Примеры решения задач (1094491), страница 10
Текст из файла (страница 10)
Êðèòåðèè ñîãëàñèÿ71Âàðèàöèîííûå ðÿäû íàáëþäåíèéÒàáëèöà 96.1 (I)Òàáëèöà 96.2 (II)Òàáëèöà 96.3 (III)ixiy = F ý(x)xiy = F ý(x)xiy = F ý(x)10,480,072,981/151,680,0720,550,133,032/151,830,1330,760,203,173/152,290,2040,830,273,224/152,460,2751,340,333,575/154,020,3361,390,403,596/154,190,4071,390,473,957/156,540,4781,940,533,968/156,640,5392,050,604,039/157,170,60102,240,674,1610/158,300,67112,520,734,3511/1510,080,73122,710,804,4712/1511,460,80132,810,874,5413/1512,210,87143,440,934,9614/1517,780,93154,521,005,01118,601,0096.2. Íà ðèñ. 12 äëÿ ïðèâåä¼ííûõ â òàáë.
96.3 äàííûõèçîáðàæåíû ãðàôèêè ýìïèðè÷åñêîé ôóíêöèè ðàñïðåäåëåíèÿ è àïïðîêñèìèðóþùåé ïðÿìîé â êîîðäèíàòàõ (u; v) (ôîðìóëû (96.5)). Èç ãðàôèêà âèäíî: k = 0,12 ⇐⇒ λ = 0,12. Äëÿ óäîáñòâà ïîñòðîåíèÿ ãðàôèêîâ íà âåðòèêàëüíîé îñè Ov íàíåñåíû ñîîòâåòñòâóþùèå v çíà÷åíèÿy ïî ôîðìóëå: v = − ln(1 − y).Ïðèìåð96.2. Êðèòåðèé Ïèðñîíà ïðîâåðêè ãèïîòåçû î âèäåçàêîíà ðàñïðåäåëåíèÿÏóñòü èìååòñÿ ñëó÷àéíàÿ âûáîðêà, ñîñòîÿùàÿ èç n ýëåìåíòîâ. Òðåáóåòñÿ íàéòè çàêîí ðàñïðåäåëåíèÿ èçó÷àåìîé ñëó÷àéíîé âåëè÷èíû ξ(èëè, êàê óñëîâèëèñü ãîâîðèòü, ãåíåðàëüíîé ñîâîêóïíîñòè), îïðåäåëèòü åãî ïàðàìåòðû è îöåíèòü ñîãëàñèå âûáîðêè ñ ïðèíÿòûì çàêîíîìðàñïðåäåëåíèÿ.Íà îñíîâàíèè ñòàòèñòè÷åñêîãî ìàòåðèàëà ïðîâåðÿåòñÿ ãèïîòåçà H0 ,ñîñòîÿùàÿ â òîì, ÷òî ñëó÷àéíàÿ âåëè÷èíà ξ ïîä÷èíÿåòñÿ íåêîòîðîìó72Ëåêöèÿ 96.
Êðèòåðèè ñîãëàñèÿyvk=1,0b=-1,8I0,992,0II0,950,90k=1,4b=-5,7III1,00,800,700,600,5000,4012345678910 11 12 13u=x0,300,20-1,00,100,05-2,00,01−1v=Φ (y-0,5)Ðèñ. 11. Âåðîÿòíîñòíàÿ áóìàãà äëÿ íîðìàëüíîãî ðàñïðåäåëåíèÿçàêîíó ðàñïðåäåëåíèÿ. Äëÿ òîãî ÷òîáû ïðèíÿòü èëè îòâåðãíóòü ãèïîòåçó H0 , ðàññìàòðèâàåòñÿ âåëè÷èíà U ñòåïåíü ðàñõîæäåíèÿ òåîðåòè÷åñêîãî è ñòàòèñòè÷åñêîãî ðàñïðåäåëåíèÿ. Çà U ïðèíèìàþò ñóììó êâàäðàòîâ (ñ íåêîòîðûìè êîýôôèöèåíòàìè) îòêëîíåíèé òåîðåòè÷åñêèõ âåðîÿòíîñòåé Pi îò ñîîòâåòñòâóþùèõ ÷àñòîò Pi∗ (êðèòåðèé χ2 ).Ñõåìà ðàñ÷¼òîâ ñ ïîìîùüþ êðèòåðèÿ Ïèðñîíà (êðèòåðèÿ χ2 ) ñëåäóþùàÿ.(1) Íà îñíîâàíèè âûáîðêè âûáèðàåì â êà÷åñòâå ïðåäïîëàãàåìîãîêàêîéòî çàêîí ðàñïðåäåëåíèÿ èçó÷àåìîé âåëè÷èíû (íàïðèìåð, ñ ïîìîùüþ âåðîÿòíîñòíîé áóìàãè) è îöåíèâàåì åãî ïàðàìåòðû, êàê îïèñàíî â ëåêöèÿõ 96 è 97.(2) Âñ¼ ìíîæåñòâî íàáëþäåíèé ðàçáèâàåì íà s èíòåðâàëîâ âèäà(aj−1 ; aj ] è ïîäñ÷èòûâàåì ýìïèðè÷åñêèå ÷àñòîòû êîëè÷åñòâîËåêöèÿ 96.
Êðèòåðèè ñîãëàñèÿy73vv = -ln (1-y)III2,00,9k=0,120,71,00,60,50,40,30,20,112345678910 11 12 13 14 15 16 17u=xÐèñ. 12. Âåðîÿòíîñòíàÿ áóìàãà äëÿ ýêñïîíåíöèàëüíîãî ðàñïðåäåëåíèÿíàáëþäåíèé mj , ïîïàâøèõ â j -ûé èíòåðâàë (ñì. ï. 96.3). Îòíîñèòåëüíàÿ ÷àñòîòà íàáëþäåíèé, ïîïàâøèõ â j -ûé èíòåðâàë,mjðàâíà Pj∗ =, (m1 + . . . + ms = n), ñóììà âñåõ ÷àñòîò,nî÷åâèäíî, ðàâíà åäèíèöå.(3) Îïðåäåëÿåì òåîðåòè÷åñêèå ÷àñòîòû m′j äëÿ j -ãî èíòåðâàëà(aj−1 ; aj ]:()′mj = F (aj ) − F (aj−1 ) · n,ãäå F (x) òåîðåòè÷åñêàÿ ôóíêöèÿ ðàñïðåäåëåíèÿ, íàéäåííàÿíà ýòàïå 1.(4) Âû÷èñëÿåì êðèòåðèé χ2íàáë (êðèòåðèé Ïèðñîíà):2χíàáëS∑(mj − m′j )2=.m′jj=1(96.6)Èç ýòîãî âûðàæåíèÿ âèäíî, ÷òî χ2íàáë ðàâíî íóëþ ëèøüïðè ñîâïàäåíèè âñåõ ñîîòâåòñòâóþùèõ ýìïèðè÷åñêèõ è òåîðåòè÷åñêèõ ÷àñòîò: mi = m′i (i = 1, 2, . .
. , l).  ïðîòèâíîì74Ëåêöèÿ 96. Êðèòåðèè ñîãëàñèÿñëó÷àå χ2íàáë îòëè÷íî îò íóëÿ è òåì áîëüøå, ÷åì áîëüøå ðàñõîæäåíèå ìåæäó ÷àñòîòàìè. Âåëè÷èíà χ2 , îïðåäåëÿåìàÿ ðàâåíñòâîì ( 96.6), ÿâëÿåòñÿ ñëó÷àéíîé, è (ïðè áîëüøèõ n) èìååòχ2 ðàñïðåäåëåíèå ñ k ñòåïåíÿìè ñâîáîäû (ïðèíèìàåòñÿ áåçäîêàçàòåëüñòâà).(5) Îïðåäåëÿåì ÷èñëî ñòåïåíåé ñâîáîäû k ñëó÷àéíîé âåëè÷èíû χ2 :k = s − 1 − r,(96.7)ãäå r ÷èñëî ïàðàìåòðîâ çàêîíà ðàñïðåäåëåíèÿ (äëÿ íîðìàëüíîãî çàêîíà ðàñïðåäåëåíèÿ r = 2), s ÷èñëî èíòåðâàëîâ.(6) Ïî çàäàííîìó óðîâíþ çíà÷èìîñòè α è ÷èñëó ñòåïåíåé ñâîáîäû k ïî òàáëèöå êðèòè÷åñêèõ òî÷åê ðàñïðåäåëåíèÿ χ2 (òàáëèöà ïðèëîæåíèÿ 4) íàõîäèì êðèòè÷åñêóþ òî÷êó χ2êð (α; k).Åñëè χ2íàáë < χ2êð (α; k) íåò îñíîâàíèé îòâåðãíóòü ãèïîòåçó î ïðèíÿòîì (íîðìàëüíîì) çàêîíå ðàñïðåäåëåíèÿ.
Åñëèχ2íàáë > χ2êð (α; k) ãèïîòåçó îòâåðãàþò ñ óðîâíåì çíà÷èìîñòè α.96.3. Ñ ïîìîùüþ êðèòåðèÿ Ïèðñîíà ïðîâåðèòü ãèïîòåçó î íîðìàëüíîì ðàñïðåäåëåíèè âûáîðêè, ïðåäñòàâëåííîé â òàáëèöå96.2 ïðèìåðà 96.1.ÏðèìåðÐ å ø å í è å: Ðàçîáü¼ì âñ¼ ìíîæåñòâî çíà÷åíèé âûáîðêè òàáë. 96.2íà 6 èíòåðâàëîâ, ãðàíèöû êîòîðûõ çàíåñåíû âî âòîðîé ñòîëáåö òàáë. 96.4.Òàáëèöà 96.4j0123456Ðåøåíèå ïðèìåðà 96.2ajmjF (aj )m′j2,510,01550,9693,030,08002,6593,540,25734,2464,040,54043,9484,520,80362,1375,010,94600,6735,50,9909 òðåòèé ñòîëáåö òàáë.
96.4 çàíîñèì êîëè÷åñòâî íàáëþäåíèé mj ,ïîïàâøèõ â j -ûé èíòåðâàë. Ïî ôîðìóëàì (93.2), (93.10), (93.5) îïðåäåëÿåì ïàðàìåòðû íîðìàëüíîãî ðàñïðåäåëåíèÿ x̄ è S ∗ äëÿ âûáîðêè èçòàáë. 96.2:x̄ = 3, 933; S ∗ = 0,664Ëåêöèÿ 96. Êðèòåðèè ñîãëàñèÿ75è íàõîäèì çíà÷åíèÿ òåîðåòè÷åñêîé ôóíêöèè ðàñïðåäåëåíèÿ F (aj ). Â( a − x̄ )jäàííîì ïðèìåðå F (aj ) = Φ+ 0,5.  ïÿòûé ñòîëáåö çàíîñèìS∗′òåîðåòè÷åñêèå ÷àñòîòû mj , âû÷èñëÿåìûå, êàê óêàçàíî âûøå.Ïî ôîðìóëå (96.6) íàõîäèì çíà÷åíèå χ2íàáë = 0,228. Ïî òàáëèöåïðèëîæåíèÿ 4 äëÿ α = 0,05 è k = 6 − 1 − 2 = 3 íàõîäèì êðèòè÷åñêóþòî÷êó χ2êð (0,05; 3) = 7,8. Ïîñêîëüêó χ2íàáë < χ2êð (0,05; 3), íåò îñíîâàíèé îòâåðãàòü ãèïîòåçó H0 î íîðìàëüíîì ðàñïðåäåëåíèè âûáîðêè èçòàáëèöû 96.2.
Çàìåòèì, ÷òî ýòîò ðåçóëüòàò õîðîøî ñîãëàñóåòñÿ ñ äàííûìè âåðîÿòíîñòíîé áóìàãè (ãðàôèê I íà ðèñ. 11).Ïîêàæåì èñïîëüçîâàíèå êðèòåðèÿ Ïèðñîíà ñ ïðèìåíåíèåì ïðîãðàìì â ðàìêàõ ïàêåòîâ Mathcad è Maxima.96.4. Äëÿ èññëåäîâàíèÿ âèäà íåêîòîðîé çàâèñèìîñòè ïðîèçâåäåíî 100 èñïûòàíèé. Ðåçóëüòàòû ïîëó÷åííûõ èñïûòàíèé ðàçáèëè íà 9 äèàïàçîíîâ, ãðàíèöû êîòîðûõ çàïèñàíû â ìàññèâ a:ÏðèìåðMathCad-ïðîãðàììà:a := ( 69.2 69.8 70.4 71.0 71.6 72.2 72.8 73.4 74.0 74.6 )T ìàññèâå m ïðåäñòàâëåíî êîëè÷åñòâî íàáëþäåíèé, ïîïàâøèõ âñîîòâåòñòâóþùèé äèàïàçîí.m := ( 1 4 11 21 27 22 10 3 1 )TÑ ïîìîùüþ êðèòåðèÿ Ïèðñîíà ïðîâåðèòü ãèïîòåçó H0 î íîðìàëüíîì ðàñïðåäåëåíèè âûáîðêè ñ óðîâíåì çíà÷èìîñòè α = 0,05.Ð å ø å í è å:∑ORIGIN := 1 s := 9 j := 1 .
. . s n :=mj n = 100j/* Íàéäåì êîîðäèíàòû ñåðåäèí èíòåðâàëîâ (ìàññèâ U ):*/aj + aj+12/* Íàéäåì âûáîðî÷íóþ ñðåäíþþ M x, èñïðàâëåííóþ âûáîðî÷íóþäèñïåðñèþ S2: */[]1 ∑n1 ∑M x := ·mj ·Uj M x = 71.876 S2 =··mj · (Uj )2 − M x2n jn−1 n jUj :=/* è èñïðàâëåííîå ñðåäíååêâàäðàòè÷åñêîå îòêëîíåíèå S : */√S2 = 0.8067S := S2S = 0.8982./* Ôóíêöèþ Ëàïëàñà*/76Ëåêöèÿ 96. Êðèòåðèè ñîãëàñèÿ1Ô(x) := √ ·2π∫xz2e− 2 dz0/* Ïîëó÷àåì, èñïîëüçóÿ âñòðîåííóþ ôóíêöèþ íîðìàëüíîãî ðàñïðåäåëåíèÿ pnorm(x, M x, σ), ïðè çíà÷åíèè ìàòåìàòè÷åñêîãî îæèäàíèÿ M x = 0 è ñðåäíåêâàäðàòè÷åñêîì îòêëîíåíèè σ = 1 */Ô := pnorm(x, 0, 1) − 0.5/* Ïîëó÷àåì òåîðåòè÷åñêèå ÷àñòîòû m1j äëÿ êàæäîãî èíòåðâàëà.
Ïðè ýòîì â ïåðâûé èíòåðâàë âêëþ÷àåì ïðîìåæóòîê îò −∞ äîX2 , à â ïîñëåäíèé èíòåðâàë îò Xs äî +∞ */()()aj+1 − M xaj − M xPj := Ô−ÔSS()()a2 − M x−∞ − M xP1 := Ô−ÔSS()()∑∞ − Mxas − M xPs := Ô−Ôm1j := n · Pjm1j = 100SSjP T = (0.0104 0.0398 0.1145 0.2146 0.2615 0.2074 0.10693 0.0359 0.0090)m1T = (1.041 3.9748 11.454 21.461 26.154 20.7354 10.6927 3.5848 0.9019)/* Îïðåäåëÿåì ÷èñëî ñòåïåíåé ñâîáîäû k äëÿ äàííîé çàâèñèìîñòè*/k := s − 1 − 2α := 0.05/* Íàéäåì òåïåðü çíà÷åíèå χ2 íàáëþäàåìîå (χ2íàáë ).*/χ2íàáë :=∑ (mj − m1j )2m1jjχ2íàáë = 0.2851/* Èñïîëüçóÿ âñòðîåííóþ ôóíêöèþ qchisq äëÿ χ2 ðàñïðåäåëåíèÿ,ïîëó÷àåì êðèòè÷åñêîå çíà÷åíèå χ2êð : */χ2êð := qchisq(1 − α, k)χ2êð = 12.5916Òàê êàê χíàáë < χêð , òî íåò îñíîâàíèÿ îòâåðãàòü ïîñòàâëåííóþãèïîòåçó H0 ñ óðîâíåì çíà÷èìîñòè α = 0,05.Íà ïðàêòèêå ÷àùå âñåãî âûáîðêà áîëüøîãî îáú¼ìà çàïèñûâàåòñÿâ òåêñòîâûé ôàéë. Çàòåì äàííûå ñ÷èòûâàþòñÿ ïðîãðàììîé îáðàáîòêè, è ïîëó÷åííàÿ âûáîðêà àíàëèçèðóåòñÿ.
Ïîýòîìó ìû òàêæå ðàçîáüåì çàäà÷ó íà äâå ïîäçàäà÷è.  ïåðâîé ñãåíåðèðóåì âûáîðêó, à âîâòîðîé å¼ îáðàáîòàåì. Äëÿ ãåíåðàöèè âûáîðêè èñïîëüçóåì êîìàíäó22Ëåêöèÿ 96. Êðèòåðèè ñîãëàñèÿ77random_normal(M, σ, n), âîçâðàùàþùóþ ñïèñîê èç n ïñåâäîñëó÷àéíûõ ÷èñåë, áëèçêèõ ê íîðìàëüíîìó çàêîíó ðàñïðåäåëåíèÿ ñ ìàòåìàòè÷åñêèì îæèäàíèåì M è ñðåäíåêâàäðàòè÷åñêèì îòêëîíåíèåì σ . Ïîëó÷åííûé ñïèñîê x1 çàïèøåì â ôàéë pirson.txt, íàõîäÿùèéñÿ íà äèñêåD â ïàïêå mymaxima./* Ïåðâàÿ ïðîãðàììà, ãåíåðèðóþùàÿ âûáîðêó îáú¼ìà n=100 è çàïèñûâàþùàÿ å¼ â òåêñòîâûé ôàéë "D:/mymaxima/pirson.txt").*/(%i1)(%i3)(%i4)(%i5)n:100$ fpprintprec:3$load(distrib)$x1:random_normal(120, 25, n)$write_data(x1, "D:/mymaxima/pirson.txt")$96.5. Äëÿ èññëåäîâàíèÿ âèäà íåêîòîðîé çàâèñèìîñòè ïðîèçâåäåíî 100 èñïûòàíèé, ðåçóëüòàòû êîòîðûõ çàïèñàíû â òåêñòîâûé ôàéë "D:/mymaxima/pirson.txt").
Ñ ïîìîùüþ êðèòåðèÿ Ïèðñîíàïðîâåðèòü ãèïîòåçó î íîðìàëüíîì ðàñïðåäåëåíèè ïîëó÷åííîé âûáîðêè.ÏðèìåðMaxima-ïðîãðàììà:(%i1) fpprintprec:3$ n:100$ numer:true$(%i4) load(distrib)$/* Ñ÷èòûâàåì âûáîðêó â ñïèñîê x1.*/(%i5) x1:read_list("D:/mymaxima/pirson.txt");/*Ñîðòèðóåì ñïèñîê x1 â ïîðÿäêå âîçðàñòàíèÿ çíà÷åíèé.*/(%i6) x2:sort(x1);/* Ðàçáèâàåì âûáîðêó íà s èíòåðâàëîâ ïîñòîÿííîé äëèíû delta.*/(%i7) s:9; delta:(x2[n] -x2[1])/s;/*Êîîðäèíàòû ãðàíèö ýëåìåíòîâ.*/(%i9) a:makelist(x2[1]+(i-1)*delta, i, 1, s+1);(%o9) [66.9, 79.6, 92.3, 105., 118., 130., 143., 156.,168., 181.]/* Êîîðäèíàòû ñåðåäèí ýëåìåíòîâ.*/(%i10) U:makelist((a[j]+a[j+1])/2, j, 1, s);(%o10) [73.2, 85.9, 98.6, 111., 124., 137., 149., 162., 175.]/*Îïðåäåëåíèå ÷àñòîòû íàáëþäåíèé ïî èíòåðâàëàì.*/(%i11) m:makelist(0, i, 1, s); for j:1 while j<=n do(k:fix((x2[j] -x2[1])/delta)+1,if k>s then k:s,m[k]:m[k]+1);78Ëåêöèÿ 96.
Êðèòåðèè ñîãëàñèÿ/* Âûâîä çíà÷åíèé ýìïèðè÷åñêîé ÷àñòîòû íàáëþäåíèé ïî èíòåðâàëàì.*/(i13) m;(%o13) [3, 9, 23, 11, 19, 16, 11, 5, 3]/* Êîíòðîëü îáú¼ìà âûáîðêè.*/(%i14) sum(m[i], i, 1, s);(%o14) 100/* Ñòðîèì ãðàôèê.*/(%i15) wxplot2d([['discrete,makelist([U[j], m[j]], j, 1, s)]],[style,[lines, 3, 5]], [gnuplot_preamble,"set grid"],[ylabel,""])$(%i16) Mx:sum(m[j]*U[j], j, 1, s)/n;(%o16) 117.(%i17) S2:n/(n-1)*(sum(m[j]*U[j]^2, j, 1, s)/n-Mx^2);(%o17) 648.(%i18) S:sqrt(S2);(%o18) 25.5(%i19) F(x):=cdf_normal (x, 0, 1) -0.5;(%o19) F(x):=cdf_normal(x, 0, 1) -0.5/*Òåîðåòè÷åñêèå âåðîÿòíîñòè.*/(%i20) P:makelist(F((a[j+1] -Mx)/S) -F((a[j] -Mx)/S), j, 1, s);P[1]:F((a[2] -Mx)/S)+0.5;P[s]:0.5-F((a[s] -Mx)/S);(%o20) [0.006, 0.03, 0.09, 0.2, 0.2, 0.2, 0.1, 0.06, 0.02](%o21) 0.007(%o22) 0.02/*Òåîðåòè÷åñêèå ÷àñòîòû.*/(%i23) m1:makelist(n*P[j], j, 1, s);(%o23) [0.7, 2.81, 8.55, 17.6, 24.4, 22.9, 14.6, 6.25, 2.22]/*Êîíòðîëü îáú¼ìà âûáîðêè äëÿ òåîðåòè÷åñêèõ ÷àñòîò.*/(%i24) sum(m1[j], j, 1, s);(%o24) 100/* Âû÷èñëåíèå χ2íàáë .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
