Chapter V Correlato ad Regresso Aalss R. 5.. So far we have cosdered ol uvarate dstrbutos. Ma a tme, however, we come across problems whch volve two or more varables. Ths wll be the subject matter of the curret chapter. D. 5.. (Correlato) The correlato meas the stud of estece, magtude ad drecto of the relato betwee two or more varables. R. 5.. (Tpes of Correlato) We dstgush. Postve ad egatve correlatos If two varables chage the same drecto, the ths s called a postve correlato. (For eample: prce ad suppl) If two varables chage the opposte drecto, the the correlato s called a egatve Correlato. (For eample: prce ad demad). Lear ad o-lear correlatos. R. 5. 3. (Degree of Correlato) Degrees Postve Negatve Absece of correlato Perfect correlato + -,.75 Hgh degree ].75, [ ] [ Moderate degree ].5,.75 [ ]-.75, -.5[ Low degree ],.5[ ].5, [ R. 5. 4. (Methods of Determg Correlato) We shall cosder the followg most commol used methods: () Scatter Plot () Karl Pearso s coeffcet of correlato (3) Spearma s rak correlato coeffcet.
R. 5. 5. (Scatter Plot or Dot Dagram) I ths method the values of the two varables are plotted o a graph paper. Oe s take alog the horzotal ( ) as ad the other alog the vertcal ( ) as. B plottg the data, we get pots (dots) o the graph whch are geerall scattered ad hece the ame scatter plot. The maer whch these pots are scattered, suggest the degree ad the drecto of correlato. Let the degree of correlato be deoted br. Its drecto s gve b the sgs postve ad egatve. ) If all pots le o a rsg straght le the correlato s perfectl postve ad r =+ ) If all pots le o a fallg straght le the correlato s perfectl egatve ad r = ) If the pots le a arrow strp, rsg upwards, the correlato s hgh degree of postve. v) If the pots le a arrow strp, fallg dowwards, the correlato s hgh degree of egatve. v) If the pots are spread wdel over a broad strp, rsg upwards, the correlato s low degree postve. v) If the pots are spread wdel over a broad strp, fallg dowwards, the correlato s low degree egatve. v) If the pots are spread (scattered) wthout a specfc patter, the correlato s abset,.e. r = Though ths method s smple ad gves a rough dea about the estece ad the degree of correlato, t s ot relable. As t s ot a eact mathematcal method, t caot measure the degree of correlato. D. 5.. (Karl Pearso s Coeffcet of Correlato) Let X ad Y be two varates. = r : =. = = E. 5.. The followg table shows the aual proft [Mo ] ad aual costs of leasg computer equpmet [ ] of 5 frms:
( ) ( ) ( ) ( ) - 3-7 34 4 89 5-5 3-7 55 5 89 3 5-5 - 5 5 4-5 -5 5 5 5 - - 6 5-5 8-6 5 44 7 3 5-5 5 8 3-9 3 5 5 5 35 5 8 - - 5 4 35 5 33 3 65 5 69 4 3 45 5 4 3 5 4 4 5 5 3 6 4 9 5 5 6 4 8 4 6 Sum 45 3 8 5 457 45 3 = = 3, = = 5 5 8 r : =.88. 5 457 3
D. 5. 3. (Spearma s Rak Coeffcet of Correlato) Let R, =,,..., : raks of the characterstc X ' R, =,,..., : raks of the characterstc Y, 6 = ρ : = ' ( R R) ( ) ( + ) : E. 5.. The followg table shows the advertsemet costs (Y ) ad the reveues (X ) of a frm: Advertsemet Costs Reveue.4.8.9 4.4 4.8 3 3. 4 3.6 4 4. 48 Fd ad terpret the Spearma s rak coeffcet of correlato. Soluto: Advertsemet Costs Reveue R ' R ( R R ).4...8...9 4 3 3.5.5.4 4 4 3.5.5.8 3 5 5.. 3. 4 6 6.. 3.6 4 7 7.. 4. 48 8 8...5 8 ' = 8, ( R R) =.5, = 6.5 ρ : =.994. 7.8.9 There s therefore a strog degree of correlato betwee X ady. ' 4
D. 5. 4. (Regresso Fucto) A regresso fucto descrbes the relatoshp betwee depedet varable Y ad (at least oe) depedet or eplaator varable X : * = f ( ) R. 5. 6. (Least Square Method) The coeffcets of a regresso fucto ca be estmated b usg the Least Square Method: = ( ) S(...) = * M! R. 5. 7. (Smple Lear Regresso) The coeffcets of a smple lear regresso fucto * = a + a ca be obtaed b solvg the followg sstem of lear equatos: a a + a + a = = = = = = = D. 5. 5. (Coeffcets of Correlato ad Determato). The coeffcet of correlato of a lear regresso fucto s defed b = = = r : =, = = = = = = r +. r ( r ) s called coeffcet of determato. E. 5. 3. I a stud of how the productvt 4 frms depeds o the degree of automato, the followg data have bee made avalable : The term regresso was frst used b Sr Fracs Galto (8-9), who studed the relatoshp betwee heghts of chldre ad heghts of ther parets. 5
Frm Productvt Degree of Automato ( %) 3 4 3 3 8 36 4 3 4 5 3 4 6 33 47 7 34 56 8 37 54 9 38 6 4 55 4 6 43 67 3 45 69 4 48 76. Plot a scatter dagram.. Fd a approprate regresso fucto. 3. Calculate the coeffcets of correlato ad determato ad terpret them. 4. Predct the level of productvt for a automato degree of 8%. Soluto:. 5 45 4 35 3 5 5 3 4 5 6 7 8 6
. 3 64 4 4 4 3 7 9 576 3 8 36 8 96 784 4 3 4 6 9 5 3 4 7 68 96 6 33 47 55 9 89 7 34 56 94 336 56 8 37 54 998 96 369 9 38 6 8 36 444 4 55 35 6 4 6 5 37 68 43 67 88 4489 849 3 45 69 35 476 5 4 48 76 3648 5776 34 total 49 74 697 434 838 4a + 74a 74a + 434a = = 49 697 The soluto of the above sstems elds: a = 7.356, 5435 a =.. Therefore, we have the regresso fucto: * = 7.356+.5435 6 5 4 3 4 6 8 7
3. 4 697 74 49 r=.9687 ( 4 434 74 74) ( 4 834 49 49) r =.9384 Because of r > the productvt s drectl depedet o the degree of automato. Chages the productvt are up to 94% due to chages the degree of automato. 3. *( 8) = 7.356+.5435 8= 5 D. 5. 6. (Tred Fucto) A tred fucto s a specal case of a regresso fucto where the depedet varable s the factor tme. D. 5. 7 (Coeffcet of Determato) The coeffcet of determato s defed as: r : = SSR SST where: = * SSR= ( ) : sum of square due to regresso = SST = ( ) : total sum of squares R. 5. 8. The formula gve for the coeffcet of correlato gve D. 5. 5. s a specal case of the formula D. 5. 7. for lear regresso. E. 5. 4. The followg table shows the cosumpto per head of butter ( kg) a certa coutr the ears 998-4: Year 998 999 3 4 Cosumpto per head 9. 9.5 9.7 9.6 9.9.5.8. Ft a lear tred b regresso aalss.. Predct the cosumpto per head of butter the ear 5. 8
Soluto. ear 998-3 9. 9-7.6 999-9.5 4-9. - 9.7-9.7 9.6. 9.9 9.9 3.5 4. 4 3.8 9 3.4 69. 8 7 a = 69. 9.886 7 =, 7 a. 8 5 = =. We, therefore, have the tred fucto:. * =.885+.5. *( 4) =.885+.5 4=.9 E. 5. 4. (Learsato of a No-Lear Fucto) The followg table shows the umber of cattle a certa coutr the ears 99-4: Year 99 994 995 998 4 No. of cattle (( 5 ). 3.3 4.6 9.6 3.7 33.. Ft a tred b the regresso fucto * = a a.. Predct the umber of cattle the ear 5. 3. Calculate ad terpret the coeffcet of determato for the regresso. Soluto: Let * = a a lg * = lga + lga lg * = lga + lga. Y*: = lg *, A = lga, A = lga. The we have: 9
Y* = A + A A = + A A + A 6A + 37A 37A + 35A = Y = = = = Y = ear lg lg 99..439.439 994 3 3.3.385 9 3.3755 995 4 4.6.6435 6 4.6574 998 7 9.6.96 49 9.458 9 3.7.37475 8.3775 4 3 33..5983 69 9.75779 37 5.3 7.5643 35 5.467 = = 7.5 5.5 The soluto of the above sstems elds:. e. A =.645439, 43443 A =., a =.49777, 965659 a =. Therefore, we have the regresso fucto: * =.5.97 6 5 4 3 5 5. 4 5 *(4) =.5.97 = 37. ( ) (Last revsed: 7..)