Chap. 13. Exercise A few exercises from Chap. 13. n = 89; c = 2; OC B [p_] = CDF[BinomialDistribution[n, p], c];

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1 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 () OC B ()

2 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 () Exercise = 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)= > 2 1 =10. Do not use accetance number c=1 because r(c)= > 2 1 =10. Use accetance number c=2 because r(c)= =10. Use the samle size n=54

3 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 () OC B ()

4 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}

5 Ex-Ca13.nb 5 OC A () OC A ()

6 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 () OC B ()

7 Ex-Ca13.nb 7 Show[G1, G2, G3, G4, DislayFunction $DislayFunction] OC B ()

8 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 Z 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, }, {0.01, }, {0.015, }, {0.02, }, {0.025, 5322}, {0.03, }, {0.035, }, {0.04, 3153}, {0.045, }, {0.05, 8350}, {0.055, 1626}, {0.06, }, {0.065, }, {0.07, 5172}, {0.075, 0957}, {0.08, }, {0.085, }, {0.09, }, {0.095, }, {0.1, }, {0.105, }, {0.11, }, {0.115, }, {0.12, }, {0.125, }, {0.13, }, {0.135, }, {0.14, }, {0.145, }, {0.15, }}

9 Ex-Ca13.nb 9 {{0.005, }, {0.01, }, {0.015, }, {0.02, }, {0.025, 6499}, {0.03, }, {0.035, }, {0.04, 5434}, {0.045, }, {0.05, }, {0.055, 4303}, {0.06, }, {0.065, }, {0.07, 7600}, {0.075, 3219}, {0.08, }, {0.085, }, {0.09, }, {0.095, }, {0.1, }, {0.105, }, {0.11, }, {0.115, }, {0.12, }, {0.125, }, {0.13, }, {0.135, }, {0.14, }, {0.145, }, {0.15, }} {{0.005, }, {0.01, }, {0.015, 9737}, {0.02, 3359}, {0.025, }, {0.03, 9412}, {0.035, 2454}, {0.04, }, {0.045, 9462}, {0.05, 3612}, {0.055, }, {0.06, }, {0.065, 9010}, {0.07, 5106}, {0.075, 1646}, {0.08, }, {0.085, }, {0.09, }, {0.095, }, {0.1, }, {0.105, }, {0.11, }, {0.115, }, {0.12, }, {0.125, }, {0.13, }, {0.135, }, {0.14, }, {0.145, }, {0.15, }} {{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, }, {0.085, }, {0.09, }, {0.095, }, {0.1, }, {0.105, }, {0.11, }, {0.115, }, {0.12, }, {0.125, }, {0.13, }, {0.135, }, {0.14, }, {0.145, }, {0.15, }}

10 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() { , { }} { , { }} AOQ rel. reduction

11 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) ATI()

12 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

13 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() Exercise 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 14 Ex-Ca13.nb ASN() ASN()

15 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()

16 16 Ex-Ca13.nb Exercise = 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()

17 Ex-Ca13.nb 17 Exercise 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() { , { }}

18 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() = 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 }]

19 Ex-Ca13.nb 19 Do not use accetance number c=0 because r(c)= > 2 1 =6. Do not use accetance number c=1 because r(c)= > 2 1 =6. Do not use accetance number c=2 because r(c)= > 2 1 =6. Use accetance number c=3 because r(c)= =6. Use the samle size n=112 ASN() Exercise = 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 = because P a [ 1 ]= and P a [ 2 ]=

20 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}

21 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 22 Ex-Ca13.nb Exercise 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 = because P a [ 1 ]= and P a [ 2 ]=

23 Ex-Ca13.nb 23 In[14]:= n σ = 12; k σ = ; u = 3 n σ k σ ; v = 3 n σ 2 k σ 2 ; n s = Ceiling n σ + u + u v 12 ; k s = 3 n s 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 i - 4 k σ, 1 < 1 - α PVarDesc i, 3 i i - 4 k σ, 2 > β, Print "Do not use samle size n s =", i, " and accetance constant k s =", 3 i i - 4 k σ, " because P a [ 1 ]=", PVarDesc i, 3 i i - 4 k σ, 1, "<", 1 - α, " or P a [ 2 ]=", PVarDesc i, 3 i i - 4 k σ, 2, ">", β ; i++ Print "Use samle size n s =", i, " and accetance constant k s =", 3 i i - 4 k σ, " because P a [ 1 ]=", PVarDesc i, 3 i i - 4 k σ, 1, " ", 1 - α, " and P a [ 2 ]=", PVarDesc i, 3 i i - 4 k σ, 2, " ", β Use samle size n s =34 and accetance constant k s = because P a [ 1 ]= and P a [ 2 ]=

24 24 Ex-Ca13.nb In[30]:= G1 = Plot[PVarDesc[34, , ], {, 0.001, 0.15}, PlotStyle RGBColor[1, 0, 0], AxesLabel {"", ""}, DislayFunction Identity]; (* RED * ) G2 = Plot CDF NoncentralStudentTDistribution 34-1, 34. Ω[], , {, 0.001, 0.15}, PlotStyle RGBColor[0, 1, 0], DislayFunction Identity ; (* GREEN * ) G3 = Plot Φ 12 ( 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]=

25 Ex-Ca13.nb 25 Exercise 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, , ], {, 0.001, 0.15}, PlotStyle RGBColor[1, 0, 0], AxesLabel {"", ""}, DislayFunction Identity]; (* RED * ) G2 = Plot CDF NoncentralStudentTDistribution 34-1, 34. Ω[], , {, 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[], , {, 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 26 Ex-Ca13.nb Out[51]= Out[52]=