Cha. 13 A few exercises from Cha. 13 Exercise 13.2 n = 89; c = 2; OC B [_] = CDF[BinomialDistribution[n, ], c]; TableForm[Table[{, OC B []}, {, 0.01, 0.05, 0.01}], TableHeadings {None, {"", "OC B ()"}}] Plot[OC B [], {, 0,.1}, AxesLabel {"", "OC B ()"}] (* OC curve tye B* ) OC B () 0.01 0.93969 0.02 0.736578 0.03 98483 0.04 0.304158 0.05 0.172077 OC B () 0.02 0.04 0.06 0.08 0.10
2 Ex-Ca13.nb 1 = 0.05; idealoc[_] = If[ 1, 1, 0]; G1 = Plot[idealOC[], {, 0, 1 }, PlotStyle {RGBColor[0, 1, 0], Thickness[0.008]}, AxesLabel {"", "Ideal P a ()"}, PlotRange {{0, 1}, {0, 1.1}}, DislayFunction Identity]; G2 = Plot[idealOC[], {, 1, 1}, PlotStyle {RGBColor[1, 0, 0], Thickness[0.02]}, PlotStyle Thickness[0.008], PlotRange {{0, 1}, {0, 1.1}}, DislayFunction Identity]; Show[G1, G2, DislayFunction $DislayFunction] Ideal P a () 0.0 0.0 Exercise 13.3 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.1; (* LTPD * ) β = 0.10; (* consumer's risk * ) Q[c_, x_] = Quantile[ChiSquareDistribution[2 (c + 1)], x]; N[Q[c, 1 - β], 5] r[c_] = N[Q[c, α], 5] ; i = 0; While r[i] > 2 1, Print "Do not use accetance number c=", i, " because r(c)=", r[i], "> 2 =", 2 ; 1 1 i++ Print "Use accetance number c=", i, " because r(c)=", r[i], " 2 1 =", 2 1 Q[i, 1 - β] n[c_] = Ceiling ; 2 2 Print["Use the samle size n=", n[i]] Do not use accetance number c=0 because r(c)=44.8906> 2 1 =10. Do not use accetance number c=1 because r(c)=10.9458> 2 1 =10. Use accetance number c=2 because r(c)=6.50896 2 1 =10. Use the samle size n=54
Ex-Ca13.nb 3 OC B [_] = CDF[BinomialDistribution[n[i], ], i]; TableForm[Table[{, OC B []}, {, 0.005, 0.15, 0.005}], TableHeadings {None, {"", "OC B ()"}}] G1 = Plot[OC B [], {, 0,.15}, AxesLabel {"", "OC B ()"}, PlotStyle RGBColor[1, 0, 0], DislayFunction Identity]; riskoints = {{ 1, 1 - α}, { 2, β}}; G2 = ListPlot[riskoints, PlotStyle PointSize[0.014], PlotStyle RGBColor[0, 1, 0], DislayFunction Identity]; Show[G1, G2, DislayFunction $DislayFunction] (* OC curve tye B obtain via Wetherill and Brown's method; roducer's risk oint (left) and consumer's risk oint (right). * ) OC B () 0.005 0.997437 0.01 0.98302 0.015 0.952447 0.02 0.906268 0.025 4742 0.03 0.779716 0.035 0.706983 0.04 32605 0.045 0.559329 0.05 89225 0.055 23727 0.06 0.363724 0.065 0.309664 0.07 6165 0.075 19535 0.08 0.182999 0.085 0.151615 0.09 0.124894 0.095 0.102326 0.1 0.0834077 0.105 0.0676558 0.11 0.0546237 0.115 0.0439056 0.12 0.0351398 0.125 0.0280082 0.13 0.022235 0.135 0.0175838 0.14 0.0138534 0.145 0.0108746 0.15 0.00850593 OC B ()
4 Ex-Ca13.nb bdist[n_, t_] := BinomialDistribution[n, t]; dist[n_, t_] := PoissonDistribution[n * t]; hdist[n_, t_, ng_] := HyergeometricDistribution[n, Round[t * ng], ng]; bin[x_, n_, t_, ng_] := PDF[bdist[n, t], x]; oi[x_, n_, t_, ng_] := PDF[dist[n, t], x]; hi[x_, n_, t_, ng_] := PDF[hdist[n, t, ng], x]; c Pa[_, {n_, c_, ng_}, f_] := f[d, n,, ng]; d=0 Paatr[_, {{a_, b_}, {e_, d_}}, ng_, f_] := Pa[, {lanoamosatrib[{{a, b}, {e, d}}, ng, f][[1]], lanoamosatrib[{{a, b}, {e, d}}, ng, f][[2]], ng}, f] lanoamosatrib[{{a_, b_}, {e_, d_}}, ng_, f_] := Module[{n, c}, j = 0; t = 0; While[t 0, i = 2; While[i ng && Pa[a, {i, j, ng}, f] 1 - b, i = i + 1] If[Pa[e, {i - 1, j, ng}, f] d, t = 1, t = 0]; j = j + 1]; While[ Pa[a, {i - 1, j - 1, ng}, f] 1 - b && Pa[e, {i - 1, j - 1, ng}, f] d, i = i - 1]; {n = i, c = j - 1}] n tot = 800; (* lot size * ) 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.1; (* LTPD * ) β = 0.10; (* consumer's risk * ) Print["Use the samle size and accetance number, (n,c)=", lanoamosatrib[{{ 1, α}, { 2, β}}, n tot, hi]] (* By using the exact distribution, both n and c are smaller than the one obtained via the Poisson aroximation * ) listoca = Table[{, N[CDF[HyergeometricDistribution[37, Round[n tot ], n tot ], 1], 5]}, {, 0.005, 0.15, 0.005}]; TableForm[listOCA, TableHeadings {None, {"", "OC A ()"}}] G3 = ListPlot[listOCA, AxesLabel {"", "OC A ()"}, PlotStyle RGBColor[1, 0, 1], DislayFunction Identity]; riskoints = {{ 1, 1 - α}, { 2, β}}; G2 = ListPlot[riskoints, PlotStyle PointSize[0.014], DislayFunction Identity]; Show[G3, G2, DislayFunction $DislayFunction] (* OC curve tye B, roducer's risk oint (left) and consumer's risk oint (right). * ) Use the samle size and accetance number, (n,c)={37, 1}
Ex-Ca13.nb 5 OC A () 0.005 0.98822 0.01 0.95111 0.015 9737 0.02 3359 0.025 0.76465 0.03 9412 0.035 2454 0.04 0.55766 0.045 9462 0.05 3612 0.055 0.38249 0.06 0.33384 0.065 9010 0.07 5106 0.075 1646 0.08 0.18597 0.085 0.15924 0.09 0.13593 0.095 0.11568 0.1 0.098171 0.105 0.083084 0.11 0.070132 0.115 0.059048 0.12 0.049594 0.125 0.041554 0.13 0.034737 0.135 0.028972 0.14 0.024111 0.145 0.020021 0.15 0.016589 OC A ()
6 Ex-Ca13.nb Print["Use the samle size and accetance number, (n,c)=", lanoamosatrib[{{ 1, α}, { 2, β}}, n tot, bin]] (* Both n and c are smaller than the one obtained via the Poisson aroximation * ) OC B [_] = CDF[BinomialDistribution[52, ], 2]; TableForm[Table[{, OC B []}, {, 0.005, 0.15, 0.005}], TableHeadings {None, {"", "OC B ()"}}] G4 = Plot[OC B [], {, 0,.15}, AxesLabel {"", "OC B ()"}, DislayFunction Identity]; riskoints = {{ 1, 1 - α}, { 2, β}}; G2 = ListPlot[riskoints, PlotStyle PointSize[0.014], DislayFunction Identity]; Show[G4, G2, DislayFunction $DislayFunction] (* OC curve tye B, roducer's risk oint (left) and consumer's risk oint (right). * ) Use the samle size and accetance number, (n,c)={52, 2} OC B () 0.005 0.997699 0.01 0.984647 0.015 0.9567 0.02 0.914066 0.025 59182 0.03 0.795393 0.035 0.726159 0.04 54624 0.045 0.583414 0.05 0.51457 0.055 49569 0.06 0.389391 0.065 0.334593 0.07 85401 0.075 41786 0.08 03539 0.085 0.170325 0.09 0.141737 0.095 0.117325 0.1 0.0966333 0.105 0.0792127 0.11 0.0646382 0.115 0.0525163 0.12 0.0424896 0.125 0.0342391 0.13 0.0274835 0.135 0.0219777 0.14 0.0175108 0.145 0.0139021 0.15 0.0109987 OC B ()
Ex-Ca13.nb 7 Show[G1, G2, G3, G4, DislayFunction $DislayFunction] OC B ()
8 Ex-Ca13.nb Exercise 13.4 n tot = 800; (* lot size * ) 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.1; (* LTPD * ) β = 0.10; (* consumer's risk * ) (* Single samling lans (n,c) according to: * ) (* (54,2), Binomial Distribution aroximation + Wetherhill and Brown method RED * ) (* (52,2), Binomial Distribution aroximation GREEN* ) (* (37,1), exact HyerGeometric Distribution BLUE * ) (* (80,2), Norm ANSI/ ASQC Z1.4-1981 MAGENTA * ) OC Wetherill [_] = CDF[BinomialDistribution[54, ], 2]; OC B [_] = CDF[BinomialDistribution[52, ], 2]; listocawetherill = Table[{, N[CDF[HyergeometricDistribution[54, Round[n tot ], n tot ], 2], 5]}, {, 0.005, 0.15, 0.005}] listocabinomial = Table[ {, N[CDF[HyergeometricDistribution[52, Round[n tot ], n tot], 2], 5]}, {, 0.005, 0.15, 0.005}] listoca = Table[{, N[CDF[HyergeometricDistribution[37, Round[n tot ], n tot ], 1], 5]}, {, 0.005, 0.15, 0.005}] listocanorm = Table[{, N[CDF[HyergeometricDistribution[80, Round[n tot ], n tot ], 1], 2]}, {, 0.005, 0.15, 0.005}] riskoints = {{ 1, 1 - α}, { 2, β}}; G1 = Plot[OC Wetherill [], {, 0,.15}, PlotStyle {RGBColor[0, 1, 0], Thickness[0.002]}, DislayFunction Identity]; G2 = Plot[OC B [], {, 0,.15}, PlotStyle {RGBColor[1, 0, 0], Thickness[0.002]}, AxesLabel {"", "OC B ()"}, DislayFunction Identity]; G3 = ListPlot[listOCA, PlotStyle {RGBColor[0, 0, 1], Thickness[0.01]}, DislayFunction Identity]; G4 = ListPlot[listOCAnorm, PlotStyle {RGBColor[1, 0, 1], Thickness[0.01]}, DislayFunction Identity]; G5 = ListPlot[riskoints, PlotStyle PointSize[0.014], DislayFunction Identity]; Show[G1, G2, G3, G4, G5, DislayFunction $DislayFunction] (* OC curve associated to the Norm far from riskoints * ) {{0.005, 0.99889}, {0.01, 0.98721}, {0.015, 0.95858}, {0.02, 0.91292}, {0.025, 5322}, {0.03, 0.78366}, {0.035, 0.70851}, {0.04, 3153}, {0.045, 0.55577}, {0.05, 8350}, {0.055, 1626}, {0.06, 0.35499}, {0.065, 0.30013}, {0.07, 5172}, {0.075, 0957}, {0.08, 0.17328}, {0.085, 0.14235}, {0.09, 0.11625}, {0.095, 0.094391}, {0.1, 0.076233}, {0.105, 0.061253}, {0.11, 0.048977}, {0.115, 0.038978}, {0.12, 0.030882}, {0.125, 0.024361}, {0.13, 0.019137}, {0.135, 0.014973}, {0.14, 0.011668}, {0.145, 0.0090579}, {0.15, 0.0070053}}
Ex-Ca13.nb 9 {{0.005, 0.99901}, {0.01, 0.98849}, {0.015, 0.96245}, {0.02, 0.92047}, {0.025, 6499}, {0.03, 0.79965}, {0.035, 0.72827}, {0.04, 5434}, {0.045, 0.58077}, {0.05, 0.50980}, {0.055, 4303}, {0.06, 0.38149}, {0.065, 0.32575}, {0.07, 7600}, {0.075, 3219}, {0.08, 0.19402}, {0.085, 0.16112}, {0.09, 0.13302}, {0.095, 0.10921}, {0.1, 0.089198}, {0.105, 0.072488}, {0.11, 0.058629}, {0.115, 0.047204}, {0.12, 0.037839}, {0.125, 0.030205}, {0.13, 0.024012}, {0.135, 0.019014}, {0.14, 0.014999}, {0.145, 0.011787}, {0.15, 0.0092291}} {{0.005, 0.98822}, {0.01, 0.95111}, {0.015, 9737}, {0.02, 3359}, {0.025, 0.76465}, {0.03, 9412}, {0.035, 2454}, {0.04, 0.55766}, {0.045, 9462}, {0.05, 3612}, {0.055, 0.38249}, {0.06, 0.33384}, {0.065, 9010}, {0.07, 5106}, {0.075, 1646}, {0.08, 0.18597}, {0.085, 0.15924}, {0.09, 0.13593}, {0.095, 0.11568}, {0.1, 0.098171}, {0.105, 0.083084}, {0.11, 0.070132}, {0.115, 0.059048}, {0.12, 0.049594}, {0.125, 0.041554}, {0.13, 0.034737}, {0.135, 0.028972}, {0.14, 0.024111}, {0.145, 0.020021}, {0.15, 0.016589}} {{0.005, 0.95}, {0.01, 1}, {0.015, 6}, {0.02, 0.51}, {0.025, 0.39}, {0.03, 9}, {0.035, 1}, {0.04, 0.15}, {0.045, 0.11}, {0.05, 0.075}, {0.055, 0.052}, {0.06, 0.036}, {0.065, 0.025}, {0.07, 0.017}, {0.075, 0.011}, {0.08, 0.0077}, {0.085, 0.0052}, {0.09, 0.0034}, {0.095, 0.0023}, {0.1, 0.0015}, {0.105, 0.00098}, {0.11, 0.00064}, {0.115, 0.00042}, {0.12, 0.00027}, {0.125, 0.00017}, {0.13, 0.00011}, {0.135, 0.000071}, {0.14, 0.000045}, {0.145, 0.000029}, {0.15, 0.000018}}
10 Ex-Ca13.nb Exercise 13.6 n tot = 800; (* lot size * ) AOQ[n_, c_, _] = 1 n tot (n tot - n) CDF[BinomialDistribution[n, ], c]; (* Average Outgoing Quality (AOQ) or ercentage of defective due to rectifying insection in a single samling lan and using the binomial aroximation to the accetance robability * ) Plot[{AOQ[80, 2, ], AOQ[37, 1, ], }, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0], RGBColor[0, 0, 1]}, AxesLabel {"", "AOQ()"}] (* AOQ of single samling lan using the Norm RED * ) (* AOQ of single samling lan obtained by solving (13.5) with the exact Hyergeometric distribution GREEN * ) FindMaximum[AOQ[80, 2, ], {, 0.001, 1}] FindMaximum[AOQ[37, 1, ], {, 0.001, 1}] AOQ[80, 2, ] AOQ[37, 1, ] Plot 1-100, 1-100, {, 0.001, 0.15}, AxesLabel {"", "AOQ rel. reduction"}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0]} (* Associated relative reduction of the ercentage of defective * ) AOQ() 0.15 0.10 0.05 {0.0154001, { 0.0280931}} {0.0214757, { 0.0427029}} AOQ rel. reduction 100 80 60 40 20
Ex-Ca13.nb 11 Exercise 13.7 n tot = 800; (* lot size * ) ATI[n_, c_, _] = n tot + (n - n tot ) CDF[BinomialDistribution[n, ], c]; (* Average Total Insection (ATI) or ercentage of defective due to rectifying insection in a single samling lan and using the binomial aroximation to the accetance robability * ) ATItable = Table[{, ATI[80, 2, ], ATI[37, 1, ]}, {, 0, 0.15, 0.01}]; TableForm[ATItable, TableHeadings {None, {"", "ATI(n=80,c=2)", "ATI(n=37,c=1)"}}] Plot[{ATI[80, 2, ], ATI[37, 1, ], n tot }, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0], RGBColor[0, 0, 1]}, AxesLabel {"", "ATI()"}] (* ATI of single samling lan using the Norm RED * ) (* ATI of single samling lan obtained by solving (13.5) with the exact Hyergeometric distribution GREEN * ) ATI(n=80,c=2) ATI(n=37,c=1) 0. 80 37 0.01 113.518 77.3459 0.02 235.218 165.85 0.03 390.951 269.885 0.04 530.153 371.768 0.05 633.953 462.92 0.06 703.199 540.095 0.07 745.976 603.006 0.08 770.928 652.864 0.09 784.838 691.512 0.1 792.308 720.93 0.11 796.193 742.977 0.12 798.158 759.28 0.13 799.128 771.188 0.14 799.595 779.793 0.15 799.816 785.946 ATI() 800 600 400 200
12 Ex-Ca13.nb Exercise 13.8 n 1 = 50; (* Collect a first samle of size n 1 * ) c 1 = 1; (* Accet the lot if D 1 c 1, reject if D1>c2 * ) n 2 = 100; (* Collect a second samle of size n 2 if c 1 <D 1 c 2 * ) c 2 = 3; (* Accet the lot if D 1 +D 2 c 2, reject otherwise * ) I[_] = CDF[BinomialDistribution[n 1, ], c 1 ]; (* RED * ) II[_] = c 2 PDF[BinomialDistribution[n 1, ], k] k=c 1 +1 CDF[BinomialDistribution[n 2, ], c 2 - k]; (* GREEN * ) a[_] = I[] + II[]; (* BLUE * ) OC B [_] = CDF[BinomialDistribution[75, ], 2]; (* MAGENTA * ) Plot[{I[], II[], a[], OC B []}, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0], RGBColor[0, 0, 1], RGBColor[1, 0, 1]}, AxesLabel {"", ""}] Null
Ex-Ca13.nb 13 Exercise 13.9 n 1 = 50; (* Collect a first samle of size n 1 * ) c 1 = 2; (* Accet the lot if D 1 c 1, reject if D1>c2 * ) n 2 = 100; (* Collect a second samle of size n 2 if c 1 <D 1 c 2 * ) c 2 = 6; (* Accet the lot if D 1 +D 2 c 2, reject otherwise * ) ASN[_] = n 1 + n 2 CDF[BinomialDistribution[n 1, ], c 2 ] - CDF[BinomialDistribution[n 1, ], c 1 ] ; (* RED * ) (* Average Samle Number * ) Plot[{ASN[], 79}, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, AxesLabel {"", "ASN()"}] ASN() 120 110 100 90 80 70 60 Exercise 13.11 n 1 = 60; (* Collect a first samle of size n 1 * ) c 1 = 2; (* Accet the lot if D 1 c 1, reject if D1>c2 * ) n 2 = 120; (* Collect a second samle of size n 2 if c 1 <D 1 c 2 * ) c 2 = 3; (* Accet the lot if D 1 +D 2 c 2, reject AS SOON AS D 1 +D 2 >c 2, * ) ASN[_] = n 1 + c 2 PDF[BinomialDistribution[n 1, ], j] j=c 1 +1 n 2 CDF[BinomialDistribution[n 2, ], c 2 - j] + c 2 - j + 1 PDF[BinomialDistribution[n 2 + 1, ], c 2 - j + 2] ; (* Average Samle Number - double samling lan with curtailment * ) TableForm[Table[{, ASN[]}, {, 0.005, 0.15, 0.005}], TableHeadings {None, {"", "ASN()"}}] Plot[ASN[], {, 0.001, 0.15}, AxesLabel {"", "ASN()"}]
14 Ex-Ca13.nb ASN() 0.005 6756 0.01 61.1169 0.015 61.8346 0.02 62.0549 0.025 61.8529 0.03 61.4504 0.035 6265 0.04 6734 0.045 6161 0.05 6449 0.055 60.1384 0.06 60.0756 0.065 60.04 0.07 60.0206 0.075 60.0104 0.08 60.0051 0.085 60.0025 0.09 60.0012 0.095 60.0005 0.1 60.0003 0.105 60.0001 0.11 60.0001 0.115 60. 0.12 60. 0.125 60. 0.13 60. 0.135 60. 0.14 60. 0.145 60. 0.15 60. ASN() 62.0 61.5 6 60.5 60.0
Ex-Ca13.nb 15 I[_] = CDF[BinomialDistribution[n 1, ], c 1 ]; II[_] = c 2 PDF[BinomialDistribution[n 1, ], k] * k=c 1 +1 CDF[BinomialDistribution[n 2, ], c 2 - k]; a[_] = I[] + II[]; (* RED * ) OC B [_] = CDF[BinomialDistribution[89, ], 2]; (* GREEN * ) Plot[{a[], OC B []}, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, AxesLabel {"", ""}] Plot[{ASN[], 89}, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}, AxesLabel {"", "ASN()"}] ASN() 90 85 80 75 70 65 60
16 Ex-Ca13.nb Exercise 13.12 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.06; (* LTPD * ) β = 0.10; (* consumer's risk * ) n 1 = 60; (* Collect a first samle of size n 1 * ) c 1 = 1; (* Accet the lot if D 1 c 1, reject if D1>c2 * ) n 2 = 2 n 1 ; (* Collect a second samle of size n 2 if c 1 <D 1 c 2 * ) c 2 = 3; (* Accet the lot if D 1 +D 2 c 2, reject otherwise * ) I[_] = CDF[BinomialDistribution[n 1, ], c 1 ]; II[_] = c 2 PDF[BinomialDistribution[n 1, ], k] * k=c 1 +1 CDF[BinomialDistribution[n 2, ], c 2 - k]; a[_] = I[] + II[]; Plot[a[], {, 0.001, 0.15}, AxesLabel {"", "P a ()"}] (* Primary OC curve of tye B of a double samling lan (without rectifying insection or curtailment) * ) ASN[_] = n 1 + n 2 CDF[BinomialDistribution[n 1, ], c 2 ] - CDF[BinomialDistribution[n 1, ], c 1 ] ; (* RED * ) (* Average Samle Number * ) Plot[ASN[], {, 0.001, 0.15}, AxesLabel {"", "ASN()"}] P a () ASN() 120 110 100 90 80 70 60
Ex-Ca13.nb 17 Exercise 13.13 n tot = 800; (* lot size * ) 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.06; (* LTPD * ) β = 0.10; (* consumer's risk * ) n 1 = 60; (* Collect a first samle of size n 1 * ) c 1 = 1; (* Accet the lot if D 1 c 1, reject if D1>c2 * ) n 2 = 2 n 1 ; (* Collect a second samle of size n 2 if c 1 <D 1 c 2 * ) c 2 = 3; (* Accet the lot if D 1 +D 2 c 2, reject otherwise * ) I[_] = CDF[BinomialDistribution[n 1, ], c 1 ]; II[_] = c 2 PDF[BinomialDistribution[n 1, ], k] * k=c 1 +1 CDF[BinomialDistribution[n 2, ], c 2 - k]; a[_] = I[] + II[]; AOQ[_] = 1 ((n tot - n 1 ) I[] + (n tot - n 1 - n 2 ) II[]); n tot Plot[AOQ[], {, 0.001, 0.15}, AxesLabel {"", "AOQ()"}] (* AOQ of a double samling (WITH rectifying insection and NO curtailment) * ) FindMaximum[AOQ[], {, 0.001, 1}] AOQ() 0.014 0.012 0.010 0.008 0.006 0.004 0.002 {0.0139338, { 0.0252902}}
18 Ex-Ca13.nb ATI[_] = n 1 I[] + (n 1 + n 2 ) II[] + n tot (1 - a[]); Plot[{ATI[], 800}, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}, AxesLabel {"", "ATI()"}, PlotRange {0, n tot }] (* ATI of a double samling lan(with rectifying insection and NO curtailment) * ) ATI() 800 600 400 200 0.00 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.06; (* LTPD * ) β = 0.10; (* consumer's risk * ) Q[c_, x_] = Quantile[ChiSquareDistribution[2 (c + 1)], x]; N[Q[c, 1 - β], 5] r[c_] = N[Q[c, α], 5] ; i = 0; While r[i] > 2 1, Print "Do not use accetance number c=", i, " because r(c)=", r[i], "> 2 =", 2 ; 1 1 i++ Print "Use accetance number c=", i, " because r(c)=", r[i], " 2 1 =", 2 1 Q[i, 1 - β] n[c_] = Ceiling ; 2 2 Print["Use the samle size n=", n[i]] ATISingle[_] = n tot + n[i] - n tot CDF[BinomialDistribution[n[i], ], i]; (* ATI of a single samling lan(with rectifying insection) * ) Plot[{ATI[], ATISingle[], 800}, {, 0.001, 0.15}, PlotStyle {RGBColor[1, 0, 0], RGBColor[0, 1, 0], RGBColor[0, 0, 1]}, AxesLabel {"", "ASN()"}, PlotRange {0, n tot }]
Ex-Ca13.nb 19 Do not use accetance number c=0 because r(c)=44.8906> 2 1 =6. Do not use accetance number c=1 because r(c)=10.9458> 2 1 =6. Do not use accetance number c=2 because r(c)=6.50896> 2 1 =6. Use accetance number c=3 because r(c)=4.88962 2 1 =6. Use the samle size n=112 ASN() 800 600 400 200 0.00 Exercise 13.14 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.07; (* LTPD * ) β = 0.10; (* consumer's risk * ) gdist = NormalDistribution[0, 1]; Φ[x_] := CDF[gdist, x]; Ω[x_] := Quantile[gdist, x]; n σ = Ceiling Ω[1 - α] - Ω[β] Ω[ 2 ] - Ω[ 1 ] 2 ; k σ = Ω[ 2] Ω[1 - α] - Ω[ 1 ] Ω[β] ; Ω[β] - Ω[1 - α] PVar[n_, _] = Φ n (- k σ - Ω[]) ; i = n σ ; While[PVar[i, 1 ] < 1 - α PVar[i, 2 ] > β, Print["Do not use samle size n σ =", i, " because P a [ 1 ]=", PVar[i, 1 ], "<", 1 - α, " or P a [ 2 ]=", PVar[i, 2 ], ">", β]; i++] Print["Use samle size n σ =", i, " and accetance constant k σ =", k σ, " because P a [ 1 ]=", PVar[i, 1 ], " ", 1 - α, " and P a [ 2 ]=", PVar[i, 2 ], " ", β] Use samle size n σ =12 and accetance constant k σ = 1.84827 because P a [ 1 ]=0.951149 0.95 and P a [ 2 ]=0.0984707 0.1
20 Ex-Ca13.nb bdist[n_, t_] := BinomialDistribution[n, t]; dist[n_, t_] := PoissonDistribution[n * t]; hdist[n_, t_, ng_] := HyergeometricDistribution[n, Round[t * ng], ng]; bin[x_, n_, t_, ng_] := PDF[bdist[n, t], x]; oi[x_, n_, t_, ng_] := PDF[dist[n, t], x]; hi[x_, n_, t_, ng_] := PDF[hdist[n, t, ng], x]; c Pa[_, {n_, c_, ng_}, f_] := f[d, n,, ng]; d=0 Paatr[_, {{a_, b_}, {e_, d_}}, ng_, f_] := Pa[, {lanoamosatrib[{{a, b}, {e, d}}, ng, f][[1]], lanoamosatrib[{{a, b}, {e, d}}, ng, f][[2]], ng}, f] lanoamosatrib[{{a_, b_}, {e_, d_}}, ng_, f_] := Module[{n, c}, j = 0; t = 0; While[t 0, i = 2; While[i ng && Pa[a, {i, j, ng}, f] 1 - b, i = i + 1] If[Pa[e, {i - 1, j, ng}, f] d, t = 1, t = 0]; j = j + 1]; While[ Pa[a, {i - 1, j - 1, ng}, f] 1 - b && Pa[e, {i - 1, j - 1, ng}, f] d, i = i - 1]; {n = i, c = j - 1}] n tot = 500; (* lot size * ) 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.07; (* LTPD * ) β = 0.10; (* consumer's risk * ) Print["Use the samle size and accetance number, (n,c)=", lanoamosatrib[{{ 1, α}, { 2, β}}, n tot, hi]] Use the samle size and accetance number, (n,c)={72, 2}
Ex-Ca13.nb 21 G1 = Plot[PVar[12, ], {, 0.001, 0.15}, PlotStyle RGBColor[0, 1, 0], AxesLabel {"", ""}, DislayFunction Identity]; (* GREEN * ) listoca = Table[{, N[CDF[HyergeometricDistribution[72, Round[n tot ], n tot ], 2], 5]}, {, 0.005, 0.15, 0.005}]; TableForm[listOCA, TableHeadings {None, {"", "OC A ()"}}]; G2 = ListPlot[listOCA, PlotStyle RGBColor[1, 0, 0], DislayFunction Identity]; (* RED * ) riskoints = {{ 1, 1 - α}, { 2, β}}; G3 = ListPlot[riskoints, PlotStyle PointSize[0.014], DislayFunction Identity]; Show[G1, G2, G3, DislayFunction $DislayFunction] (* OC curve tye B, roducer's risk oint (left) and consumer's risk oint (right). * )
22 Ex-Ca13.nb Exercise 13.15 In[1]:= 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.07; (* LTPD * ) β = 0.10; (* consumer's risk * ) gdist = NormalDistribution[0, 1]; Φ[x_] := CDF[gdist, x]; Ω[x_] := Quantile[gdist, x]; n σ = Ceiling Ω[1 - α] - Ω[β] Ω[ 2 ] - Ω[ 1 ] 2 ; k σ = Ω[ 2] Ω[1 - α] - Ω[ 1 ] Ω[β] ; Ω[β] - Ω[1 - α] PVar[n_, _] = Φ n (- k σ - Ω[]) ; i = n σ ; While[PVar[i, 1 ] < 1 - α PVar[i, 2 ] > β, Print["Do not use samle size n σ =", i, " because P a [ 1 ]=", PVar[i, 1 ], "<", 1 - α, " or P a [ 2 ]=", PVar[i, 2 ], ">", β]; i++] Print["Use samle size n σ =", i, " and accetance constant k σ =", k σ, " because P a [ 1 ]=", PVar[i, 1 ], " ", 1 - α, " and P a [ 2 ]=", PVar[i, 2 ], " ", β] Use samle size n σ =12 and accetance constant k σ = 1.84827 because P a [ 1 ]=0.951149 0.95 and P a [ 2 ]=0.0984707 0.1
Ex-Ca13.nb 23 In[14]:= n σ = 12; k σ = 1.848273; u = 3 n σ k σ 2-2 + 8; v = 3 n σ 2 k σ 2 ; n s = Ceiling n σ + u + u2 + 24 v 12 ; k s = 3 n s - 3 3 n s - 4 k σ; Ω[1 - ] - k PVarDesc[n_, k_, _] = Φ 1+ 3 n k2 6 n- 8 n 3 n- 4 3 n- 3 ; (* PVarDesc[n_,k_,_]= CDF NoncentralStudentTDistribution n- 1, n Ω[],- n k ; * ) i = n s ; While PVarDesc i, 3 i - 3 3 i - 4 k σ, 1 < 1 - α PVarDesc i, 3 i - 3 3 i - 4 k σ, 2 > β, Print "Do not use samle size n s =", i, " and accetance constant k s =", 3 i - 3 3 i - 4 k σ, " because P a [ 1 ]=", PVarDesc i, 3 i - 3 3 i - 4 k σ, 1, "<", 1 - α, " or P a [ 2 ]=", PVarDesc i, 3 i - 3 3 i - 4 k σ, 2, ">", β ; i++ Print "Use samle size n s =", i, " and accetance constant k s =", 3 i - 3 3 i - 4 k σ, " because P a [ 1 ]=", PVarDesc i, 3 i - 3 3 i - 4 k σ, 1, " ", 1 - α, " and P a [ 2 ]=", PVarDesc i, 3 i - 3 3 i - 4 k σ, 2, " ", β Use samle size n s =34 and accetance constant k s = 1.85768 because P a [ 1 ]=0.952257 0.95 and P a [ 2 ]=0.0969855 0.1
24 Ex-Ca13.nb In[30]:= G1 = Plot[PVarDesc[34, 1.857679, ], {, 0.001, 0.15}, PlotStyle RGBColor[1, 0, 0], AxesLabel {"", ""}, DislayFunction Identity]; (* RED * ) G2 = Plot CDF NoncentralStudentTDistribution 34-1, 34. Ω[], - 34 1.857679, {, 0.001, 0.15}, PlotStyle RGBColor[0, 1, 0], DislayFunction Identity ; (* GREEN * ) G3 = Plot Φ 12 (- 1.848273 - Ω[]), {, 0.001, 0.15}, PlotStyle RGBColor[0, 0, 1], DislayFunction Identity ; (* BLUE * ) riskoints = {{ 1, 1 - α}, { 2, β}}; G4 = ListPlot[riskoints, PlotStyle PointSize[0.014], DislayFunction Identity]; Show[G1, G2, G3, G4, DislayFunction $DislayFunction] Out[35]=
Ex-Ca13.nb 25 Exercise 13.17 In[36]:= 1 = 0.01; (* AQL * ) α = 0.05; (* roducer's risk * ) 2 = 0.07; (* LTPD * ) β = 0.10; (* consumer's risk * ) gdist = NormalDistribution[0, 1]; Φ[x_] := CDF[gdist, x]; Ω[x_] := Quantile[gdist, x]; Ω[1 - ] - k PVarDesc[n_, k_, _] = Φ 1+ 3 n k2 6 n- 8 n 3 n- 4 3 n- 3 ; G1 = Plot[PVarDesc[34, 1.857679, ], {, 0.001, 0.15}, PlotStyle RGBColor[1, 0, 0], AxesLabel {"", ""}, DislayFunction Identity]; (* RED * ) G2 = Plot CDF NoncentralStudentTDistribution 34-1, 34. Ω[], - 34 1.857679, {, 0.001, 0.15}, PlotStyle RGBColor[0, 1, 0], DislayFunction Identity ; (* GREEN * ) G3 = Plot[PVarDesc[25, 1.85, ], {, 0.001, 0.15}, PlotStyle RGBColor[0, 0, 1], DislayFunction Identity]; (* BLUE * ) G4 = Plot CDF NoncentralStudentTDistribution 25-1, 25 Ω[], - 25 1.85, {, 0.001, 0.15}, PlotStyle RGBColor[1, 0, 1], DislayFunction Identity ; (* MAGENTA * ) riskoints = {{ 1, 1 - α}, { 2, β}}; G5 = ListPlot[riskoints, PlotStyle PointSize[0.014], DislayFunction Identity]; Show[G1, G3, G5, DislayFunction $DislayFunction] Show[G2, G4, G5, DislayFunction $DislayFunction] Show[G1, G2, G3, G4, G5, DislayFunction $DislayFunction] Out[50]=
26 Ex-Ca13.nb Out[51]= Out[52]=