LECTURE NOTES Course No: STCA-101 STATISTICS TIRUPATI. Prepared By

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1 LECTURE NOTES Course No: STCA-0 STATISTICS TIRUPATI Karl Pearso R. A. Fsher Karl Pearso (é Carl Pearso) Bor: 7 March 857 Islgto, Lodo, Eglad Sr Roald Aylmer Fsher (890-96) Bor: 7 February 890 East Fchley, Lodo, Eglad Prepared By Dr. G. MOHAN NAIDU M.Sc., Ph.D., Assstat Professor & Head Dept. of Statstcs & Mathematcs S.V. Agrcultural College TIRUPATI ACHARYA N.G. RANGA AGRICULTURAL UNIVERSITY

2 LECTURE OUTLINE Course No. STCA-0 Credts: (+) Course Ttle: STATISTICS THEORY S. No. Topc/Lesso Itroducto to Statstcs, Defto, Advatages ad Lmtatos. Frequecy dstrbuto: Costructo of Frequecy Dstrbuto table. 3 Measures of Cetral Tedecy: Defto, Characterstcs of Satsfactory average. 4 Arthmetc Mea, Meda, Mode for grouped ad ugrouped data Merts ad Demerts of Arthmetc Mea. 5 Measures of Dsperso: Defto, stadard devato, varace ad coeffcet of varato. 6 Normal Dstrbuto ad ts propertes. Itroducto to Samplg: Radom samplg, cocept of stadard error of Mea. 7 Tests of Sgfcace: Itroducto, Types of errors, Null hypothess, level of sgfcace ad degrees of freedom, steps testg of hypothess. 8 Large sample tests: Test for Meas Z-test, Oe sample ad Two samples wth populato S.D. kow ad Ukow. 9 Small sample tests: Test for Meas Oe sample t test, Two samples t-test ad Pared t-test. 0 Ch-Square test cotgecy table wth Yate s correcto, F-test. Correlato: Defto, types, propertes, Scatter dagram, calculato ad testg. Regresso: Defto, Fttg of two les Y o X ad X o Y, Propertes, ter relato betwee correlato ad regresso. 3 Itroducto to Epermetal Desgs, Basc Prcples, ANOVA ts assumptos. 4 Completely Radomzed Desg: Layout, Aalyss wth equal ad uequal replcatos. 5 Radomzed Block Desg: Layout ad Aalyss. 6 Lat Square Desg: Layout ad Aalyss.

3 3 PRACTICALS S.No. Topc Costructo of Frequecy Dstrbuto tables Computato of Arthmetc Mea for Grouped ad U-grouped data 3 Computato of Meda for Grouped ad U-grouped data 4 Computato of Mode for Grouped ad U-grouped data 5 Computato of Stadard Devato ad varace for grouped ad ugrouped data 6 Computato of coeffcet of varato for grouped ad ugrouped data 7 SND (Z) test for sgle sample, Populato SD kow ad Ukow 8 SND (Z) test for two samples, Populato SD kow ad Ukow 9 Studet s t-test for sgle ad two samples 0 Pared t-test ad F-test Ch-square test cotgecy table wth Yate s correcto Computato of correlato coeffcet ad ts testg 3 Fttg of smple regresso equatos Y o X ad X o Y 4 Completely Radomzed Desg: Aalyss wth equal ad uequal replcatos 5 Radomzed Block Desg: Aalyss 6 Lat Square Desg: Aalyss

4 S T A T I S T I C S Statstcs has bee defed dfferetly by dfferet authors from tme to tme. Oe ca fd more tha hudred deftos the lterature of statstcs. Statstcs ca be used ether as plural or sgular. Whe t s used as plural, t s a systematc presetato of facts ad fgures. It s ths cotet that majorty of people use the word statstcs. They oly meat mere facts ad fgures. These fgures may be wth regard to producto of food gras dfferet years, area uder cereal crops dfferet years, per capta come a partcular state at dfferet tmes etc., ad these are geerally publshed trade jourals, ecoomcs ad statstcs bullets, ews papers, etc., Whe statstcs s used as sgular, t s a scece whch deals wth collecto, classfcato, tabulato, aalyss ad terpretato of data. The followg are some mportat defto of statstcs. Statstcs s the brach of scece whch deals wth the collecto, classfcato ad tabulato of umercal facts as the bass for eplaatos, descrpto ad comparso of pheomeo - Lovtt The scece whch deals wth the collecto, aalyss ad terpretato of umercal data - Corto & Cowde The scece of statstcs s the method of judgg collectve, atural or socal pheomeo from the results obtaed from the aalyss or eumerato or collecto of estmates - Kg Statstcs may be called the scece of coutg or scece of averages or statstcs s the scece of the measuremet of socal orgasm, regarded as whole all ts mafestatos - Bowley Statstcs s a scece of estmates ad probabltes -Boddgto Statstcs s a brach of scece, whch provdes tools (techques) for decso makg the face of ucertaty (probablty) - Walls ad Roberts Ths s the moder defto of statstcs whch covers the etre body of statstcs All deftos clearly pot out the four aspects of statstcs collecto of data, aalyss of data, presetato of data ad terpretato of data. Importace: Statstcs plays a mportat role our daly lfe, t s useful almost all sceces socal as well as physcal such as bology, psychology, educato, ecoomcs, busess maagemet, agrcultural sceces etc.,. The statstcal methods ca be ad are

5 beg followed by both educated ad ueducated people. I may staces we use sample data to make fereces about the etre populato.. Plag s dspesable for better use of ato s resources. Statstcs are dspesable plag ad takg decsos regardg eport, mport, ad producto etc., Statstcs serves as foudato of the super structure of plag.. Statstcs helps the busess ma the formulato of polces wth regard to busess. Statstcal methods are appled market ad producto research, qualty cotrol of maufactured products 3. Statstcs s dspesable ecoomcs. Ay brach of ecoomcs that requre comparso, correlato requres statstcal data for salvato of problems 4. State. Statstcs s helpful admstrato fact statstcs are regarded as eyes of admstrato. I collectg the formato about populato, mltary stregth etc., Admstrato s largely depeds o facts ad fgures thud t eeds statstcs 5. Bakers, stock echage brokers, surace compaes all make etesve use of statstcal data. Isurace compaes make use of statstcs of mortalty ad lfe premum rates etc., for bakers, statstcs help decdg the amout requred to meet day to day demads. 6. Problems relatg to poverty, uemploymet, food storage, deaths due to dseases, due to shortage of food etc., caot be fully weghted wthout the statstcal balace. Thus statstcs s helpful promotg huma welfare 7. Statstcs are a very mportat part of poltcal campags as they lead up to electos. Every tme a scetfc poll s take, statstcs are used to calculate ad llustrate the results percetages ad to calculate the marg for error. I agrcultural research, Statstcal tools have played a sgfcat role the aalyss ad terpretato of data.. I makg data about dry ad wet lads, lads uder taks, lads uder rrgato projects, rafed areas etc.,. I determg ad estmatg the rrgato requred by a crop per day, per base perod. 3. I determg the requred doses of fertlzer for a partcular crop ad crop lad.

6 3 4. I sol chemstry also statstcs helps classfyg the sols basg o ther aalyss results, whch are aalyzed wth statstcal methods. 5. I estmatg the losses curred by partcular pest ad the yeld losses due to sect, brd, or rodet pests statstcs s used etomology. 6. Agrcultural ecoomsts use forecastg procedures to determe the future demad ad supply of food ad also use regresso aalyss the emprcal estmato of fucto relatoshp betwee quattatve varables. 7. Amal scetsts use statstcal procedures to ad aalyzg data for decso purposes. 8. Agrcultural egeers use statstcal procedures several areas, such as for rrgato research, modes of cultvato ad desg of harvestg ad cultvatg machery ad equpmet. Lmtatos of Statstcs:. Statstcs does ot study qualtatve pheomeo. Statstcs does ot study dvduals 3. Statstcs laws are ot eact laws 4. Statstcs does ot reveal the etre formato 5. Statstcs s lable to be msused 6. Statstcal coclusos are vald oly o average base Types of data: The data are of two types ) Prmary Data ad ) Secodary Data ) Prmary Data: It s the data collected by the prmary source of formato.e by the vestgator hmself. ) Secodary Data: It s the data collected from secodary sources of formato, lke ews papers, trade jourals ad statstcal bullets, etc., Varables ad Attrbutes: Varablty s a commo characterstc bologcal sceces. A quattatve or qualtatve characterstc that vares from observato to observato the same group s called a varable. I case of quattatve varables, observatos are made usg terval scales whereas case of qualtatve varables omal scales are used. Covetoally, the quattatve varables are termed as varables ad qualtatve varables are termed as attrbutes. Thus, yeld of a crop, avalable troge sol, daly temperature, umber of leaves per plat ad umber of eggs lad by sects are all varables. The crop varetes, sol types, shape of seeds, seasos ad se of sects are attrbutes.

7 4 The varable tself ca be classfed as cotuous varable ad dscrete varable. The varables for whch fractoal measuremets are possble, at least coceptually, are called cotuous varables. For eample, the rage of 7 kg to 0 yeld of a crop, yeld mght be 7.5 or 7.04kg. Hece, yeld s a cotuous varable. The varables for whch such factoal measuremets are ot possble are called dscrete or dscotuous varables. For eample, the umber of gras per pacle of paddy ca be couted full umbers lke 79, 80, 8 etc. Thus, umber of gras per pacle s a dscrete varable. The varables, dscrete or cotuous are deted by captal letters lke X ad Y. Costructo of Frequecy Dstrbuto Table: I statstcs, a frequecy dstrbuto s a tabulato of the values that oe or more varables take a sample. Each etry the table cotas the frequecy or cout of the occurreces of values wth a partcular group or terval, ad ths way the table summarzes the dstrbuto of values the sample. The followg steps are used for costructo of frequecy table Step-: The umber of classes are to be decded The approprate umber of classes may be decded by Yule s formula, whch s as follows: Number of classes =.5 /4. where s the total umber of observatos Step-: The class terval s to be determed. It s obta by usg the relatoshp Mamum value the gve data Mmum value the gve data C.I = Number of classes Step-3: The frequeces are couted by usg Tally marks Step-4: The frequecy table ca be made by two methods a) Eclusve method b) Iclusve method a) Eclusve method: I ths method, the upper lmt of ay class terval s kept the same as the lower lmt of the just hgher class or there s o gap betwee upper lmt of oe class ad lower lmt of aother class. It s cotuous dstrbuto

8 5 E: C.I. Tally marks Frequecy (f) b) Iclusve method: There wll be a gap betwee the upper lmt of ay class ad the lower lmt of the just hgher class. It s dscotuous dstrbuto E: C.I. Tally marks Frequecy (f) To covert dscotuous dstrbuto to cotuous dstrbuto by subtractg 0.5 from lower lmt ad by addg 0.5 to upper lmt Note: The arragemet of data to groups such that each group wll have some umbers. These groups are called class ad umber of observatos agast these groups are called frequeces. Each class terval has two lmts. Lower lmt ad. Upper lmt The dfferece betwee upper lmt ad lower lmt s called legth of class terval. Legth of class terval should be same for all the classes. The average of these two lmts s called md value of the class. Eample: Costruct a frequecy dstrbuto table for the followg data 5, 3, 45, 8, 4, 4,,, 9, 5, 6, 35, 3, 4, 47, 8, 44, 37, 7, 46, 38, 4, 43, 46, 0,, 36, 45,, 8. Soluto: Number of observatos () = 30 Number of classes =.5 /4 =.5 30 /4 =.5.3 =

9 6 Class terval = Ma. value M. value No. of. classes = = = Iclusve method: C.I. Tally marks Frequecy (f) Total 30 Eclusve method: C.I. Tally marks Frequecy (f) Total 30 MEASURES OF CENTRAL TENDENCY Oe of the most mportat aspects of descrbg a dstrbuto s the cetral value aroud whch the observatos are dstrbuted. Ay mathematcal measure whch s teded to represet the ceter or cetral value of a set of observatos s kow as measure of cetral tedecy (or ) The sgle value, whch represets the group of values, s termed as a measure of cetral tedecy or a measure of locato or a average.

10 7 Characterstcs of a Satsfactory Average:. It should be rgdly defed. It should be easy to uderstad ad easy to calculate 3. It should be based o all the observatos 4. It should be least affected by fluctuatos samplg 5. It should be capable of further algebrac treatmet 6. It should ot be affected much by the etreme values 7. It should be located easly Types of average:. Arthmetc Mea. Meda 3. Mode 4. Geometrc Mea 5. Harmoc Mea Arthmetc Mea (A.M): It s defed as the sum of the gve observatos dvded by the umber of observatos. A.M. s measured wth the same uts as that of the observatos. Ugrouped data: Drect Method: Let,,, be observatos the the A.M s computed from the formula: A.M. =... where = sum of the gve observatos = Number of observatos Lear Trasformato Method or Devato Method: Whe the varable costtutes large values of observatos, computato of arthmetc mea volves more calculatos. To overcome ths dffculty, Lear Trasformato Method s used. The value s trasformed to d. ad A.M. = A+ where A = Assumed mea whch s geerally take as class md pot of mddle class or the class where frequecy s large. d

11 8 d = A = devatos of the th value of the varable take from a assumed mea ad = umber of observatos Grouped Data: Let f, f,., f be frequeces correspodg to the md values of the class tervals,, the ad A.M. = f f f f... f f f... f N f (drect method) A.M. =A+ fd f d f f where d = devato = f f d C A fd C (drect method) N A ; f = frequecy; C = class terval; c = md values of classes. Arthmetc mea, whe computed for the data of etre populato, s represeted by the symbol. Where as whe t s computed o the bass of sample data, t s represeted as X, whch s the estmate of. Propertes of A.M.: ) The algebrac sum of the devatos take from arthmetc mea s zero.e. (-A.M.) = 0 ) Let be the mea of observatos, be the mea of the observatos k be the mea of k observatos the the mea of = ( k) observatos s gve by = Merts: It s well defed formula defed. It s easy to uderstad ad easy to calculate 3. It s based upo all the observatos k k k = k k

12 9 4. It s ameable to further algebrac treatmets, provded the sample s radomly obtaed. 5. Of all averages, arthmetc mea s affected least by fluctuatos of samplg Demerts:. Caot be determed by specto or t ca be located graphcally. Arthmetc mea caot be obtaed f a sgle observato s mssg or lost 3. Arthmetc mea s affected very much by etreme values 4. Arthmetc mea may lead to wrog coclusos f the detals of the data from whch t s computed are ot gve 5. I etremely asymmetrcal (skewed) dstrbuto, usually arthmetc mea s ot a sutable measure of locato Eamples: ) Ugrouped data: If the weghts of 7 ear-heads of sorghum are 89, 94, 0, 07, 08, 5 ad 6 g. Fd arthmetc mea by drect ad devato methods Soluto: ) Drect Method: A.M. = ) Devato Method: A.M. = Where = umber of observatos; A d = gve values; A = arbtrary mea (assumed value); d = devato = -A d = A = = = = = = = 4 74 d 9 here A = assumed value = 0 ) A.M. = () A.M. = A d

13 0 ) Grouped Data: = = 0 7 = g = g. The followg are the 405 soybea plat heghts collected from a partcular plot. Fd the arthmetc mea of the plats by drect ad drect method: Plat heght (Cms) No. of plats( f ) Soluto: a) Drect Method: A.M. = b) Devato Method: Where A.M. = f N = md values of the correspodg classes f N = Total frequecy = f = frequecy f d A N C Where d = devato (.e. d A ) C Legth of class terval (C ) = 5; Assumed value (A) = 30 A d C.I f f C Total N = 405 Σf = 70 Σf d = 4

14 a) Drect Method: b) Devato Method: A.M. = cms A.M. = = cms. MEDIAN The meda s the mddle most tem that dvdes the dstrbuto to two equal parts whe the tems are arraged ascedg order. Ugrouped data: If the umber of observatos s odd the meda s the mddle value after the values have bee arraged ascedg or descedg order of magtude. I case of eve umber of observatos, there are two mddle terms ad meda s obtaed by takg the arthmetc mea of the mddle terms. I case of dscrete frequecy dstrbuto meda s obtaed by cosderg the cumulatve frequeces. The steps for calculatg meda are gve below: ) Arrage the data ascedg or descedg order of magtude ) Fd out cumulatve frequeces ) Apply formula: Meda = Sze of N, where N= f v) Now look at the cumulatve frequecy colum ad fd, that total whch s ether equal to N or et hgher to that ad determe the value of the varable correspodg to t, whch gves the value of meda. Cotuous frequecy dstrbuto: If the data are gve wth class tervals the the followg procedure s adopted for the calculato of meda. ) fd ) ) N, where N = f see the (less tha) cumulatve frequecy just greater tha the correspodg value of s meda N

15 I the case of cotuous frequecy dstrbuto, the class correspodg to the cumulatve frequecy just greater tha N obtaed by the followg formula: Eamples: Meda = l + s called the meda class ad the value of meda s N m C f Where l s the lower lmt of meda class f s the frequecy of the meda class m s the cumulatve frequecy of the class precedg the meda class C s the class legth of the meda class N = total frequecy Case-) whe the umber of observatos () s odd: The umber of rus scored by players of a crcket team of a school are 5, 9, 4,, 50, 30,, 0, 5, 36, 7 To compute the meda for the gve data, we proceed as follows: I case of ugrouped data, f the umber of observatos s odd the meda s the mddle value after the values have bee arraged ascedg or descedg order of magtude. Let us arrage the values ascedg order: 0, 5,, 9,, 7, 30, 36, 4, 50, 5 Meda = th value = value th = 6 th value Now the 6 th value the data s 7. Meda = 7 rus Case-) whe the umber of observatos () s eve: Fd the meda of the followg heghts of plats Cms: 6, 0, 4, 3, 9,,, 8

16 3 I case of eve umber of observatos, there are two mddle terms ad meda s obtaed by takg the arthmetc mea of the mddle terms. Let us arrage the gve tems ascedg order 3, 4, 6, 9, 0,, 8, I ths data the umber of tems = 8, whch s eve. Meda = average of th ad th terms. Meda = 9.5 Cms. Grouped Data: = average of 9 ad 0 Fd out the meda for the followg frequecy dstrbuto of 80 sorghum ear-heads. Weght of ear-heads ( g) No.of ear-heads N m Soluto: Meda = l C () f Where l s the lower lmt of the meda class f s the frequecy of the meda class m s the cumulatve frequecy of the class precedg the meda class C s the class terval of the meda class ad N = f = Total umber of observatos 80-00

17 4 Here Weght of earheads ( g) N No. of ear-heads Cumulatve Frequecy (CF) m f 4 (Meda class) N = f =80 = Cumulatve frequecy just greater tha 90.5 s 69 ad the correspodg class s The meda class s N = ; L =40; f = 4; m = 39 ad C = 0 Substtutg the above values equato (), we get Meda = = g Merts ad Demerts of Meda: Merts:. It s rgdly defed.. It s easly uderstood ad s easy to calculate. I some cases t ca be located merely by specto. 3. It s ot at all affected by etreme values. 4. It ca be calculated for dstrbutos wth ope-ed classes Demerts:. I case of eve umber of observatos meda caot be determed eactly. We merely estmate t by takg the mea of two mddle terms.

18 5. It s ot ameable to algebrac treatmet 3. As compared wth mea, t s affected much by fluctuatos of samplg. MODE Mode s the value whch occurs most frequetly a set of observatos or mode s the value of the varable whch s predomat the seres. I case of dscrete frequecy dstrbuto mode s the value of correspodg to mamum frequecy I case of cotuous frequecy dstrbuto, mode s obtaed from the formula: ( f Mode = l + f f f) f Where l s the lower lmt of modal class C s class terval of the modal class f the frequecy of the modal class f ad f are the frequeces of the classes precedg ad succeedg the modal class respectvely If the dstrbuto s moderately asymmetrcal, the mea, meda ad mode obey the emprcal relatoshp: Mode = 3 Meda Mea Eample: Fd the mode value for the followg data: 7, 8, 30, 33, 3, 35, 34, 33, 40, 4, 55, 46, 3, 33, 36, 33, 4, 33. Soluto: As see from the above data, the tem 33 occurred mamum umber of tmes.e. 5 tmes. Hece 33 s cosdered to be the modal value of the gve data. Grouped Data: Eample: The followg table gves the marks obtaed by 89 studets Statstcs. Fd the mode. Marks C No. of studets Soluto: Mode = ( f l f f f) f C Where l = the lower lmt of the modal class ; C = legth of the modal class

19 6 f = the frequecy of the modal class f = the frequecy of the class precedg modal class f = the frequecy of the class succeedg modal class Sometmes t so happeed that the above formula fals to gve the mode. I ths case, the modal value les a class other tha the oe cotag mamum frequecy. I such cases we take the help of the followg formula; Mode = l f f f C Where f, c, f ad f have usual meags. Marks No. of studets (f) f f f From the above table t s clear that the mamum frequecy s ad t les the class Thus the modal class s Here L = 9.5, c = 5, f =, f =6, f = 8 6 Mode = 30 * 5 * 6 8 = = m

20 7 Merts ad Demerts of Mode: Merts:. Mode s readly comprehesble ad easy to calculate.. Mode s ot at all affected by etreme values. 3. Mode ca be coveetly located eve f the frequecy dstrbuto has classtervals of uequal magtude provded the modal class ad the classes precedg ad succeedg t are of the same magtude. Ope-ed classes also do ot pose ay problem the locato of mode Demerts:. Mode s ll defed. It s ot always possble to fd a clearly defed mode. I some cases, we may come across dstrbutos wth two modes. Such dstrbutos are called b-modal. If a dstrbuto has more tha two modes, t s sad to be multmodal.. It s ot based upo all the observatos. 3. It s ot capable of further mathematcal treatmet. 4. As compared wth mea, mode s affected to a greater etet by fluctuatos of samplg. Dsperso Dsperso meas scatterg of the observatos amog themselves or from a cetral value (Mea/ Meda/ Mode) of data. We study the dsperso to have a dea about the varato. Suppose that we have the dstrbuto of the yelds (kg per plot) of two Groud ut varetes from 5 plots each. The dstrbuto may be as follows: Varety : Varety : It ca be see that the mea yeld for both varetes s 50 k.g. But we ca ot say that the performaces of the two varetes are same. There s greater uformty of yelds the frst varety where as there s more varablty the yelds of the secod varety. The frst varety may be preferred sce t s more cosstet yeld performace.

21 8 Measures of Dsperso: These measures gve us a dea about the amout of dsperso a set of observatos. They gve the aswers the same uts as the uts of the orgal observatos. Whe the observatos are klograms, the absolute measure s also klograms. If we have two sets of observatos, we caot always use the absolute measures to compare ther dsperso. The absolute measures whch are commoly used are:. The Rage. The Quartle Devato 3. The Mea Devato 4. The Stadard Devato ad Varace 5. Coeffcet of Varato 6. Stadard Error Stadard Devato: It s defed as the postve square root of the arthmetc mea of the squares of the devatos of the gve values from arthmetc mea. The square of the stadard devato s called varace. Ugrouped data: Let,,., be observatos the the stadard devato s gve by the formula S.D. = ( A. M.) Smplfyg the above formula, we have where A.M. = where = o. of observatos., or S.D. = by lear trasformato method, we have

22 9 = d d where d = A ; A= Assumed value; = Gve values Cotuous frequecy dstrbuto: (grouped data): Let f, f,,f be the frequeces correspodg to the md values of the classes,,, respectvely, the the stadard devato s gve by S.D. = N A M f.). ( where N f Smplfyg the above formula, we have S.D. = N f f N by lear trasformato method, we have S.D. = C N d f d f N ) ( where d = C A ; A = assumed value; ad C = class terval S.D. for populato data s represeted by the symbol Ugrouped data: Eample: Calculate S.D. for the kapas yelds ( kg per plot) of a cotto varety recorded from seve plots 5, 6, 7, 7, 9, 4, 5 ) Drect method: S.D. =

23 0 ) Devato Method: S.D. = d d Where = gve values Assumed value (A) = 7 d = devato (.e. d = -A) d A d = = = = = = =- 4 Σ = 43 Σ = 8 Σd = -6 Σ d = ) Drect method: ) Devato Method: S.D. = S.D. = 8 7 =.55 kg. 7 =.55 kg

24 Grouped Data: Eample: The followg are the 38soybea plat heghts Cms collected from a partcular plot. Fd the Stadard devato of the plats by drect ad devato method: Soluto: ) Drect method: Plat heghts (Cms) No. of Plats (f ) A.M. = f ; where N = N f ) Devato Method: S.D. = N f f N A.M. = A f d N C

25 S.D. = C N f d f d N C.I. f f f A d C f d f d N =38 Σf = Σf = Σf d = 348 Σ f d = 60 ) Drect method: A.M. = S.D. = = 9.96 Cms = =. 506 =.3 Cms. ) Devato Method: 348 A.M. = = 9.96 Cms 38

26 3 Measures of Relatve Dsperso: S.D. = = =.3 Cms. These measures are calculated for the comparso of dsperso two or more tha two sets of observatos. These measures are free of the uts whch the orgal data s measure. If the orgal s dollar or klometers, we do ot use these uts wth relatve measure of dsperso. These are a sort of rato ad are called coeffcets. Suppose that the two dstrbutos to be compared are epressed the same uts ad ther meas are equal or early equal. The ther varablty ca be compared drectly by usg ther stadard devatos. However, f ther meas are wdely dfferet or f they are epressed dfferet uts of measuremet. We ca ot use the stadard devatos as such for comparg ther varablty. We have to use the relatve measures of dsperso such stuatos. Coeffcet of Varato (C.V.) Coeffcet of varato s the percetage rato of stadard devato ad the arthmetc mea. It s usually epressed percetage. The formula for C.V. s, S. D. C.V. = 00 Mea Where S.D. = ad Mea The coeffcet of varato wll be small f the varato s small of the two groups, the oe wth less C.V. sad to be more cosstet. Note:. Stadard devato s absolute measure of dsperso. Coeffcet of varato s relatve measure of dsperso.

27 4 Eample: Cosder the dstrbuto of the yelds (per plot) of two groud ut varetes. For the frst varety, the mea ad stadard devato are 8 kg ad 6 kg respectvely. For the secod varety, the mea ad stadard devato are 55 kg ad 8 kg respectvely. The we have, for the frst varety For the secod varety 6 C.V. = 00 = 9.5% 8 8 C.V. = 00 = 4.5% 55 It s apparet that the varablty secod varety s less as compared to that the frst varety. But terms of stadard devato the terpretato could be reverse. Eample: Below are the scores of two crcketers 0 gs. Fd who s more cosstet scorer by Idrect method. A B Soluto: Let the player A = Ad the player B = y Coeffcet of varato of = ( C. V.) = X00 Where = A Stadard devato of = d where d d d A ad coeffcet of varato of y = ( C. V.) y = X00 y Stadard devato of y = y y d y dy Where y = d B y where d y y B

28 5 Here A = 50 ad B = 90 y d = A d y =y B d Σ d = -704 Σd y = -959 Σd = Σd y =0789 dy = 50 y = = 79.6 rus = 94. rus y = = = 55.4 rus = 39.0 rus 55.4 ( C. V.) = X y 39.0 ( C. V.) = = 69.4% = 4.46% Coeffcet of varato of A coclude that coeffcet of player B s more cosstet s greater tha coeffcet of varato of B ad hece we

29 6 NORMAL DISTRIBUTION The Normal Dstrbuto (N.D.) was frst dscovered by De-Movre as the lmtg form of the bomal model 733, later depedetly worked Laplace ad Gauss. The Normal dstrbuto s probably the most mportat dstrbuto statstcs. It s a probablty dstrbuto of a cotuous radom varable ad s ofte used to model the dstrbuto of dscrete radom varable as well as the dstrbuto of other cotuous radom varables. The basc from of ormal dstrbuto s that of a bell, t has sgle mode ad s symmetrc about ts cetral values. The fleblty of usg ormal dstrbuto s due to the fact that the curve may be cetered over ay umber o the real le ad t may be flat or peaked to correspod to the amout of dsperso the values of radom varable. Defto: A radom varable X s sad to follow a Normal Dstrbuto wth parameter ad ad f ts desty fucto s gve by the probablty law f() = ( ) e - < < ; - < < ; > 0 where = a mathematcal costat equalty = /7 e = Napera base equalg.783 = populato mea = populato stadard devato = a gve value of the radom varable the rage - < < Characterstcs of Normal dstrbuto ad ormal curve: The ormal probablty curve wth mea ad stadard devato s gve by the equato f() = ( ) e ; - < < ad has the followg propertes. The curve s bell shaped ad symmetrcal, about the mea. The heght of ormal curve s at ts mamum at the mea. Hece the mea ad mode of ormal dstrbuto cocdes. Also the umber of observatos below the mea a ormal dstrbuto s equal to the umber of observatos about the mea. Hece mea ad meda of N.D. cocdes. Thus, N.D. has Mea = meda = mode

30 7. As creases umercally, f() decreases rapdly, the mamum probablty occurrg at the pot =, ad gve by p[()] ma = 3 v. Skewess = 3 = 0 4 v. Kurtoss = = v. v. v. All odd cetral momets are zero s.e = 3 (,, ad 3 4 are called cetral momets) The frst ad thrd quartles are equdstat from the meda Lear combato of depedet ormal varates s also a ormal varate. The pots of fleo of the curve s gve by. If, f ( ) e f ( ) d the the area uder the ormal curve s dstrbuted as follows ) - < < + covers 68.6% of area ) ) - < < + covers 95.44% of area -3 < < +3 coves 99.73% of area Area uder Normal curve

31 8 The Normal Curve: The graph of the ormal dstrbuto depeds o two factors - the mea ad the stadard devato. The mea of the dstrbuto determes the locato of the ceter of the graph, ad the stadard devato determes the heght ad wdth of the graph. Whe the stadard devato s large, the curve s short ad wde; whe the stadard devato s small, the curve s tall ad arrow. All ormal dstrbutos look lke a symmetrc, bell-shaped curve, as show below. The curve o the left s shorter ad wder tha the curve o the rght, because the curve o the left has a bgger stadard devato. Stadard Normal Dstrbuto: If X s a ormal radom varable wth Mea ad stadard devato, the Z = stadard devato =. X s a stadard ormal varate wth zero mea ad The probablty desty fucto of stadard ormal varate z s e z f(z) = ad f ( z) dz = A graph represetg the desty fucto of the Normal probablty dstrbuto s also kow as a Normal Curve or a Bell Curve (see Fgure below). To draw such a curve, oe eeds to specfy two parameters, the mea ad the stadard devato. The graph below has a mea of zero ad a stadard devato of,.e., (m =0, s =). A Normal dstrbuto wth a mea of zero ad a stadard devato of s also kow as the Stadard Normal Dstrbuto. Stadard Normal Dstrbuto

32 9 Testg of Hypothess Itroducto: The estmate based o sample values do ot equal to the true value the populato due to heret varato the populato. The samples draw wll have dfferet estmates compared to the true value. It has to be verfed that whether the dfferece betwee the sample estmate ad the populato value s due to samplg fluctuato or real dfferece. If the dfferece s due to samplg fluctuato oly t ca be safely sad that the sample belogs to the populato uder questo ad f the dfferece s real we have every reaso to beleve that sample may ot belog to the populato uder questo. The followg are a few techcal terms ths cotet. Hypothess: The assumpto made about ay ukow characterstcs s called hypothess. It may or may be true. E:. =.3; be the populato mea. =. ; be the populato stadard devato Populato follows Normal Dstrbuto. There are two types of hypothess, amely ull hypothess ad alteratve hypothess. Null Hypothess: Null hypothess s the statemet about the parameters. Such a hypothess, whch s usually a hypothess of o dfferece s called ull hypothess ad s usually deoted by H 0. or ay statstcal hypothess uder test s called ull hypothess. It s deoted by H 0.. H 0 : = 0. H 0 : = Alteratve Hypothess: Ay hypothess, whch s complemetary to the ull hypothess, s called a alteratve hypothess, usually deoted by H. E:. H : # 0. H : # Parameter: A characterstcs of populato values s kow as parameter. For eample, populato mea () ad populato varace ( ).

33 30 I practce, f parameter values are ot kow ad the estmates based o the sample values are geerally used. Statstc: A Characterstcs of sample values s called a statstc. For eample, sample mea ( ), sample varace (s ) where = ad s =... Samplg dstrbuto: The dstrbuto of a statstc computed from all possble samples s kow as samplg dstrbuto of that statstc. Stadard error: The stadard devato of the samplg dstrbuto of a statstc s kow as ts stadard error, abbrevated as S.E. S.E.( ) = ; where = populato stadard devato ad = sample sze Sample: A fte subset of statstcal objects a populato s called a sample ad the umber of objects a sample s called the sample sze. Populato: I a statstcal vestgato the terest usually les the assessmet of the geeral magtude ad the study of varato wth respect to oe or more characterstcs relatg to objects belogg to a group. Ths group of objects uder study s called populato or uverse. Radom samplg: If the samplg uts a populato are draw depedetly wth equal chace, to be cluded the sample the the samplg wll be called radom samplg. It s also referred as smple radom samplg ad deoted as SRS. Thus, f the populato cossts of N uts the chace of selectg ay ut s /N. A theoretcal defto of SRS s as follows: Suppose we draw a sample of sze from a populato sze N; the there are equal chace, samplg. N N possble samples of sze. If all possble samples have a of beg draw, the the samplg s sad to be smple radom Smple Hypothess: A hypothess s sad to be smple f t completely specfes the dstrbuto of the populato. For stace, case of ormal populato wth mea

34 3 ad stadard devato, a smple ull hypothess s of the form H 0 : = 0, s kow, kowledge about would be eough to uderstad the etre dstrbuto. For such a test, the probablty of commttg the type- error s epressed as eactly. Composte Hypothess: If the hypothess does ot specfy the dstrbuto of the populato completely, t s sad to be a composte hypothess. Followg are some eamples: H 0 : 0 ad s kow H 0 : 0 ad s kow All these are composte because oe of them specfes the dstrbuto completely. Hece, for such a test the LOS s specfed ot as but as at most. Types of Errors: I testg of statstcal hypothess there are four possble types of decsos. Rejectg H 0 whe H 0 s true. Rejectg H 0 whe H 0 s false 3. Acceptg H 0 whe H 0 s true 4. Acceptg H 0 whe H 0 s false ad 4 th possbltes leads to error decsos. Statstca gves specfc ames to these cocepts amely Type-I error ad Type-II error respectvely. the above decsos ca be arraged the followg table Rejectg H 0 H 0 s true Type-I error (Wrog decso) H 0 s false Correct Acceptg H 0 Correct Type-II error Type-I error: Rejectg H 0 whe H 0 s true Type-II error: Acceptg H 0 whe H 0 s false The probabltes of type-i ad type-ii errors are deoted by ad respectvely. Degrees of freedom: It s defed as the dfferece betwee the total umber of tems ad the total umber of costrats. If s the total umber of tems ad k the total umber of costrats the the degrees of freedom (d.f.) s gve by d.f. = -k Level of sgfcace(los): The mamum probablty at whch we would be wllg to rsk a type-i error s kow as level of sgfcace or the sze of Type-I error s level of sgfcace. The level of sgfcace usually employed testg of hypothess are 5%

35 3 ad %. The Level of sgfcace s always fed advace before collectg the sample formato. LOS 5% meas the results obtaed wll be true s 95% out of 00 cases ad the results may be wrog s 5 out of 00 cases. Crtcal value: whle testg for the dfferece betwee the meas of two populatos, our cocer s whether the observed dfferece s too large to beleve that t has occurred just by chace. But the the questo s how much dfferece should be treated as too large? Based o samplg dstrbuto of the meas, t s possble to defe a cut-off or threshold value such that f the dfferece eceeds ths value, we say that t s ot a occurrece by chace ad hece there s suffcet evdece to clam that the meas are dfferet. Such a value s called the crtcal value ad t s based o the level of sgfcace. Steps volved test of hypothess:. The ull ad alteratve hypothess wll be formulated. Test statstc wll be costructed 3. Level of sgfcace wll be fed 4. The table (crtcal) values wll be foud out from the tables for a gve level of sgfcace 5. The ull hypothess wll be rejected at the gve level of sgfcace f the value of test statstc s greater tha or equal to the crtcal value. Otherwse ull hypothess wll be accepted. 6. I the case of rejecto the varato the estmates wll be called sgfcat varato. I the case of acceptace the varato the estmates wll be called otsgfcat.

36 33 STANDARD NORMAL DEVIATE TESTS OR LARGE SAMPLE TESTS If the sample sze >30 the t s cosdered as large sample ad f the sample sze < 30 the t s cosdered as small sample. SND Test or Oe Sample (Z-test) Case-I: Populato stadard devato () s kow Assumptos:. Populato s ormally dstrbuted. The sample s draw at radom Codtos:. Populato stadard devato s kow. Sze of the sample s large (say > 30) Procedure: Let,,, be a radom sample sze of from a ormal populato wth mea ad varace. Let be the sample mea of sample of sze Null hypothess (H 0 ): populato mea (µ) s equal to a specfed value 0 Uder H 0, the test statstc s.e. H 0 : = 0 Z = 0 ~ N(0,).e the above statstc follows Normal Dstrbuto wth mea 0 ad varace. If the calculated value of Z < table value of Z at 5% level of sgfcace, H 0 s accepted ad hece we coclude that there s o sgfcat dfferece betwee the populato mea ad the oe specfed H 0 as 0. Case-II: If s ot kow Assumptos:. Populato s ormally dstrbuted. Sample s draw from the populato should be radom 3. We should kow the populato mea

37 34 Codtos:. Populato stadard devato s ot kow. Sze of the sample s large (say > 30) Null hypothess (H 0 ) : = 0 uder H 0, the test statstc Z = 0 s ~ N(0,) where s = = Sample mea; = sample sze If the calculated value of Z < table value of Z at 5% level of sgfcace, H 0 s accepted ad hece we coclude that there s o sgfcat dfferece betwee the populato mea ad the oe specfed H 0 otherwse we do ot accept H 0. The table value of Z at 5% level of sgfcace =.96 ad table value of Z at % level of sgfcace =.58. Two sample Z-Test or Test of sgfcat for dfferece of meas Case-I: whe s kow Assumptos:. Populatos are dstrbuted ormally. Samples are draw depedetly ad at radom Codtos:. Populatos stadard devato s kow. Sze of samples are large Procedure: Let be the mea of a radom sample of sze from a populato wth mea ad varace Let be the mea of a radom sample of sze from aother populato wth mea ad varace Null hypothess H 0 : = Alteratve Hypothess H :.e. The ull hypothess states that the populato meas of the two samples are detcal. Uder the ull hypothess the test statstc becomes

38 35 Z = ~ N(0,) ().e the above statstc follows Normal Dstrbuto wth mea 0 ad varace. If = = (say).e both samples have the same stadard devato the the test statstc becomes Z = ~ N(0,) () If the calculated value of Z < table value of Z at 5% level of sgfcace, H 0 s accepted otherwse rejected. If H 0 s accepted meas, there s o sgfcat dfferece betwee two populato meas of the two samples are detcal. Eample: The Average pacle legth of 60 paddy plats feld No. s 8.5. cms ad that of 70 paddy plats feld No. s 0.3 cms. Wth commo S.D..5 cms. Test whether there s sgfcat dfferece betwee tow paddy felds w.r.t pacle legth. Soluto: Null hypothess: H 0 : There s o sgfcat dfferece betwee the meas of two paddy felds w.r.t. pacle legth. H 0 : Uder H 0, the test statstc becomes Z = ~ N (0,) () Where frst feld sample mea = 8.5 ches secod feld sample mea = 0.3 ches = frst sample sze = 60 = secod sample sze = 70 = commo S.D. =.5 ches Substtute the gve values equato (), we get

39 36 Z = = = 8.89 Calculated value of Z = 5. Cal. Value of Z > table value of Z at 5% LOS(.96), H 0 s rejected. Ths meas, there s hghly sgfcat dfferece betwee two paddy felds w.r.t. pacle legth. Case-II: whe s ot kow Assumptos:. Populatos are ormally dstrbuted. Samples are draw depedetly ad at radom Codtos:. Populato stadard devato s ot kow. Sze of samples are large Null hypothess H 0 : = Uder the ull hypothess the test statstc becomes Z = s s ~ N(0,) () Where = st sample mea, = d sample mea s = st sample varace, s = d sample varace = st sample sze, = d sample sze If the calculated value of Z < table value of Z at 5% level of sgfcace, H 0 s accepted otherwse rejected. Eample: A breeder clams that the umber of flled gras per pacle s more a ew varety of paddy ACM.5 compared to that of a old varety ADT.36. To verfy hs clam a radom sample of 50 plats of ACM.5 ad 60 plats of ADT.36 were selected from the epermetal felds. The followg results were obtaed: (ForACM.5) (For ADT.36) 39.4-gras/pacle.9 gras/pacle s = s = = 50 = 60 Test whether the clam of the breeder s correct.

40 37 Sol: Null hypothess H 0 : (.e. the average umber of flled gras per pacle s the same for both ACM.5 ad ADT.36) Uder H 0, the test statstc becomes Z = s s ~ N (0,) () Where frst varety sample mea = 39.4 gras/pacle Secod varety sample mea =.9 gras/pacle s = frst sample stadard devato = s = secod sample stadard devato = = frst sample sze = 50 ad = secod sample sze = 60 Substtute the gve values equato (), we get Z = = (6.864) (0.096) 60 = 4.76 Calculated value of Z > Table value of Z at 5% LOS (.96), H 0 s rejected. We coclude that the umber of flled gras per pacle s sgfcatly greater ACM.5 tha ADT.36

41 38 SMALL SAMPLE TESTS The etre large sample theory was based o the applcato of ormal test. However, f the sample sze s small, the dstrbuto of the varous statstcs, e.g., Z = 0 are far from ormalty ad as such ormal test caot be appled f s small. I such cases eact sample tests, we use t-test poeered by W.S. Gosset (908) who wrote uder the pe ame of studet, ad later o developed ad eteded by Prof. R.A. Fsher. Studet s t-test: Let,,., be a radom sample of sze has draw from a ormal populato wth mea ad varace the studet s t s defed by the statstc t = 0 s where ad s = ths test statstc follows a t dstrbuto wth (-) degrees of freedom (d.f.). To get the crtcal value of t we have to refer the table for t-dstrbuto agast (-) d.f. ad the specfc level of sgfcace. Comparg the calculated value of t wth crtcal value, we ca accept or reject the ull hypothess. The Rage of t dstrbuto s - to +. Oe Sample t test Oe sample t-test s a statstcal procedure that s used to kow the populato mea ad the kow value of the populato mea. I oe sample t-test, we kow the populato mea. We draw a radom sample from the populato ad the compare the sample mea wth populato mea ad make a statstcal decso as to whether or ot the sample mea s dfferet from the populato mea. I oe sample t-test, sample sze should be less tha 30. Assumptos:. Populato s ormally dstrbuted Codtos:. Sample s draw from the populato ad t should be radom 3. We should kow the populato mea. Populato S.D. s ot kow. Sze of the sample s small (<30).

42 39 Procedure: Let : Let,,., be a radom sample of sze has draw from a ormal populato wth mea ad varace. Null hypothess (H 0 ): populato mea (µ) s equal to a specfed value 0 Uder H 0, the test statstc becomes.e. H 0 : = 0 t = 0 ad follows studet s t dstrbuto wth (-) degrees of freedom. where s ad s = We ow compare the calculated value of t wth the tabulated value at certa level of sgfcace If calculated value of t < table value of t at (-) d.f., the ull hypothess s accepted ad hece we coclude that there s o sgfcat dfferece betwee the populato mea ad the oe specfed H 0 as 0. Eample: Based o feld epermets, a ew varety of greegram s epected to gve a yeld of 3 qutals per hectare. The varety was tested o radomly selected farmer felds. The yelds (qutal/hectare) were recorded as 4.3,.6, 3.7, 0.9,3.7,.0,.4,.0, 3.,.6, 3.4 ad 3.. Do the results coform the epectato? Soluto: Null Hypothess: H 0 : = 0 = 3.e. the results coform the epectato The test statstc becomes Where t = 0 s ~t (-) d.f. ad s = ( ) s a ubased estmate of

43 40 Let yeld = (say) Σ = 5.8 Σ = =.73 s = =.0 t = (5.8) = = 0.93 qa/h. t-table value at (-) = d.f. at 5 percet level of sgfcace s.0. Calculated value of t < table value of t, H 0 s accepted ad we may coclude that the results coform to the epectato. t-test for Two Samples Assumptos:. Populatos are dstrbuted ormally Codtos:. Samples are draw depedetly ad at radom. Stadard devatos the populatos are same ad ot kow. Sze of the sample s small Procedure: If two depedet samples ( =,,., ) ad y j ( j =,,.., ) of szes ad have bee draw from two ormal populatos wth meas ad respectvely. Null hypothess H 0 : =

44 4 The ull hypothess states that the populato meas of the two groups are detcal, so ther dfferece s zero. Uder H 0, the test statstcs s t = Where S = = or S y y y S = pooled varace = ( ) s ( ) s where s ad s are the varaces of the frst ad secod samples respectvely. ad ad y j y j ; where ad y are the two sample meas. Whch follows Studet s t dstrbuto wth ( + -) d.f. If calculated value of t < table value of t wth ( + -) d.f. at specfed level of sgfcace, the the ull hypothess s accepted otherwse rejected. Eample: Two vertes of potato plats (A ad B) yelded tubes are show the followg table. Does the mea umber of tubes of the varety A sgfcatly dffer from that of varety B? Soluto: Tuber yeld, kg/plat Varety-A Varety-B Hypothess H 0 : =.e the mea umber of tubes of the varety A sgfcatly dffer from the varety B Statstc t = S y ~ t ( ) d. f = st sample sze; = d sample sze = Mea of the frst sample; y = mea of the secod sample

45 4 = Where S = ad y y y y y Σy = Σy = Σ = 4.70 Σ = = y 7.40 y 0 =.5 Kg =.74 Kg Where (4.70) (7.40) S = = = 0.09 Kg S = S = 0.3 Kg. Test statstc t = = Calculated value of t = 3.77 = Table value of t for 9 d.f. at 5 % level of sgfcace s.09 Sce the calculated value of t > table value of t, the ull hypothess s rejected ad hece we coclude that the mea umber of tubes of the varety A sgfcatly ot dffer from the varety B

46 43 Pared t test The pared t-test s geerally used whe measuremets are take from the same subject before ad after some mapulato such as jecto of a drug. For eample, you ca use a pared t test to determe the sgfcace of a dfferece blood pressure before ad after admstrato of a epermetal pressor substace. Assumptos:. Populatos are dstrbuted ormally. Samples are draw depedetly ad at radom Codtos:. Samples are related wth each other. Szes of the samples are small ad equal 3. Stadard devatos the populatos are equal ad ot kow Hypothess H 0 : 0 Uder H 0, the test statstc becomes, d t = d S t(-) d.f. where d d ad S = where S s varace of the devatos d = sample sze; where d = -y ( =,,,) If calculated value of t < table value of t for (-)d.f. at α% level of sgfcace, the the ull hypothess s accepted ad hece we coclude that the two samples may belog to the same populato. Otherwse, the ull hypothess rejected. Eample: The average umber of seeds set per pod Lucere were determed for top flowers ad bottom flowers te plats. The values observed were as follows: Top flowers Bottom flowers Test whether there s ay sgfcat dfferece betwee the top ad bottom flowers wth respect to average umbers of seeds set per pod. Soluto: Null Hypothess H 0 : d = 0 d

47 44 Uder H 0 becomes, the test statstc s Where d d t d s ~ ) t( d. f. ad d s d y d=-y d Σd = Σd = d 7.90 d = 0 = 0.79 s (7.90) =.7 s = s. 7 t = =.65 Calculated value of t =.65 Table value of t for 9 d.f. at 5% level of sgfcace s.6 Calculated value of t < table value of t, the ull hypothess s accepted ad we coclude that there s o sgfcat dfferece betwee the top ad bottom flowers wth respect to average umbers of seeds set per pod.

48 45 F Test I agrcultural epermets the performace of a treatmet s assessed ot oly by ts mea but also by ts varablty. Hece, t s of terest to us to compare the varablty of two populatos. I testg of hypothess the equalty of varaces, the greater varace s always placed the Numerator ad smaller varace s placed the deomator. F- test s used to test the equalty of two populato varaces, equalty of several regresso coeffcets, ANOVA. F- test was dscovered by G.W. Sedecor. The rage of F : 0 to Let,,......, ad y, y,... y be the two depedet radom samples of szes ad draw from two ormal populatos N(, ) ad N(, ) respectvely. S ad S are the sample varaces of the two samples. Null hypothess H 0 : Uder H 0, the test statstc becomes S F = S where, S > S Whch follows F-dstrbuto wth ( -, -)d.f. Where S ad S = y y S or F = S where S > S Whch follows F-dstrbuto wth ( -, -)d.f. If calculated value of F < table value of F wth ( -, -)d.f at specfed level of sgfcace, the the ull hypothess s accepted ad hece we coclude that the varaces of the populatos are homogeeous otherwse heterogeeous. Eample: The heghts meters of red gram plats wth two types of rrgato two felds are as follows: Tap water () Sale water (y) Test whether the varaces of the two system of rrgato are homogeeous. Soluto: H 0 : The varaces of the two systems of rrgato are homogeeous.

49 46.e. Uder H 0, the test statstc becomes Where ad S F = S S ; ( S ) S = fst sample varace = S = secod sample varace = y y y y y = y ad S = (37.9) S = (9.7) =.4 mt = 0.79 mt S F = S F calculated value =.78 = =.78 Table value of F 0.05 for ( -, -) d.f. = 3.3 Calculated value of F < Table value of at 5% level of sgfcace, H 0 s accepted ad hece we coclude that the varaces of the two systems of rrgato are homogeeous. F or F F F

50 47 Ch-square ( ) test The varous tests of sgfcace studed earler such that as Z-test, t-test, F-test were based o the assumpto that the samples were draw from ormal populato. Uder ths assumpto the varous statstcs were ormally dstrbuted. Sce the procedure of testg the sgfcace requres the kowledge about the type of populato or parameters of populato from whch radom samples have bee draw, these tests are kow as parametrc tests. But there are may practcal stuatos whch the assumpto of ay kd about the dstrbuto of populato or ts parameter s ot possble to make. The alteratve techque where o assumpto about the dstrbuto or about parameters of populato s made are kow as o-parametrc tests. Ch-square test s a eample of the o parametrc test. Ch-square dstrbuto s a dstrbuto free test. If X N(0,) the ~ Ch-square dstrbuto was frst dscovered by Helmert 876 ad later depedetly by Karl Pearso 900. The rage of ch-square dstrbuto s 0 to. Measuremetal data: the data obtaed by actual measuremet s called measuremetal data. For eample, heght, weght, age, come, area etc., Eumerato data: the data obtaed by eumerato or coutg s called eumerato data. For eample, umber of blue flowers, umber of tellget boys, umber of curled leaves, etc., test s used for eumerato data whch geerally relate to dscrete varable where as t-test ad stadard ormal devate tests are used for measure metal data whch geerally relate to cotuous varable. test ca be used to kow whether the gve objects are segregatg a theoretcal rato or whether the two attrbutes are depedet a cotgecy table. The epresso for test for goodess of ft ( O E ) = E where O = observed frequeces E = epected frequeces = umber of cells( or classes) Whch follows a ch-square dstrbuto wth (-) degrees of freedom

51 48 The ull hypothess H 0 = the observed frequeces are agreemet wth the epected frequeces If the calculated value of < Table value of wth (-) d.f. at specfed level of sgfcace (), we accept H 0 otherwse we do ot accept H 0. Codtos for the valdty of test: The valdty of -test of goodess of ft betwee theoretcal ad observed, the followg codtos must be satsfed. ) The sample observatos should be depedet ) Costrats o the cell frequeces, f ay, should be lear o = e ) N, the total frequecy should be reasoably large, say greater tha 50 v) If ay theoretcal (epected) cell frequecy s < 5, the for the applcato of ch-square test t s pooled wth the precedg or succeedg frequecy so that the pooled frequecy s more tha 5 ad fally adjust for the d.f. lost poolg. Applcatos of Ch-square Test:. testg the depedece of attrbutes. to test the goodess of ft 3. testg of lkage geetc problems 4. comparso of sample varace wth populato varace 5. testg the homogeety of varaces 6. testg the homogeety of correlato coeffcet Test for depedece of two Attrbutes of () Cotgecy Table: A characterstc whch ca ot be measured but ca oly be classfed to oe of the dfferet levels of the character uder cosderato s called a attrbute. cotgecy table: Whe the dvduals (objects) are classfed to two categores wth respect to each of the two attrbutes the the table showg frequeces dstrbuted over classes s called cotgecy table.

52 49 Suppose the dvduals are classfed accordg to two attrbutes say tellgece (A) ad colour (B). The dstrbuto of frequeces over cells s show the followg table. A B A A Row totals B a B R +(a+b) B c D R = (c+d) Colum total C = (a+c) C = (b+d) N = (R +R ) or (C +C ) Where R ad R are the margal totals of st row ad d row C ad C are the margal totals of st colum ad d colum N = grad total The ull hypothess H 0 : the two attrbutes are depedet ( f the colour s ot depedet o tellget) Based o above H 0, the epected frequeces are calculated as follows. R C R C R C R C E(a) = ; E(b) = ; E(c) = ; E(d) = N N N N Where N = a+b+c+d To test ths hypothess we use the test statstc ( O E ) = E the degrees of freedom for m cotgecy table s (m-)(-) the degrees of freedom for cotgecy table s (-)(-) = Ths method s appled for all rc cotgecy tables to get the epected frequeces. The degrees of freedom for rc cotgecy table s (r-)(c-) If the calculated value of < table value of at certa level of sgfcace, the H 0 s accepted otherwse we do ot accept H 0 The alteratve formula for calculatg cotgecy table s = ( ad bc) N R R C C Eample: Eame the followg table showg the umber of plats havg certa characters, test the hypothess that the flower colour s depedet of the shape of leaf. Flower colour Shape of leaf Totals Flat leaves Curled leaves Whte flowers 99 (a) 36 (b) R = 35 Red flowers 0( c) 5 (d) R = 5 Totals C = 9 C = 4 N = 60

53 50 Soluto: Null hypothess H 0 : attrbutes flower colour ad shape of leaf are depedet of each other. Uder H 0 the statstc s ( o e ) e where o = observed frequecy ad Epected frequeces are calculated as follows. E(a) = E(b) = E( c ) = R R * C N * C N R * C N e = epected frequecy 35*9 = where R ad R = Row totals 60 35* 4 =34.59 C ad C = colum totals 60 5*9 60 =8.59 N= Grad totals E(d) = R * C N o Drect Method: 5* 4 = e o e (o - e ) (o - e ) e Calculated value of Statstc: ( o e ) = 0.49 = e N( ad bc) R R C C here a = 99, b = 36, c = 0 ad d = 5 ad N = 60 = 60(99*5 36*0) 35* 5*9*4 ( o e e ) =0.49

54 5 Table value of Calculated value of 60*5065 = = = Calculated value of = 0.40 for (-) (-) = d.f. s 3.84 < Table value of at 5% LOS for d.f., Null hypothess s accepted ad hece we coclude that two characters, flower colour ad shape of leaf are depedet of each other. Yates correcto for cotuty a cotgecy table: I a cotgecy table, the umber of d.f. s (-)(-) =. If ay oe of Epected cell frequecy s less tha 5, the we use of poolg method for test results wth `0 d.f. (sce d.f. s lost poolg) whch s meagless. I ths case we apply a correcto due to Yates, whch s usually kow a Yates correcto for cotuty. Yates correcto cossts of the followg steps; () add 0.5 to the cell frequecy whch s the least, () adjust the remag cell frequeces such a way that the row ad colum totals are ot chaged. It ca be show that ths correcto wll result the formula. (corrected) = N N ad bc R R C C Eample: The followg data are observed for hybrds of Datura. Flowers volet, fruts prckly =47 Flowers volet, fruts smooth = Flowers whte, fruts prckly = Flowers whte, fruts smooth = 3. Usg ch-square test, fd the assocato betwee colour of flowers ad character of fruts. Sol: H 0 : The two attrbutes colour of flowers ad fruts are depedet. We caot use Yate s correcto for cotuty based o observed values. epected frequecy less tha 5, we use Yates s correcto for cotuty. If oly The test statstc s

55 5 (corrected) = N N ad bc R R C C Flowers Volet Flowers whte Total Fruts Prckly 47(48.34) (9.66) 68 Fruts smooth (0.66) 3(4.34) 5 Total The fgures the brackets are the epected frequeces ( corrected ) = Calculated value of Table value of Calculated value of = = (47 *3) (*) 68*5*59* *5*59* = 0.8 = 0.8 for (-) (-) = d.f. s 3.84 < table value of colour of flowers ad character of fruts are ot assocated, H 0 s accepted ad hece we coclude that CORRELATION Whe there are two cotuous varables whch are cocomtat ther jot dstrbuto s kow as bvarate ormal dstrbuto. If there are more tha two such varables ther jot dstrbuto s kow as multvarate ormal dstrbutos. I case of bvarate or multvarate ormal dstrbutos, we may be terested dscoverg ad measurg the magtude ad drecto of the relatoshp betwee two or more varables. For ths purpose we use the statstcal tool kow as correlato. Defto: If the chage oe varable affects a chage the other varable, the two varables are sad to be correlated ad the degree of assocato shp (or etet of the relatoshp) s kow as correlato. Types of correlato: a). Postve correlato: If the two varables devate the same drecto,.e., f the crease (or decrease) oe varable results a correspodg crease (or decrease) the other varable, correlato s sad to be drect or postve.

56 53 E: () Heghts ad weghts () Household come ad epedture () Amout of rafall ad yeld of crops (v) Prces ad supply of commodtes (v) Feed ad mlk yeld of a amal (v) Soluble troge ad total chlorophyll the leaves of paddy. b). Negatve correlato: If the two varables costatly devate the opposte drecto.e., f crease (or decrease) oe varable results correspodg crease) the other varable, correlato s sad to be verse or egatve. E: () Prce ad demad of a goods () Volume ad pressure of perfect gas () Sales of woole garmets ad the day temperature (v) Yeld of crop ad plat festato decrease (or c) No or Zero Correlato: If there s o relatoshp betwee the two varables such that the value of oe varable chage ad the other varable rema costat s called o or zero correlato. Fgures: Methods of studyg correlato:. Scatter Dagram. Karl Pearso s Coeffcet of Correlato 3. Spearma s Rak Correlato 4. Regresso Les. Scatter dagram: It s the smplest way of the dagrammatc represetato of bvarate data. Thus for the bvarate dstrbuto (,y ); = j =,,, If the values of the varables X ad Y be plotted alog the X-as ad Y-as respectvely the y-plae, the dagram of dots so obtaed s kow as scatter dagram. From the scatter dagram, f the pots are very close to each other, we should epect a farly good amout of correlato

57 54 betwee the varables ad f the pots are wdely scattered, a poor correlato s epected. Ths method, however, s ot sutable f the umber of observatos s farly large. If the plotted pots shows a upward tred of a straght le the we say that both the varables are postvely correlated Postve Correlato Whe the plotted pots shows a dowward tred of a straght le the we say that both the varables are egatvely correlated Negatve Correlato If the plotted pots spread o whole of the graph sheet, the we say that both the varables are ot correlated. No Correlato