Testing Statistical Hypotheses for Compare. Means with Vague Data
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1 Iteratioal Mathematical Forum 5 o Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach Ghaemshahr Ira e_baloui8@yahoo.com. Abstract I this paper we cosider two-sample hypotheses tests for meas i ormal populatios with fuzzy data. I this fuzzy test we will make a fuzzy decisio for rejectio or acceptace ull hypothesis. A fuzzy decisio shows a degree of acceptability ad degree of rejectio of the ull hypothesis. This test is well-defied sice if the data are precise we get a classical statistical test. Fially we also provide a example. Mathematics Subject Classificatio: 6F3 Keywords: Hypothesis testig Fuzzy data Fuzzy test Itroductio Test hypotheses are oe of the most importat compoets of statistical iferece. I traditioal testig the observatios of sample are crisp ad a statistical test leads to the biary decisio. However i the real word ofte observatios caot be measured as exact umerical value. The fuzzy set theory is a well- kow tool i modelig the imprecise data. We are goig to apply fuzzy sets theory to the statistical test by cosiderig fuzzy umber for vague observatio. Casals et al. [3] So et al. [] ad Grzegorzewski [5] cosidered statistical hypothesis with vague data. Delgado et al. [4] Saade ad Schwarzlader [9] Saade [8] Wataabe ad Imaizumi[] ad Moteegro et al. [7] cosidered fuzzy hypothesis testig with vague data. Kruse ad Meyer [6] discussed the problem of cofidece iterval uder fuzzy radom variables. Arold [] coducted statistical tests uder fuzzy costraits o the type I ad type II errors. We provide some defiitio i the ext sectio. Fuzzy tests itroduce i sectio 3. I sectio 4 we deal some of umerical example.
2 66 E. Baloui Jamkhaeh ad A. adi Ghara Prelimiaries Suppose a radom experimet is described as usual by a probability space ( Ω Α Ρ) where Ω is a set of all possible outcomes of the experimet Α is a algebra of subset of Ω ad Ρ is a probability measure. Defiitio: [] The fuzzy subset of real lie IR with the membership fuctio : IR [] is a fuzzy umber if ad oly if (a) is ormal (b) is fuzzy covex (c) is upper semi cotiuous (d) supp ( ) is bouded. Defiitoi: [] The -cut of a fuzzy umber is a o-fuzzy set defied as [ ] = { x IR; ( x) }. Hece we have [ ] = [ [ ] [ ]] where [ ] = if { x IR; ( x) } [ ] = sup { x IR; ( x) }. The set of all these fuzzy umbers is deoted by F(IR). Defiitio3: [5] A mappig : Ω F( IR) is called a fuzzy radom variable if it satisfies the followig properties: (a) { ; [] } is a set represetatio of (ω) for all ω Ω (b) for each [] both = = if = = sup are usual real valued radom variables o ( Ω Α Ρ). Thus a fuzzy radom variable is cosidered as a perceptio of a ukow usual radom variable V : Ω IR called a origial of. et χ deote a set of all possible origials of. We ca cosider a fuzzy set o χ with a membership fuctio V : χ [] give as follows: ν ( V ) = if { ( ω ) ( V ) ω Ω} () I fact this iterpretig is used whe the origi of radom variable is defied as classic but the observatio of radom variable is imprecise. Similarly -dimesioal fuzzy radom sample... may be treated as a fuzzy perceptio of the usual radom sample V V... V (wherev V... V are idepedet ad idetically distributed crisp radom variables). A set χ of all possible origials of that radom sample is i fact a fuzzy set with membership fuctio: ν V... V ) = mi = if ω ( V ) ω Ω () i ( i.. ( ) i { } Fuzzy tests et... ad Y Y... Ym deote two fuzzy samples i.e. two fuzzy perceptio of the usual radom samples V V... V ad... m from the
3 Testig statistical hypotheses 67 populatios with the distributio ormal with parameters ( ) ad ( ) respectively where i i = are kow. ϕ is called a fuzzy test for the hypothesis H o the sigificace level () if sup [ ] P { ω Ω ϕ ( Y ) { } H } (3) where ϕ is the cut set of ϕ ( Y ). m Defiitio4: [5] A fuctio : ( F( IR)) F{ } Theorem: et defie = [ ] for all [] as = Y Z ad m m The fuctio : ( F( IR)) F{ } = Y Z m ϕ with followig cuts {} d ( ( )) {} d (( ) ) ϕ ( Y i =... j =... m) = (5) i j {} d ( ( )) Φ d ( ( )) Is a fuzzy test for H : = d agaist H : d o the sigificace level. Proof: Kruse ad Meyer [6] itroduced the otio fuzzy cofidece iterval for the parameterθ. emma. et [ ] deote two-sided symmetrical cofidece iterval for d o the level. et i i =.. ad Y i i =.. m deote fuzzy radom samples which are fuzzy perceptio of the crisp radom samplesv i i =.. ad i i =... m. The a covex ad ormal fuzzy set with the membership fuctio x y ) sup I ( x y ) [] (6) { } ( = [ ] where ad are defied by (4) is called a fuzzy cofidece iterval for d o the cofidece level i.e. P{ ω Ω Λ ( d) } [] (7) where = [ ] = ( ( ω)... Y... Ym ) = ( ( ω)... Y... Ym ) (4)
4 68 E. Baloui Jamkhaeh ad A. adi Ghara ad Λ ( d ) is a fuzzy perceptio of the parameter do give by (8 ). Proof: For the proof we refer the reader to [6]. et us take ay []. The we have P{ Ω ϕ ( Y ) {} H }= P ω Ω ϕ ( ( ) Y ω ) { Φ{}} H = P{ ω Ω ϕ ( ( ) Y ω ) {{}{}} H } = P Ω ϕ ( Y ) {{}{}} H ω { } { ω } { ω Ω d ( ω } = P ) H (8) If the ull hypothesis H : = d is fulfilled the d Λ ( d) for all []. By usig from (7) we get P { ω Ω d }. Thus from (8) we get sup [ ] P { ω Ω ϕ ( Y ) { } H } (9) which completes the proof. By theorem give above we get a membership fuctio of the fuzzy test cosidered above for the hypothesis H : = d agaist H : d is ϕ = ( d) I{} ( d) I{} = ( d) I{} ( ( d)) I{ }( t) t {} () It is easily see that our fuzzy set lead to the fuzzy decisio. We get ϕ = where [] (i) if = the we should accept H (ii) if = the we should reject H (iii) otherwise we should accept (with degree ) or reject (with degree ) the hypothesis H. Fig shows this idea. degree of rejectio degree of acceptability Fig.. x ϕ ( x y ) = Similarly we ca obtai fuzzy tests for oe-sided hypothesis of H : d agaist H : > d ad H : d agaist H : < d
5 Testig statistical hypotheses 69 sufficiet that i the Theorem ad emma will substitute with ad respectively that ad where = [ ] = Y Z m () = [ ] = Y Z () m ad are the oe-sided fuzzy cofidece iterval for d. 3 umerical examples We have two kids of tire for automobile (A ad B) ad we set up each oe o the some taxi ad the we request from taxi drivers to record cosumptio of the petrol. The data are recorded as triagular fuzzy umber that are as followig. Suppose the radom variables have distributio ormal ad the variace of both populatios are kow ad equal with oe. We ivestigate the effect of tire s kid o cosumptio of the petrol i sigificat level.5. A (456) ( ) (55.56) ( ) (345) B (56.58) (456) ( ) (567) (678) (67.59) H = H : : A B A B A : Mea of cosumptio i A B : Mea of cosumptio i B After simple calculatio we obtai a fuzzy sample average ad Y with membership fuctio x < x < 5..9 = 6 x 5. < x < 6.9 Y y = 7.75 y < y < < y < 7.75 = [ ] = Y Z m = Y Z m (3) with substitutig Z = Z =.64 = 5 m = 6 ad = i to (3) we.95 =
6 6 E. Baloui Jamkhaeh ad A. adi Ghara coclude that = [ ] = [ ] (4) ad substitutig d = ito () ad usig (5) we get ϕ ( x y ) = that is we should accept (with degree.388) or reject (with degree.6) the hypothesis H also may be iterpreted as approximately H should be rejected. Refereces [] B.F. Arold Statistical tests optimally meetig certai fuzzy requiremets o the power fuctio ad o the sample size Fuzzy Set ad System 75 (995) [] E. Baloui Jamkhaeh B. Sadeghpour Gildeh GH. Yari Importat criteria of rectifyig ispectio for sigle samplig pla with fuzzy parameter Iteratioal Joural Cotempt Mathematics Scieces 36 (9) [3] R. Casals M.A. Gill A ote o the operativeess of eyma-pearso tests with fuzzy iformatio Fuzzy Set ad System 3 (989) [4] M. Delgado J.. Verdegay ad M.A. Vila Testig fuzzy hypotheses: A Bayesia approach i: M.M. Gupta et al.(eds.) Approximate Reasoig i Expert Systems Elsevier Amsterdam (985) [5] P.Grzegorzewski Testig statistical hypotheses with vague data Fuzzy Set ad System () 5-5. [6] R. Kruse K.D. Meyer Cofidece itervals for the parameter of a liguistic radom variable i: J. Kasprzyk M. Fedriz(Eds.) Combiig Fuzzy Imprecisio with Probabilistic certaity i Decisio Makig Spriger Berli (988) 3-3. [7] M. Moteegro M.R. Casals M.A. ubiao ad M.A. Gil Two-sample hypothesis tests of meas of a fuzzy radom variable Iformatio Scieces 33 () [8] J. Saade Extesio of fuzzy hypothesis testig with hybrid data Fuzzy Set ad System 63 (994) [9] J. Saade H. Schwarzlader Fuzzy hypothesis testig with hybrid data Fuzzy Set ad System 35 (99) [] J. CH. So I. Sog H.Y. Kim A fuzzy decisio problem based o the geeralized eyma-pearso criteria Fuzzy Set ad System 47 (99) []. Wataabe T. Imaizumi A fuzzy statistical test of fuzzy hypotheses Fuzzy Set ad System 53 (993) Received: August 9
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