ; m (a) m i i i 7. i i m. Ri i R. m UCL A R..77(.). LCL A R..77(.) 7.9 UCL D R.(.).7 R LCL D R (.) R m ˆ R/ d./.. i. pˆ Pr{ LSL} (.).9. -

41 Range Chapter Eercise Solutions - (-7). (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > R Under Options, Estimate select Rbar as method to estimate standard deviation. R Chart of Detent (E-Det) UCL=.7 R=.7 LCL= 7 9 Test Results for R Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points: Process is not in statistical control -- sample # eceeds the upper control limit on the Range chart. -

42 Range Chapter Eercise Solutions - continued Ecluding Number : MTB > Stat > Control Charts > Variables Charts for Subgroups > R Under Options, Estimate omit subgroup and select Rbar. R Chart of Detent (E-Det) Ecluded from Calculations UCL=.9 R=. LCL= 7 9 Test Results for R Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points: (c) Without sample #: ˆ R/ d./.. (d) Assume the cigar lighter detent is normally distributed. Without sample #: ˆ USL LSL.. CP. ˆ (..) -

43 Range Mean Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Subgroups > R Under Options, Estimate use subgroups : and :, and select Rbar. Xbar-R Chart of E-Det Limits based on s -, -... UCL=. X= LCL=-. UCL=.9 R=. LCL= Test Results for Xbar Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points:,,,,, 7,,, TEST. 9 points in a row on same side of center line. Test Failed at points:, TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:,,, 7,, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 9,,,, Test Results for R Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. 9 points in a row on same side of center line. Test Failed at points:, -

44 Range Mean Chapter Eercise Solutions - continued We are trying to establish trial control limits from the first samples to monitor future production. Note that samples,,, and are out of control on the chart. If these samples are removed and the limits recalculated, sample is also out of control on the chart. Removing sample gives Xbar-R Chart of E-Det Limits based on first samples, ecluding,,, and... UCL=.9 X= LCL=-. UCL=9. R=. LCL= is now out of control on the R chart. No additional samples are out of control on the chart. While the limits on the above charts may be used to monitor future production, the fact that of samples were out of control and eliminated from calculations is an early indication of process instability. (a) Given the large number of points after sample beyond both the and R control limits on the charts above, the process appears to be unstable. -

45 Range Mean Chapter Eercise Solutions - continued Xbar-R Chart of Detent (E-Det) All s in Calculations. UCL=... X= LCL=-. UCL=. R=7. LCL= With Test only: Test Results for Xbar Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points:,,,, 7 Test Results for R Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points: -

46 Range Mean Chapter Eercise Solutions - continued Removing samples,,,, and 7 from calculations: Xbar-R Chart of Detent (E-Det) s,,,, 7 ecluded from calculations. UCL=.9.. X= LCL=-.9 UCL=. R=. LCL= With Test only: Test Results for Xbar Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points:,,,, 7, Test Results for R Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points: -

47 Range Mean Chapter Eercise Solutions - continued is now also out of control. Removing sample from calculations, Xbar-R Chart of E-Det s,,,, 7, ecluded from calculations. UCL=... X= LCL=-. UCL=. R=.7 LCL= With Test only: Test Results for Xbar Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points:,,,, 7,, Test Results for R Chart of E-Det TEST. One point more than. standard deviations from center line. Test Failed at points: is now out-of-control, for a total 7 of the samples, with runs of points both above and below the centerline. This suggests that the process is inherently unstable, and that the sources of variation need to be identified and removed. -7

48 Chapter Eercise Solutions - (-9). (a) i n ; m ; m ; R.; R.97 y, i y, i i i ˆ R / d R / m d (./ ) /.., i ˆ R / d R / m d (.97/) /.. y y y, i y i Want Pr{( y) <.9} =.. Let z = y. Then ˆ ˆ ˆ z y....9 z. ˆ z.9 z (.)..9 z.. z.(.) (-). n ;,7; R ; m (a) m i i i Ri i R. m UCL DR.(.) 9. R LCL DR(.) R m i i,7 ˆ 9. m ˆ R/ d. /. 7.7 i -

49 Chapter Eercise Solutions -7 continued (c) USL = + = ; LSL = - = ˆ USL LSL Cp.7 ˆ (7.7) 9 9 pˆ (.7) (.) (d) To minimize fraction nonconforming the mean should be located at the nominal dimension () for a constant variance. - (-). n ;,7; S ; m (a) m i i i Si i S.7 m UCL BS.(.7).99 S LCL BS(.7) S m i i,7 ˆ 9. m ˆ S / c.7 /.9. i -9

50 Chapter Eercise Solutions -9 (-). (a) n ; ; n n UCL LCL 9 k Z Z Z /./..7 UCL k k n.7. LCL k k n (-). n ; UCL ; centerline ; LCL 9; k ; 9; Pr{out-of-control signal by at least rd plot point} Pr{not detected by rd sample} [Pr{not detected}] Pr{not detected} Pr{LCL UCL } Pr{ UCL } Pr{ LCL } UCL LCL (.) (.) [Pr{not detected}] (.).79 - (-). ARL.99 Pr{not detect}. - (-). ˆ USL LSL USL LSL.-97. CP.77 ˆ Sc..9 The process is not capable of meeting specifications. -

51 beta Chapter Eercise Solutions - (-). n ; ; (a) centerline c.9() 9. S UCL B.(). S LCL B () S k Z Z Z /./..9 UCL k k n.9 9. LCL k k n (-7). n = 9; USL = + = ; LSL = = (a) ˆ USL LSL USL LSL CP. ˆ 7./.97 Rd Process is capable of meeting specifications. n 9; L ; L k n L k n for k = {,.,.7,.,.,.,.,.,.}, = {.997,.9,.77,.,.,.,.,.,.} Operating Characteristic Curve for n = 9, L = k -

52 Chapter Eercise Solutions - (-). n 7; 7; R ; m (a) m i i i i 7 Ri i i 9; R m m UCL A R 9.9() 9.7 LCL A R 9.9(). UCL D R.9() 7.9 R LCL D R.7(). R ˆ R/ d /.7.79 (c) S cˆ.99(.79).9 UCL.(.9).7 S LCL.(.9).7 S i m - (-9). n 9; ; ; =. k Z Z Z.7 /./. UCL k k n.7 9. LCL k k n (-). ˆ R/ d.9/.9 Pr{detect shift on st sample} Pr{ LCL} Pr{ UCL} Pr{ LCL} Pr{ UCL} LCL new UCL new ( ) ()

53 Chapter Eercise Solutions - (-). ARL. Pr{not detect} Pr{detect}.7-9 (-). (a) ˆ R/ d.9/.97. LCL UCL Pr{ LCL} Pr{ UCL} 7 ( ) () ˆ USL LSL ( ) CP.7 ˆ () The process is not capable of producing all items within specification. (c) new = 7 Pr{not detect on st sample} Pr{LCL UCL} UCL LCL new ˆ ˆ n n new () () (d).; k Z Z Z.7 /./. ˆ UCL k k n LCL

54 Chapter Eercise Solutions - (-). (a) ˆ R/ d./.9. S c ˆ.9(). UCL BS.(.). S LCL BS(.) S (c) LSL USL pˆ Pr{ LSL} Pr{ USL} ˆ ˆ 9 (.) (.)..9. (d) To reduce the fraction nonconforming, try moving the center of the process from its current mean of closer to the nominal dimension of. Also consider reducing the process variability. (e) Pr{detect on st sample} Pr{ LCL} Pr{ UCL} LCL new UCL new () () (f) Pr{detect by rd sample} Pr{not detect by rd sample} (Pr{not detect}) (.977). -

55 Chapter Eercise Solutions - (-). (a) ˆ 7.; ˆ S / c.7/.9.7 UNTL ˆ 7 (.7) 7. LNTL 7 (.7) 7. (c) pˆ Pr{ LSL} Pr{ USL} LSL USL ˆ ˆ (.) (.) (d) Pr{detect on st sample} Pr{ LCL} Pr{ UCL} LCL new UCL new (.) (.) (e) Pr{detect by rd sample} Pr{not detect by rd sample} (Pr{not detect}) (.99). -

56 Chapter Eercise Solutions - (-). (a) ˆ 7; ˆ S / c 7.979/.9. pˆ Pr{ LSL} Pr{ USL} LSL USL ˆ ˆ (.) (.)..99. (c) Pr{ LCL} Pr{ UCL} LCL UCL (.) (.)..99. (d) Pr{detect on st sample} Pr{ LCL} Pr{ UCL} LCL new UCL new,new,new (.) (.) (e) ARL. Pr{not detect} Pr{detect}. -

57 Percent Moving Range Individual Value Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Weight (E-Wt).7 UC L=... X=... LC L=. Observation. UC L= MR=.7. Observation There may be a sawtooth pattern developing on the Individuals chart. LC L=.; ˆ.; MR.7 MTB > Stat > Basic Statistics > Normality Test Probability Plot of Weight (E-Wt) Normal Mean. StDev. N AD.97 P-Value...7. E-Wt.. Visual eamination of the normal probability indicates that the assumption of normally distributed coffee can weights is valid. %underfilled % Pr{ oz}. % % (.99).%. -7

58 Percent Moving Range Individual Value Chapter Eercise Solutions -(-7). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Hardness (E-Har) UC L=. X=.7 LC L= Observation. UC L= MR=.. LC L= 7 9 Observation.7; ˆ.9; MR.9 MTB > Stat > Basic Statistics > Normality Test Probability Plot of Hardness (E-Har) Normal Mean.7 StDev.7 N AD. P-Value.7 E-Har Although the observations at the tails are not very close to the straight line, the p-value is greater than., indicating that it may be reasonable to assume that hardness is normally distributed. -

59 Moving Range Individual Value Percent Chapter Eercise Solutions - (-). (a) MTB > Stat > Basic Statistics > Normality Test Probability Plot of Viscosity (E-Vis) Normal Mean 99 StDev 9. N AD.9 P-Value. 7 9 E-Vis Viscosity measurements do appear to follow a normal distribution. MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Viscosity (E-Vis) UC L=.9 X=9.9 LC L=.9 Observation UC L=. MR=. Observation The process appears to be in statistical control, with no out-of-control points, runs, trends, or other patterns. LC L= (c) ˆ 9.9; ˆ.; MR. -9

60 Moving Range Individual Value Chapter Eercise Solutions - (-9). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Viscosity (E-Vis) With five net measurements UCL=.9 X=9.9 LCL=.9 Observation UCL=. MR=. LCL= Observation All points are inside the control limits. However all of the new points on the I chart are above the center line, indicating that a shift in the mean may have occurred. -

61 Percent Moving Range Individual Value Chapter Eercise Solutions -7 (-). (a) MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Oide Thickness (E-7aTh) UC L=. X=9. LC L=. 9 Observation 7 UC L=.79 MR=.7 LC L= 9 Observation 7 The process is in statistical control. MTB > Stat > Basic Statistics > Normality Test Probability Plot of Oide Thickness (E-7aTh) Normal Mean 9. StDev. N AD. P-Value. E-7aTh The normality assumption is reasonable. -

62 Moving Range Individual Value Chapter Eercise Solutions -7 continued MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Oide Thickness (E-7bTh) With new measurements and some sensitizing rules 7 UCL=. X=9. LCL=. Observation UCL=.79 MR=.7 LCL= Observation Test Results for I Chart of E-7bTh TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. 9 points in a row on same side of center line. Test Failed at points:, 9, TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:, 9, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 7,, 9, We have turned on some of the sensitizing rules in MINITAB to illustrate their use. There is a run above the centerline, several of beyond sigma, and several of beyond sigma on the chart. However, even without use of the sensitizing rules, it is clear that the process is out of control during this period of operation. -

63 Moving Range Individual Value Chapter Eercise Solutions -7 continued (c) MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Oide Thickness (E-7cTh) + New Measurements, with Sensitizing Rules On 7 UCL=. X=9. LCL=. Observation UCL=.79 MR=.7 LCL= Observation The process has been returned to a state of statistical control. -

64 Moving Range Individual Value Chapter Eercise Solutions - (-). (a) The normality assumption is a little bothersome for the concentration data, in particular due to the curve of the larger values and three distant values. MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Concentration (E-C) UC L=. X=7.7 9 Observation 7 LC L=.9 UC L=. MR=.7 LC L= 9 Observation 7 Test Results for I Chart of E-C TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points: Test Results for MR Chart of E-C TEST. One point more than. standard deviations from center line. Test Failed at points: 7 The process is not in control, with two Western Electric rule violations. -

65 Percent Chapter Eercise Solutions - continued (c) MTB > Stat > Basic Statistics > Normality Test Probability Plot of ln(concentration) (E-lnC) Normal Mean. StDev.7 N AD. P-Value E-lnC....7 The normality assumption is still troubling for the natural log of concentration, again due to the curve of the larger values and three distant values. -

66 Moving Range Individual Value Chapter Eercise Solutions - continued (d) MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of ln(concentration) (E-lnC).. UC L= X=.. LC L=.9 9 Observation 7. UC L=.... MR=.77. LC L= 9 Observation 7 Test Results for I Chart of E-lnC TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points: Test Results for MR Chart of E-lnC TEST. One point more than. standard deviations from center line. Test Failed at points: 7 The process is still not in control, with the same to Western Electric Rules violations. There does not appear to be much difference between the two control charts (actual and natural log). -

67 Moving Range Individual Value Percent Chapter Eercise Solutions -9. MTB > Stat > Basic Statistics > Normality Test Probability Plot of Velocity of Light (E-9Vel) Normal Mean 99 StDev.9 N AD.7 P-Value E-9Vel Velocity of light measurements are approimately normally distributed. MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Velocity of Light (E-9Vel) UC L=. X=99 7 Observation LC L=.9 UC L=.7 MR=. LC L= Observation I-MR Chart of E-9Vel Test Results for MR Chart of E-9Vel TEST. One point more than. standard deviations from center line. Test Failed at points: The out-of-control signal on the moving range chart indicates a significantly large difference between successive measurements (7 and ). Since neither of these measurements seems unusual, use all data for control limits calculations. There may also be an early indication of less variability in the later measurements. For now, consider the process to be in a state of statistical process control. -7

68 Moving Range Individual Value Chapter Eercise Solutions -. (a) MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR Select I-MR Options, Estimate to specify which subgroups to use in calculations I-MR Chart of Velocity of Light (E-Vel) New measurements with old limits UCL=. 7 X=99 LCL=.9 Observation UCL=.7 MR=. LCL= Observation I-MR Chart of E-Vel Test Results for I Chart of E-Vel TEST. 9 points in a row on same side of center line. Test Failed at points:, 7,, 9, Test Results for MR Chart of E-Vel TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. 9 points in a row on same side of center line. Test Failed at points:, 7 The velocity of light in air is not changing, however the method of measuring is producing varying results this is a chart of the measurement process. There is a distinct downward trend in measurements, meaning the method is producing gradually smaller measurements. Early measurements ehibit more variability than the later measurements, which is reflected in the number of observations below the centerline of the moving range chart. -

69 Percent Percent Chapter Eercise Solutions -. (a) MTB > Stat > Basic Statistics > Normality Test Probability Plot of Uniformity Determinations (E-Un) Normal Mean.7 StDev. N AD. P-Value <. E-Un The data are not normally distributed, as evidenced by the S - shaped curve to the plot points on a normal probability plot, as well as the Anderson-Darling test p-value. The data are skewed right, so a compressive transform such as natural log or square-root may be appropriate. Probability Plot of ln(uniformity) (E-lnUn) Normal Mean. StDev.9 N AD. P-Value.9... E-lnUn.. The distribution of the natural-log transformed uniformity measurements is approimately normally distributed. -9

70 Moving Range Individual Value Chapter Eercise Solutions - continued MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of ln (Uniformity) (E-lnUn). UC L=... X=... 9 Observation 7 LC L=.7 UC L= MR=.. LC L= 9 Observation 7 The etching process appears to be in statistical control. -7

71 Percent Chapter Eercise Solutions - (-). (a) MTB > Stat > Basic Statistics > Normality Test Probability Plot of Batch Purity (E-Pur) Normal Mean. StDev.7 N AD.7 P-Value < E-Pur....7 Purity is not normally distributed. -7

72 Moving Range Individual Value Chapter Eercise Solutions - continued MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Purity (E-Pur). UC L=.9.. X=...7 Observation LC L=.7. UC L= MR=.. LC L= Observation Test Results for I Chart of E-Pur TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points: 9 TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, The process is not in statistical control. (c) all data: ˆ., ˆ. without sample : ˆ., ˆ. -7

73 Moving Range Individual Value Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR Select Estimate to change the method of estimating sigma I-MR Chart of Can Weight (E-Wt).7 UC L=... X=... LC L=. Observation. UC L= MR=.. LC L= Observation There is no difference between this chart and the one in Eercise -; control limits for both are essentially the same. -7

74 Moving Range Individual Value Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR Select Estimate to change the method of estimating sigma I-MR Chart of Hardness-Coded (E-Har) UC L=. X=.7 LC L=. 7 9 Observation. UC L= MR=.9. LC L= 7 9 Observation The median moving range method gives slightly tighter control limits for both the Individual and Moving Range charts, with no practical difference for this set of observations. -7

75 Moving Range Individual Value Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR Select Estimate to change the method of estimating sigma I-MR Chart of Polymer Viscosity (E-Vis) UC L=7.7 X=9.9 LC L=. Observation UC L=. MR=.7 LC L= Observation The median moving range method gives slightly wider control limits for both the Individual and Moving Range charts, with no practical meaning for this set of observations. -7

76 Moving Range Individual Value Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR (a) I-MR Chart of Oide Thickness (E-7cTh) All Observations--Average Moving Range Method 7 UCL=.7 X=. LCL=. Observation UCL=. MR=. LCL= Observation Test Results for I Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. 9 points in a row on same side of center line. Test Failed at points:, 9, TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:, 9, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 7,, 9, Test Results for MR Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: Recall that observations on the Moving Range chart are correlated with those on the Individuals chart that is, the out-of-control signal on the MR chart for observation is reflected by the shift between observations and on the Individuals chart. Remove observation and recalculate control limits. -7

77 Moving Range Individual Value Chapter Eercise Solutions - (a) continued Ecluding observation from calculations: I-MR Chart of Oide Thickness (E-7cTh) Less Observation -- Average Moving Range Method 7 UCL=. X=.77 LCL=.9 Observation UCL=. MR=. LCL= Observation Test Results for I Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. 9 points in a row on same side of center line. Test Failed at points:, 9, TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:, 9, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 7,, 9, Test Results for MR Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: -77

78 Moving Range Individual Value Chapter Eercise Solutions - continued I-MR Chart of Oide Thickness (E-7cTh) All Observations -- Median Moving Range Method 7 UCL=. X=. LCL=.7 Observation UCL=. MR=. LCL= Observation Test Results for I Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: TEST. 9 points in a row on same side of center line. Test Failed at points:, 9, TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:, 9, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 7,, 9, Test Results for MR Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: -7

79 Moving Range Individual Value Chapter Eercise Solutions - continued Ecluding observation from calculations: I-MR Chart of Oide Thickness (E-7cTh) Ecluding Observation from Calculations -- Median Moving Range Method 7 UCL=. X=.77 LCL=.9 Observation UCL=7. MR=. LCL= Observation Test Results for I Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points:, TEST. 9 points in a row on same side of center line. Test Failed at points:, 9, TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points:, 9, TEST. out of points more than standard deviation from center line (on one side of CL). Test Failed at points:, 7,, 9, Test Results for MR Chart of E-7cTh TEST. One point more than. standard deviations from center line. Test Failed at points: (c) The control limits estimated by the median moving range are tighter and detect the shift in process level at an earlier sample,. -79

80 Moving Range Individual Value Chapter Eercise Solutions -7 (-7). (a) MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Measurements (E-7Meas) UC L=. X=.9 LC L=7.79 Observation UC L=. MR=. LC L= Observation ˆ R/ d./..7 MTB > Stat > Basic Statistics > Descriptive Statistics Descriptive Statistics: E-7Meas Total Variable Count Mean StDev Median E-7Meas.9.. ˆ S/ c./

81 Moving Range Individual Value Chapter Eercise Solutions -7 continued (c) MTB > Stat > Control Charts > Variables Charts for Individuals > I-MR I-MR Chart of Measurements (E-7Meas) Median Moving Range Method--Span = UCL=.9 X=.9 LCL=7.7 Observation UCL=.9 MR=. LCL= Observation ˆ R/ d./..7 (d) Average MR Chart: ˆ R/ d.9/.9. Average MR Chart: ˆ R/ d.9/.9. Average MR9 Chart: ˆ R/ d./.9. Average MR Chart: ˆ R/ d./.7. (e) As the span of the moving range is increased, there are fewer observations to estimate the standard deviation, and the estimate becomes less reliable. For this eample, gets larger as the span increases. This tends to be true for unstable processes. -

82 Range MR of Subgroup Mean Subgroup Mean Chapter Eercise Solutions - (-). MTB > Stat > Control Charts > Variables Charts for Subgroups > I-MR-R/S (Between/Within) Select I-MR-R/S Options, Estimate and choose R-bar method to estimate standard deviation I-MR-R (Between/Within) Chart of Vane Heights (E-v,..., E-v). UCL=.9.7 X=.7.7 LCL=. UCL=... MR=.77. LCL=. UCL=.77. R=.. LCL= E-Cast I-MR-R/S Standard Deviations of E-v,..., E-v Standard Deviations Between. Within. Between/Within. The Individuals and Moving Range charts for the subgroup means are identical. When compared to the s chart for all data, the R chart tells the same story same data pattern and no out-of-control points. For this eample, the control schemes are identical. -

83 Range Mean Chapter Eercise Solutions -9 (-9). (a) MTB > Stat > Control Charts > Variables Charts for Subgroups > Xbar-R Xbar-R Chart of Casting Diameter (E-9d,..., E-9d)..79 UC L=.79.7 X= LC L=.7.7. UC L=.9.9. R=... LC L= Xbar-R Chart of E-9d,..., E-9d Test Results for Xbar Chart of E-9d,..., E-9d TEST. One point more than. standard deviations from center line. Test Failed at points:, 7, 9,, 7 TEST. out of points more than standard deviations from center line (on one side of CL). Test Failed at points: 7 Though the R chart is in control, plot points on the chart bounce below and above the control limits. Since these are high precision castings, we might epect that the diameter of a single casting will not change much with location. If no assignable cause can be found for these out-of-control points, we may want to consider treating the averages as an Individual value and graphing between/within range charts. This will lead to a understanding of the greatest source of variability, between castings or within a casting. -