Qulity contol Finl exm: // (Thu), 9:-: Q Q Q3 Q4 Q5 YOUR NAME NOTE: Plese wite down the deivtion of you nswe vey clely fo ll questions. The scoe will be educed when you only wite nswe. Also, the scoe will be educed if the deivtion is not cle. The scoe will be dded even when you nswe is incoect but the deivtion is coect.
Q Bsic clcultions / Ove-disesion. Chis of n engineeing lstic hve vege size 45mm nd stndd devition 3mm. Ue nd lowe secifiction limits fo the chis e 48mm nd 4mm, esectively. ) Suose tht X,..., X 9 e the size of the chis. Then, wht is the obbility of X being outside the secifiction limit? ) We obtined sizes of chis (39, 44, 39, 4, 47). Find n estimte of the stndd devition by nge method (use tble). 3) Let Y, Y, } be indeendent sequence of - vibles (Benoulli tils) with { Y3 P( Y i ) with obbility.5, with obbility.5 V( X ), whee X Y Y Y. 3 ( i,,3), Comute 4) Let Y, Y, } be sequence of - vibles with { Y3 with obbility.5 i ), ( i,,3), with obbility.5 nd let Y ).8, (,). Deive Y ) Y ) Y ) 5) Comute V( X ), whee X Y Y Y3. 6) Show ove-disesion in the bove exmle.
Q. Shewht cht fo continuous vible We conduct the testing hyothesis: H : B vesus H : B fo the following dt of k=5 gous nd n=4 smles. Gou Men s i i= 6 i= 6 i=3 3 5 i=4 3 5 i=5 4 4 9 Rnge ) Conduct test fo H : (5% level). B ) Dw X -Cht (dw both ction limits nd wning limits) 3) Dw Rnge-cht (dw both ction limits nd wning limits) 3
Q3. Avege un length fo X -cht Let R be the un length fo X -Cht with both wning nd ction limits. E R [LWL, UWL] =, (in tems of E [R] ) ) X E R X [LAL, UAL] =. R X [UWL, E R X [UWL, [LAL, E R X [UWL, R X [UWL, [LAL, ( in tems of R [LAL, LWL] E [LWL, UWL] =, (in tems of E [R] ). UAL] =. [UWL, UAL] =. E LWL] =, E ) X ) Let P( LWL X UWL), P( UWL X UAL ), P( LAL X LWL). Deive E [R] in tems of, nd. 3) Deive R [UWL, UAL] E. X 4) Let X i N(, ) with / n, nd UAL= 3 / n, UWL= / n, LWL= / n, LAL= 3 / n. Clculte,, nd E [R]. Come E [R] with the vlue in Tble below. 4
Q4. Contol cht fom secifiction limit. Dw X -Cht nd deive smle size fo the following set of metes unde ction limit only nd single secifiction limit t USL=. NOTE: Denote the osition of USL, UAL, nd cente. No need to wite theoeticl deivtion. Poduce s isk oint Consume s isk oint N(, ) L L. 3.5 5 X i. In ctice, USL is not. Fo instnce, the ue secifiction limit fo the size of chis is 48mm. Why one cn lwys set USL= without loss of genelity? (this oblem hs some eoneous oint bout the osition of cente ) 5
Q5. Contol cht fo discete dt A mnufctue conducts wteoof testing fo 5 electic bods in PC. The numbe of electic cicuits nd the numbe of defective cicuits on the bod is ecoded s follows: Bod ID 3 4 5 The numbe of defectives cicuits 35 5 The numbe of cicuits. This dt tye is clled (ttibute dt / countble dt / continuous dt) since the numbe of defective cicuits follows (Binomil distibution / Poisson distibution / Noml distibution). A suitble cht is (c-cht / n-cht / X -Cht).. The disesion test is to test the hyothesis: H :. (wite fomul) 3. Comute test sttistic ( D ) nd the cut-off oint, nd then test H. 4. Dw contol cht (both ction nd wning limits). Is the dt in-contol o out-ofcontol? 6
Tble: the noml distibution Tbles z x ex dx z.5.84.5.64.33.5.7 3 3.5 4 P.695.8.843.933.95.977.99.9938.9967.9986.99977.99997 Tble: Convesion of nge to stndd devition n 3 4 5 6.8.683.59.36.534 n d Tble: Sque, sque-oot tble numbe sque sque oot 5 5.36 6 36.449 7 49.646 8 64.88 9 8 3. 3.6 3.37 44 3.464 Tble: Fctos fo constucting nge chts fom n vege nge Action limit Wning limit n D D 3 D 3 D 4 4.6.36.37.8 5...4.7 Tble: Avege un length fo vious sizes fom the tget vlue Avege n length (ARL) Devition: /( / n) 78.5.6 5.6.5 8.78 4.7.5.4 Tble: Citicl oint of F-distibution: F.5(df, df) df= 3 4 5 6 7 8 9 df= 4.96 4. 3.7 3.48 3.33 3. 3.4 3.7 3. 4.84 3.98 3.59 3.36 3. 3.9 3..95.9 4.75 3.89 3.49 3.6 3. 3..9.85.8 3 4.67 3.8 3.4 3.8 3.3.9.83.77.7 4 4.6 3.74 3.34 3..96.85.76.7.65 5 4.54 3.68 3.9 3.6.9.79.7.64.59 6 4.49 3.63 3.4 3..85.74.66.59.54 7 4.45 3.59 3..96.8.7.6.55.49 7
Degees of feedom (df) χ vlue.4..6.5.46.7.64.7 3.84 6.64.83...45.7.39.4 3. 4.6 5.99 9. 3.8 3.35.58..4.37 3.66 4.64 6.5 7.8.34 6.7 4.7.6.65. 3.36 4.88 5.99 7.78 9.49 3.8 8.47 5.4.6.34 3. 4.35 6.6 7.9 9.4.7 5.9.5 6.63. 3.7 3.83 5.35 7.3 8.56.64.59 6.8.46 7.7.83 3.8 4.67 6.35 8.38 9.8. 4.7 8.48 4.3 8.73 3.49 4.59 5.53 7.34 9.5.3 3.36 5.5.9 6. 9 3.3 4.7 5.38 6.39 8.34.66.4 4.68 6.9.67 7.88 3.94 4.86 6.8 7.7 9.34.78 3.44 5.99 8.3 3. 9.59 P vlue (Pobbility).95.9.8.7.5.3...5.. 8
IMPORTANT NOTE: This is vey simlified nswe. In the exm, you need to wite down moe detiled clcultions. Answe. ) X N(45, ), P( X 4) P( X 48) P( Z 4) P( Z 3). 4 ) sd=(47-39)/.36 = 3.44 3) V (X) =3(/)(/)=3/4=.75 4) Y ) =-.8=., {P( Y Y ) ), Y, Y Y ) =-.=.8 ) / )}/ ) (/.8/ 9 ) ) /(/ ) 5) P( X ), Y, Y3 ) (/ ).8.8. 3. P( X ).8, P( X ).8, P( X 3).3. V ( X ).3(.5).8(.5).8(.5) 6)Ove-disesion since V( X ).53.75 V( X )..3(.5)..53 Answe ) Ovell men=4, s B =46/4=.5, s W =6/5=5., F ns B / s W =465/6=8.85 Citicl oint=f(k-,(n-)k)=f(4, 5)=3.6 (5%). Reect H : B. ) LAL=4-33.4=3.8, UAL=4+33.4=4., LWL=4-3.4=7., UWL=4+3.4=.8. (Add these line nd gou mens in the cht). 3) R =3/5=4.6 LAL=.64.6=.736, UAL=.364.6=.856, LWL=.374.6=.7, UWL=.84.6=8.36. (Add these line nd nges in the cht) Answe3 E R [LWL, UWL] = + E [R] ) X E R X [LAL, UAL] =. E R X [UWL, [LWL, UWL] =+ E [R] E R X [UWL, [LAL, UAL] =. E R X [UWL, [UWL, UAL] =. E R X [UWL, [LAL, LWL] = + E R [LAL, LWL] X
) 3) E[ R ] E R X (see you notes fo deivtion) ( )( ) E[ R] [UWL, UAL] ( )( ) ( ) 4) Since X N( / n, / n), Z ( X / n) /( / n) N(, ). P( / n X / n) P( 4 Z ).5, P( Z ).843.5.343, P( 5 Z 4). E [R] =(+.343)/(-.5-.5.343)=.343/.33=4.6. This is close to 4.7 in the Tble. Answe4. =. L =3 n q / ( Zq Z ) /( q Z Z Set n =8. Z =.33, =.5 L =.33 ) Z q Z =.64 =.7, L =5 q / L =. Z q =.84. =(.7-.84)^/(.33-.64)^=(.87/.69)^=7.34. ka Z Zq / n =.33-.7/sqt(8)=.37. UAL=- k A =-.37=-.74, cente=- =- (usully, cente is below UAL).. If USL=48mm fo vible X, set new vible Y=X-48. Then, USL fo Y is. Answe 5 ) Attibute dt; Binomil distibution; n-cht. ) H : V( X i ) n( ), whee X i is the numbe of defectives with ID = i, nd is unknown obbility of being defective in the cicuit. 3) ˆ =., X =, s =35/4=87.5, s 5 7 / =5.646.44/= 9.354. D=35/(..9)=9.44 > (.95) =9.49. Reect H : V( X i ) n( ). df 4 4)UAL=+39.354=48.6, UWL=+9.354=38.78, LWL=-9.354=.9, LAL=-39.354=-8.6 (set ). In-contol (Figue omitted).