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*/(%i25) x2nabl:sum((m[j] -m1[j])^2/m1[j], j, 1, s);(%o25) 5.69Ïðàêòè÷åñêîå çàíÿòèå 96. Îáðàáîòêà ñîâîêóïíîñòè79Òàê êàê χ2íàáë =5,69 íàáëþäàåìîå ìåíüøå êðèòè÷åñêîãî ðàâíîãî,χêð =12,6, òî, íåò îñíîâàíèÿ îòâåðãàòü âûäâèíóòóþ ãèïîòåçó î íîðìàëüíîì ðàñïðåäåëåíèè èññëåäóåìîé âûáîðêè îáú¼ìà n = 100.Îòâåò: Íà áàçå ïîëó÷åííîé âûáîðêè äåëàåì âûâîä, ÷òî èññëåäóåìàÿ íåïðåðûâíàÿ ñëó÷àéíàÿ âåëè÷èíà ïîä÷èíÿåòñÿ íîðìàëüíîìó çàêîíó ðàñïðåäåëåíèÿ.2Ñàìîñòîÿòåëüíàÿ ðàáîòà1.
Äëÿ óêàçàííîãî ïðåïîäàâàòåëåì íîìåðà âàðèàíòà nv ñãåíåðèðîâàòü âûáîðêó îáú¼ìà n = 100 ïñåâäîñëó÷àéíûõ çíà÷åíèé íåïðåðûâíîé ñëó÷àéíîé âåëè÷èíû xi ñ ïàðàìåòðàìè M (ξ) = 2nv + 120 èσ(ξ) = 15 + nv . Ïîëó÷åííóþ âûáîðêó çàïèñàòü â òåêñòîâûé ôàéë.2. Ñ÷èòàòü âûáîðêó, ïîëó÷åííóþ â ïåðâîì çàäàíèè. Ðàçáèòü ñîâîêóïíîñòü ýëåìåíòîâ âûáîðêè íà s èíòåðâàëîâ è äîêàçàòü èëè îïðîâåðãíóòü ãèïîòåçó î íîðìàëüíîì ðàñïðåäåëåíèè íåïðåðûâíîé ñëó÷àéíîé âåëè÷èíû ñ óðîâíåì çíà÷èìîñòè α = 0,05 è α = 0,01.  êà÷åñòâåçíà÷åíèÿ äëÿ ïàðàìåòðà s âûáðàòü öåëóþ ÷àñòü ÷èñëà 8 + nv /5.Ïðàêòè÷åñêîå çàíÿòèå 96.
Îáðàáîòêà ïðîñòîéñòàòèñòè÷åñêîé ñîâîêóïíîñòèÏóñòü â ðåçóëüòàòå îïûòîâ ïîëó÷åíû n çíà÷åíèé äâóìåðíîé ñëó÷àéíîé âåëè÷èíû (ξ, ζ), íå ïîäâåðãíóòûå èç-çà íåáîëüøîãî îáú¼ìà âûáîðêè ïðåäâàðèòåëüíîé ãðóïïèðîâêå:(x1 , y1 ),(x2 , y2 ), ..., (xn , yn ).Äëÿ õàðàêòåðèñòèêè òàêîé ïðîñòîé ñòàòèñòè÷åñêîé ñîâîêóïíîñòè÷àùå âñåãî èñïîëüçóþò îöåíêè ìàòåìàòè÷åñêèõ îæèäàíèé è äèñïåðñèèêîìïîíåíò ξ è ζ , âûáîðî÷íîãî êîýôôèöèåíòà êîððåëÿöèè è óðàâíåíèÿðåãðåññèè. Ñðåäíèå âûáîðî÷íûå îïðåäåëÿþòñÿ êàêx̄ =n∑xi /n,ȳ =i=1n∑yi /n.i=1Âûáîðî÷íûå äèñïåðñèèSx2=n∑i=1(xi − x̄) /n,2Sy2=2∑i=1(yi − ȳ)2 /n.80Ïðàêòè÷åñêîå çàíÿòèå 96.
Îáðàáîòêà ñîâîêóïíîñòèÂûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè∗rxy=(xy − x̄ȳ).Sx SyÂûáîðî÷íîå óðàâíåíèå ïðÿìîé ðåãðåññèè ζ íà ξ èìååò âèä:∗ȳx − ȳ = rxySy(x − x̄),Sxãäå ȳx ñðåäíåå àðèôìåòè÷åñêîå çíà÷åíèé ζ ïðè ôèêñèðîâàííîì çíà÷åíèè ξ = x.Àíàëîãè÷íî îïðåäåëÿåòñÿ ðåãðåññèÿ ξ íà ζ :∗x̄y − x̄ = rxySx(y − ȳ).SyÍàéäåì âñå ýòè õàðàêòåðèñòèêè ñ ïîìîùüþ ïðîãðàììû íà Mathcadè Maxima.
Âî èçáåæàíèå ïóòàíèöû ñ ïåðåìåííûìè â óðàâíåíèÿõ ðåãðåññèè îáîçíà÷èì íàáëþäåíèÿ áîëüøèìè áóêâàìè X è Y . Ðåçóëüòàòïðåäñòàâëåí íà ðèñ. 31.:ORIGIN := 1Îáú¼ì âûáîðêè: n := 10/*Íàáëþäàåìûå çíà÷åíèÿ ñëó÷àéíûõ âåëè÷èí:*/X := ( 1.42 1.83 1.59 1.90 1.64 1.36 1.24 1.60 1.82 1.57 )TY := ( 3.12 3.42 2.94 3.63 3.18 2.90 3.71 3.15 3.42 3.33 )T/*Íàéäåì âûáîðî÷íûå ñðåäíèå: */i := 1 .
. . n1 ∑1 ∑M x := ·XiM x = 1.597M y := ·YiM y = 3.28n in iMathCad-ïðîãðàììà/*Íàéäåì âûáîðî÷íûå ñðåäíèå êâàäðàòè÷åñêèå îòêëîíåíèÿ: */1∑1∑Sx2 :=(Xi −M x)2 Sy2 :=(Yi −M y)2 Sx2 = 0.041 Sy2 = 0.066n in iSx :=√Sx2Sy :=√Sy2Sx = 0.203Sy = 0.257/*Íàéäåì âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè: */1 ∑xy − M xM yxy := ·Xi · YiR :=R = 0.251n iSx · SyÏðàêòè÷åñêîå çàíÿòèå 96. Îáðàáîòêà ñîâîêóïíîñòè81/*Íàéäåì âûáîðî÷íûå óðàâíåíèÿ ïðÿìûõ ðåãðåññèè è ïîñòðîèì èõãðàôèêè: */R · SyR · SxY x(x) :=· (x − M x) + M yXy(y) :=· (y − M y) + M xSxSy:(%i1) load (descriptive)$ fpprintprec:5$ numer:true$ n:10$(%i5) X:[1.42,1.83,1.59,1.9,1.64,1.36,1.24,1.6,1.82,1.57];(%i6) Y:[3.12,3.42,2.94,3.63,3.18,2.9,3.71,3.15,3.42,3.33];/*Âû÷èñëèòü ñðåäíèå âûáîðî÷íûå çíà÷åíèÿ ñëó÷àéíûõ âåëè÷èí.*/(%i7) Mx:mean(X); My:mean(Y);(%o7) 1.597 ; (%o8) 3.28/* Âû÷èñëèòü âûáîðî÷íûå ñðåäíèå êâàäðàòè÷åñêèå çíà÷åíèÿ ñëó÷àéíûõ âåëè÷èí.*/(%i9) Sx:std(X); Sy:std(Y);(%o9) 0.203; (%o10) 0.257(%i11) xy:sum(X[i]*Y[i], i, 1, n)/n;(%o11) 5.2513/*Âû÷èñëèòü âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè ñëó÷àéíûõ âåëè÷èí.*/(%i12) R:(xy-Mx*My)/(Sx*Sy);(%o12) 0.251/* Çàäàòü óðàâíåíèÿ âûáîðî÷íûõ óðàâíåíèé ïðÿìîé ðåãðåññèè ñëó÷àéíûõ âåëè÷èí.*/(%i13) Yx(x):=R*Sy/Sx*(x-Mx)+My;(%i14) Xy(x):=R*Sx/Sy*(x-My)+Mx;/* Íàðèñîâàòü ãðàôèêè âûáîðî÷íûõ ïðÿìûõ ðåãðåññèè ñëó÷àéíûõ âåëè÷èí.*/(%i15) wxplot2d([Yx, Xy], [x, -20, 20], [y, -5, 5], [style,[lines,3,5], [lines,2,5]],[gnuplot_preamble, "set grid;"])$Maxima-ïðîãðàììà82Ïðàêòè÷åñêîå çàíÿòèå 96.
Îáðàáîòêà ñîâîêóïíîñòè5Yx(x)2.5Xy(y)0-2.5-5-20-1001020Ðèñ. 13. Ïðÿìûå ðåãðåññèè ê ïðàêòè÷åñêîìó çàäàíèþÑàìîñòîÿòåëüíàÿ ðàáîòà òàáëèöå 96.1 ïðèâåäåíû èñõîäíûå äàííûå äëÿ òðèäöàòè âàðèàíòîâ. Òðåáóåòñÿ íàéòè îöåíêè îñíîâíûõ ÷èñëîâûõ õàðàêòåðèñòèê:ìàòåìàòè÷åñêèõ îæèäàíèé, äèñïåðñèé, êîýôôèöèåíòà êîððåëÿöèè èýìïèðè÷åñêèå ëèíèè ðåãðåññèè ζ íà ξ è ξ íà ζ .Ïðàêòè÷åñêîå çàíÿòèå 96. Îáðàáîòêà ñîâîêóïíîñòè83Òàáëèöà 96.1N12345678910111213141516xiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyi123456789101,421,831,591,901,641,361,241,601,821,573,123,422,943,633,182,903,713,153,423,336,215,184,133,523,022,642,031,281,936,427,776,185,435,674,643,912,131,436,135,224,453,433,182,732,081,360,920,440,350,370,123,425,126,168,337,349,9310,2412,3013,461,801,641,901,841,801,821,681,851,821,62727483799083748579631,001,081,221,341,391,481,621,731,821,902,012,532,913,673,964,444,875,616,086,71-2,00-1,70-1,57-1,32-1,08-0,92-0,63-0,31-0,100,1212,4110,179,016,143,771,81-0,18-1,88-3,02-4,115,166,387,778,349,0310,8312,4415,2018,3225,1231,7225,4326,4828,1320,7418,4912,308,8310,557,181,021,271,411,581,831,992,222,442,592,835,045,474,484,824,363,742,412,641,931,363,183,423,814,186,106,5310,1212,3018,5320,002,394,034,165,634,835,788,349,4112,1814,360,250,320,400,480,740,921,121,441,622,103,122,422,221,082,572,983,322,641,423,781,341,791,631,881,661,381,361,631,841,602,993,332,903,603,212,503,833,163,523,415,435,324,003,963,182,732,131,360,820,406,187,246,225,426,034,544,442,101,400,306,435,184,233,453,122,842,111,480,930,270,832,135,486,1810,328,369,9011,4312,7515,280,830,940,930,900,600,730,820,881,000,874,214,482,124,304,833,534,134,005,003,1221,226,028,331,035,236,338,240,842,344,563,068,370,169,370,1472,075,480,385,190,32,134,187,3310,4811,1814,7315,3217,3219,6022,403,067,2310,457,6817,8820,3025,0827,1225,9331,3584Ïðàêòè÷åñêîå çàíÿòèå 96.
Îáðàáîòêà ñîâîêóïíîñòèÒàáëèöà 96.1N1718192021222324252627282930xiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyixiyi123456789100,240,380,450,560,600,630,680,720,740,752,632,412,101,901,771,781,601,491,521,401,022,102,924,084,986,007,148,229,099,902,123,374,234,256,187,168,208,0210,1311,431,433,013,303,414,205,025,896,147,221,902,571,792,643,852,363,603,584,554,634,571,501,622,783,104,054,154,635,355,906,884,743,512,872,952,902,600,962,880,630,750,971,832,233,102,634,805,205,896,027,002,032,003,172,754,553,414,962,944,806,500,901,322,092,423,203,864,175,500,758,086,134,305,213,004,343,162,102,921,851,8211,0011,2223,6424,7332,7538,1044,0348,1153,4656,191,122,431,483,813,925,934,506,207,566,239,2111,8021,1322,4824,5034,4933,1844,1253,1757,188,532,647,506,084,424,106,253,064,362,5444,5053,7157,1661,0062,3777,3078,1882,1191,7398,103,264,182,417,184,367,154,678,366,156,0813,4320,0827,1929,3637,1039,1240,0844,3850,7660,007,545,467,055,637,645,463,754,524,833,921,232,173,034,234,555,607,235,989,136,813,582,114,676,184,037,834,968,548,776,650,1030,2350,2430,3300,4050,4830,4750,5500,6100,8631,671,082,854,872,774,065,833,485,436,961,021,253,273,685,025,126,186,537,737,956,164,305,423,852,664,533,170,821,853,830,101,262,442,954,625,366,080,507,008,631,053,081,733,863,956,125,556,885,256,94Ëåêöèÿ 97.
Äèñïåðñèîííûé àíàëèç85Ëåêöèÿ 97. Äèñïåðñèîííûé àíàëèçÎáùàÿ ïîñòàíîâêà çàäà÷è. Ñðàâíåíèå äâóõ äèñïåðñèé. Îäíîôàêòîðíûé äèñïåðñèîííûé àíàëèç97.1. Îáùàÿ ïîñòàíîâêà çàäà÷èÏóñòü íåçàâèñèìûå ñëó÷àéíûå âåëè÷èíû ξ1 , . . . , ξk èìåþò íîðìàëüíîå ðàñïðåäåëåíèå ñ îäèíàêîâîé äèñïåðñèåé. Ìàòåìàòè÷åñêèå îæèäàíèÿ è äèñïåðñèÿ ñëó÷àéíûõ âåëè÷èí íåèçâåñòíû. Èìåþòñÿ íàáëþäåíèÿ íàä êàæäîé èç ýòèõ ñëó÷àéíûõ âåëè÷èí.
Òðåáóåòñÿ ïðè çàäàííîì óðîâíå çíà÷èìîñòè ïðîâåðèòü íóëåâóþ ãèïîòåçó H0 : M (ξ1 ) == M (ξ2 ) = . . . = M (ξk ) î ðàâåíñòâå âñåõ ìàòåìàòè÷åñêèõ îæèäàíèéïðè àëüòåðíàòèâíîé ãèïîòåçå H1 î òîì, ÷òî íå âñå ýòè ìàòåìàòè÷åñêèåîæèäàíèÿ ðàâíû.Äðóãèìè ñëîâàìè, òðåáóåòñÿ óñòàíîâèòü, çíà÷èìî ëè îòëè÷àþòñÿâûáîðî÷íûå ñðåäíèå.
Ñëåäóåò ñðàçó îòâåðãíóòü êàê íåýôôåêòèâíóþèäåþ ïîïàðíîãî ñðàâíåíèÿ ìàòåìàòè÷åñêèõ îæèäàíèé ñ ïîìîùüþ êðèòåðèÿ, èçëîæåííîãî â ëåêöèè 98 (ï. 98.3), ò.ê. ñ âîçðàñòàíèåì ÷èñëàk ñðàâíèâàåìûõ ñðåäíèõ âîçðàñòàåò è íàèáîëüøåå ðàçëè÷èå ìåæäóíèìè (íàêàïëèâàåòñÿ îøèáêà). Ïîýòîìó äëÿ ïðîâåðêè íóëåâîé ãèïîòåçû èñïîëüçóþò ìåòîä, îñíîâàííûé íà ñðàâíåíèè äèñïåðñèé, ÷òî èîáúÿñíÿåò åãî íàçâàíèå.Íà ïðàêòèêå äèñïåðñèîííûé àíàëèç ïðèìåíÿþò ïðè ïðîâåðêå íåñêîëüêèõ âûáîðîê íà îäíîðîäíîñòü: åñëè âñå âûáîðêè èìåþò îäèíàêîâîå íîðìàëüíîå ðàñïðåäåëåíèå ñ îäèíàêîâîé äèñïåðñèåé è áóäåò óñòàíîâëåíî ðàâåíñòâî èõ ìàòåìàòè÷åñêèõ îæèäàíèé, âñå èõ ìîæíî ñ÷èòàòü íàáëþäåíèÿìè íàä îäíîé ñëó÷àéíîé âåëè÷èíîé è îáúåäèíèòü.Äðóãàÿ îáëàñòü ïðèìåíåíèÿ äèñïåðñèîííîãî àíàëèçà ïðîâåðêàçíà÷èìîñòè âëèÿíèÿ íåêîòîðîãî ôàêòîðà íà íàáëþäàåìóþ ñëó÷àéíóþâåëè÷èíó ξ .
Ïóñòü íåêîòîðûé ôàêòîð F (íàïðèìåð íàçâàíèå èçó÷àåìîãî ñòóäåíòàìè ïðåäìåòà) èìååò k óðîâíåé F1 , F2 , . . . , Fk (ñòóäåíòû èçó÷àþò k ïðåäìåòîâ). Ñëó÷àéíàÿ âåëè÷èíà ξ (óñïåâàåìîñòüñòóäåíòîâ) íàáëþäàåòñÿ ïðè êàæäîì çíà÷åíèè ôàêòîðà F (èìåþòñÿäàííûå óñïåâàåìîñòè ñòóäåíòîâ ïî êàæäîìó ïðåäìåòó). Íà îñíîâàíèè ýòèõ äàííûõ òðåáóåòñÿ óñòàíîâèòü, çíà÷èìî ëè âëèÿåò ôàêòîð Fíà ñðåäíåå çíà÷åíèå ñëó÷àéíîé âåëè÷èíû ξ èëè ðàçíèöà íàáëþäåíèéîáóñëîâëåíà ñëó÷àéíûìè êîëåáàíèÿìè (îäèíàêîâà ëè óñïåâàåìîñòü ïî86Ëåêöèÿ 97.
Äèñïåðñèîííûé àíàëèçðàçëè÷íûì ïðåäìåòàì). Åñëè óñòàíîâëåíî, ÷òî âëèÿíèå ôàêòîðà ñóùåñòâåííî, äàëüíåéøåå èññëåäîâàíèå ìîæåò ïðîâîäèòüñÿ â íàïðàâëåíèè âûÿâëåíèÿ íàèáîëåå âëèÿþùåãî óðîâíÿ ôàêòîðà F ïóò¼ì ïîïàðíîãî ñðàâíåíèÿ âûáîðîê, êàê îïèñàíî â ï. 98.3 (âûÿâëåíèå ïðåäìåòà ññàìîé íèçêîé óñïåâàåìîñòüþ). èçëîæåííîé ïîñòàíîâêå çàäà÷à íàçûâàåòñÿ îäíîôàêòîðíûì äèñïåðñèîííûì àíàëèçîì.Èíîãäà ïðèõîäèòñÿ èññëåäîâàòü âëèÿíèå íåñêîëüêèõ ôàêòîðîâ íàñëó÷àéíóþ âåëè÷èíó ìíîãîôàêòîðíûé äèñïåðñèîííûé àíàëèç. Íàïðèìåð âëèÿíèå ïðåäìåòà, ïðåïîäàâàòåëÿ è ãîäà íàáîðà ñòóäåíòîâíà óñïåâàåìîñòü.Ïðåæäå ÷åì ïåðåéòè ê èçëîæåíèþ îäíîôàêòîðíîãî äèñïåðñèîííîãî àíàëèçà, ðàññìîòðèì åù¼ îäèí êðèòåðèé ïðîâåðêè ñòàòèñòè÷åñêîéãèïîòåçû êðèòåðèé Ôèøåðà.97.2.
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