Module 7: Probability and Statistics

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1 Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to check the valdty of a hypothess about a populato parameter from a observed sample. Smlarly, t may be ecessary to check whether a observed dataset belogs to a partcular probablty dstrbuto. Frst part of ths lecture deals wth emprcal determato of probablty dstrbuto of a RV. The use of a probablty paper ths cotext alog wth ts method of costructo s dscussed here. Descrptos of Normal ad Logormal probablty paper ad probablty plot are preseted. Also, there are certa statstcal tests, kow as goodess-of-ft tests, to check the probablty dstrbuto that a dataset (sample) possbly follows. I real lfe scearos, some commo examples of goodess-of-ft tests are whether a sample of a dscrete varable follows a Posso dstrbuto, whether a sample of a cotuous varable follows a Normal dstrbuto or whether two samples are draw from detcal dstrbutos. The most commoly used tests are - Ch-square ( ) Test, Kolmogorov-Smrov K S Test ad Aderso-Darlg Test. These are dscussed at the later part of ths lecture.. Emprcal Determato of probablty dstrbuto of a RV I may real lfe scearos, the actual probablty dstrbuto of a radom process s ukow. O the bass of frequecy dstrbuto determed from observed data, some probablty dstrbuto may be assumed emprcally. Probablty papers are useful to check the assumpto of a partcular probablty dstrbuto of a radom varable. 3. Probablty Paper A probablty paper s a specally costructed plottg paper where the oe of the axes (where the RV s plotted) s a arthmetc axs ad the probablty axs s dstorted such a way that the cumulatve probablty dstrbuto of the RV plots as a straght le. It may be oted that separate probablty papers are eeded to plot the CDF of dfferet probablty dstrbutos as a straght le. For a partcular probablty dstrbuto wth dfferet parameters, oe probablty paper may be suffcet as the case of expoetal dstrbuto. For some probablty dstrbutos such as Gamma dstrbuto, separate probablty papers are eeded for each set of parameters. 4. Utltes of a probablty paper Whe a RV s expected to ft a certa probablty dstrbuto, ts observed CDF s plotted o the correspodg probablty paper. The resultg plot has the followg utltes: If the plot s a straght le, t ca be drectly cocluded that the RV follows the hypothetcal pdf.

2 If the plot devates from a straght le, the the locato of devato dcates the rego (eg. the tal, the mode etc) where the ft s ot good. If the plot s ot at all close to a straght le, t mples that the hypothetcal dstrbuto has to be rejected ad some other dstrbuto has to be tested for the ft. If the plot follows a straght le for a part of the rage ad the devates, t s the dcato of a chage dstrbuto beyod a certa rage of the RV. The slope ad tercept of the straght le plotted o the probablty paper ca be used to estmate the parameters of the dstrbuto. 5. Normal Probablty Paper The ormal probablty paper s costructed o the bass of stadard ormal probablty dstrbuto fucto. The radom varable X s represeted o the horzotal (or vertcal) axs arthmetc scale. The vertcal (or horzotal) axs represets two scales the stadard ormal varate Z X ad the cumulatve probablty values F X x ragg from 0 to. m The expermetal data pots are plotted usg Gumbel s plottg posto gve by N where N the total umber of observatos ad m rak of the data pot whe the observed values are arraged ascedg order. As the probablty scale s compressed ear the meda ad expaded towards the tals, hece a ormal varate X wth mea ad stadard devato plots as a straght le o ths paper. The straght le passes through Z X ad F X x 0. 5 ad has a slope of. Hece the parameters of the X dstrbuto ca be easly obtaed from the plot. Problem o Normal Probablty Paper Q. The observed stregths of 30 cocrete cubes are gve below. Check whether the stregth of cocrete cubes follows ormal dstrbuto or ot by plottg o ormal probablty paper. Determe the mea stregth ad the stadard devato. Sl o Stregth KN/m Sl o Stregth KN/m Sl o Stregth KN/m Sl o Stregth KN/m Sl o Stregth KN/m Sl o Stregth KN/m

3 Sol. The observed stregths of the cocrete cubes are frst arraged ascedg order. The ther m plottg postos are determed by, where N 30 ad m rak of the observed data N pot whe arraged ascedg order of ther values. The ormal probablty plot s prepared ad the data s foud to plot almost as a straght le. Thus, the stregth of the cocrete cubes follows ormal dstrbuto Normal Probablty Plot Probablty Stregth (KN/m ) From the plot, the value of stregth correspodg to cumulatve probablty of 0.5 s 3 KN / m. Thus the mea stregth s 3 KN / m. The stadard devato s gve by the verse of the slope of the straght le. It s obtaed roughly as 8 KN / m. 6. Log Normal Probablty Paper The log ormal probablty paper dffers from a ormal probablty paper as follows: The horzotal axs for the radom varable X s logarthmc scale stead of arthmetc lx / x scale. The stadard ormal varate Z s gve by Z m where x m s the meda of X.

4 7. Geeral Probablty Paper ad Probablty plot For ay probablty dstrbuto, the probablty paper may be costructed by detfyg a sutable stadard varate Z. The use of the stadard varate esures that the costructed probablty paper s depedet of the parameters of the dstrbuto. The stadard varate Z s represeted o the vertcal axs arthmetc scale. The cumulatve probablty values are also represeted o the vertcal axs. The radom varable X s represeted o the horzotal axs arthmetc scale. The plottg posto of the expermetal data pots ca be obtaed by a umber of methods (eg. Gumbel, Haze etc). As per the Gumbel method, plottg m posto s gve by where N the total umber of observatos ad m rak of the N data pot whe the observed values are arraged ascedg order. If the plotted data pots gve rse to a straght le o the paper, the the data pots belog to the partcular probablty dstrbuto for whch the paper s costructed. Depedg o the stadard varate selected, the parameters of the dstrbuto ca be obtaed from the slope, tercept etc of the plot. 8. Goodess-of-ft tests As metoed the troducto, certa statstcal tests, kow as goodess-of-ft tests, are used to check the probablty dstrbuto that a dataset (sample) possbly follows. The most commoly used tests are - Ch-square ( ) Test, Kolmogorov-Smrov K S Test ad Aderso-Darlg Test. These are dscussed ths secto oe after aother. 8. Ch-square test The Ch Square Dstrbuto s used for testg the goodess of ft of a set of data to a specfc probablty dstrbuto. For ths, observed ad hypothetcal frequeces that follow the specfc probablty dstrbuto are compared. It ca be used for the both dscrete or cotuous radom varables. Let us cosder a sample cotag observed values of a radom varable. Let O, O, O3,..., O k be the k observed frequeces of the varates ad the correspodg frequeces from a assumed theoretcal dstrbuto be E, E, E3,..., Ek. It s requred to test whether the dffereces betwee the observed ad expected frequeces are sgfcat. Thus the goodess s checked as X k O E E As k approaches to fty, the samplg dstrbuto of k s the degrees of freedom. X teds to a v dstrbuto wth The test for goodess of ft s geerally relable f 5 k ad E 5. It may be oted that most cases the parameters of theoretcal dstrbuto are ot kow. Hece, the parameters

5 should be estmated from the data tself ad the statstc remas vald f the degree of freedom s reduced by oe for every ukow parameter. If k O E E C, v where C, v s value of approxmate v dstrbuto at cumulatve probablty, the the assumed theoretcal dstrbuto s a acceptable model at the sgfcace level. Problem o ch squared dstrbuto Q. Cosder a gve stato a watershed where the severe rastorms are recorded over a perod of 70 years. Out of these 70 years, years were wthout severe rastorms ad 5, 4, 6, 3 years wth,, 3 ad 4 rastorms aually. Test whether the data ca be assumed to follow Posso dstrbuto at 5% sgfcace level. Sol. Average occurrece rate of rastorms rastorms / year Now to check the goodess of ft, we use the ch-square dstrbuto at 5% sgfcace level As the dataset s qute small, the data the class four storms/year s combed to the class of three storms/year Null hypothess H 0 : The radom varable has a Posso dstrbuto wth Alterate hypothess H : The radom varable does ot follow the dstrbuto specfed ull hypothess. Level of sgfcace: Here 4 k ; degree of freedom, k Crtcal rego v,0.05

6 From ch square table, 5. 99,0.05 No of storms/year Observed frequecy O Theoretcal frequecy E O E / E Total Thus, we get k O E ,0.05 E Hece, the Posso dstrbuto s a vald model at 5% sgfcace level Decso: The ull hypothess ca ot be rejected at 5% sgfcace level 8. Kolmogorov-Smrov (KS) Test Ths test s also most commoly used to check the valdty of the assumed model for cotuous radom varables. It relates to the CDF rather tha the pdf of cotuous varables. It compares the observed or data based cumulatve frequecy wth assumed theoretcal cumulatve dstrbuto. Let the cotuous varable be X ad x, x,... x represet the ordered sample of sze, the values arraged creasg order. Now from ths ordered set the emprcal or sample dstrbuto fucto S x s developed. Ths fucto s a step fucto. Thus the cumulatve frequecy step fucto s defed as S x s the step fucto ad x S 0 x x k x) xk x xk ; k,,..., x x ( F s the proposed theoretcal dstrbuto. The dscrepacy betwee the theoretcal model ad the observed data s computed ad the maxmum dfferece D betwee S x ad F x over the etre rage of x s obtaed.

7 D max x F x S x S (x) F(x) F(x) S (x) D x Fg. S (x) ad F(x) Thus for a specfed sgfcace level,the wth the crtcal value D K S test compares the maxmum dfferece D s defed as P D D If the observed value s less tha the crtcal value, the the proposed dstrbuto s vald at the sgfcace level. 8.. Advatage of KS Test As Ch-Square Test, dvso of data to tervals s ot ecessary ths case. The test statstc s dstrbuto free ulke that of Ch-Square Test. K S test works for o-ormal data also. However, the test may fal f the data s too far from ormal. If the sample dstrbuto s large, Smrov has gve the lmtg dstrbuto of D as lm For 50 for ad 0. 0 P D z exp k z k 8z D. 36 ad D. respectvely

8 Problem o KS test The data of fracture toughess of pla cocrete specmes made wth burt brck aggregates s show the table ext slde. The data appears to fall approxmately a straght le o a Normal probablty paper wth N 0.540,0.05. Perform the Kolmogorov-Smrov test at 5% sgfcace level to statstcally justfy the assumpto for the gve data. Fracture toughess (MPa m) of pla cocrete specmes ( creasg order) m K m IC K m IC K IC Sol. Null hypothess H 0 : The radom varable has a Normal Dstrbuto Alterate hypothess H : The radom varable does ot have the specfed dstrbuto. Level of sgfcace: Crtcal rego (from table) D D The cumulatve frequecy of the gve data s plotted the followg fgure wth respect to the equato of K S Test. The theoretcal dstrbuto fucto of ormal model s also show.

9 Cumulatve dstrbuto of fracture toughess S 0.6 N(0.540,0.05) CDF D K Fg. S (x) ad F(x) From the fgure, the maxmum dscrepacy of two fuctos, D max occurrg at K IC MPa. m.e. the maxmum dscrepacy s less tha the crtcal value Therefore the ull hypothess caot be rejected at 5% sgfcace level. Thus, the N 0.540,0.05 s a vald model at 5% sgfcace level. model 8.. Kolmogorov-Smrov two-sample test The same test used the case of oe sample test ca be used to evaluate whether two samples come from the same dstrbuto. Let the maxmum absolute dfferece betwee two emprcal dstrbuto fuctos be D m, Let the two fuctos be represeted as step fuctos samples of szes m ad, respectvely. G m x ad x S based o two Thus the dfferece becomes D m, max x G m x S x

10 F(x) D m, x Fg 3. Kolgomorov-Smrov two-sample goodess-of-ft test If the sample dstrbuto have large values of m ad, Smrov has gve the lmtg dstrbuto as lm P Dm, z, exp m m z k 8 m k z (replacg wth m m ) Problem o two-sample test Q. The table showg modulus of rupture data for two dfferet groups of tmber s show ext slde. Suppler delver tem two lots. The frst lot cossts of 50 samples ad the secod lot cossts of 30 samples. Both the lots were suppled by a same suppler who clams the secod lot to be superor to the frst lot. Apply the Kolmogorov-Smrov two-sample test to verfy whether the two samples are of same type (from the same populato).

11 LOT A (Modulus of Rupture N / mm ) LOT B (Modulus of Rupture N / mm ) Sol. Null hypothess H 0 : The radom varables sampled by the frst 50 values ad the radom varables sampled by the ext 30 values have the same dstrbuto. Alterate hypothess H : The radom varables have dfferet dstrbutos. Level of sgfcace:

12 Calculatos: The data from each sample are raked separately wth values of the step fuctos x S x. G m ad The samples are sorted creasg order ad raked accordgly for both samples ad G m x ad S x are determed as show the followg table. The the step fucto of both samples are plotted. The maxmum absolute dfferece betwee the emprcal dstrbuto s the determed. Rak x at lot A MR at A Rak x at lot A MR at A Rak x at B MR at B Rak x at B MR at B m 5030 The maxmum dfferece s D 0. ad m 50 30

13 S(x) G(x) F(x) x The crtcal value s 0.30 (from table). Fg 4. S(x) ad G(x) As 0. s less tha 0.30, the ull hypothess caot be rejected at 5% sgfcace level. 0. Cocludg Remarks I ths lecture, commoly used goodess of ft tests such as Ch-square ( ) test, Kolmogorov-Smrov K S test ad Aderso-Darlg test are dscussed. Example problems usg these tests are also preseted here. The ext lecture troduces regresso aalyses ad correlato.

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