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Ïî òàáëèöå êðèòè÷åñêèõ òî÷åê ðàñïðåäåëåíèÿ Ñòüþäåíòà ïðè óðîâíå çíà÷èìîñòè α = 0,05 è ÷èñëó ñòåïåíåéñâîáîäû k = 20 − 2 = 18 íàõîäèì têð (0,05; 18) = 2,10.Òàê êàê Tíàáë < Têð , òî íåò îñíîâàíèé îòâåðãíóòü íóëåâóþ ãèïîòåçó î ðàâåíñòâå íóëþ òåîðåòè÷åñêîãî êîýôôèöèåíòà êîððåëÿöèè. Ýòîîçíà÷àåò, ÷òî âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè íåçíà÷èìî îòëè÷àåòñÿ îò íóëÿ, ò.å.
ξ è ζ íåêîððåëèðîâàíû.Ïðèìåð 97.2. Ïî äâóì íåçàâèñèìûì âûáîðêàì, îáú¼ìû êîòîðûõn = 60 è m = 50, èçâëå÷åííûì èç íîðìàëüíûõ ãåíåðàëüíûõ ñîâîêóïíîñòåé, íàéäåíû âûáîðî÷íûå ñðåäíèå x̄ = 25 è ȳ = 23. Ñîîòâåòñòâóþùèå ãåíåðàëüíûå äèñïåðñèè σ12 = 5, σ22 = 4. Òðåáóåòñÿ ïðè óðîâíåçíà÷èìîñòè 0,01 ïðîâåðèòü íóëåâóþ ãèïîòåçó H0 : M (ξ) = M (ζ) ïðèêîíêóðèðóþùåé ãèïîòåçå H1 : M (ξ) ̸= M (ζ).Ïðàêòè÷åñêîå çàíÿòèå 97. Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåç93Ð å ø å í è å: Íàéäåì íàáëþäàåìîå çíà÷åíèå êðèòåðèÿ ïî ôîðìóëå(95.2):25 − 23Zíàáë = √≈ 54,436.5/60 + 4/50Òàê êàê êîíêóðèðóþùàÿ ãèïîòåçà M (ξ) ̸= M (ζ), òî êðèòè÷åñêàÿîáëàñòü äâóñòîðîííÿÿ.
Ñîãëàñíî (95.3), ïðàâóþ êðèòè÷åñêóþ òî÷êóíàéäåì èç ðàâåíñòâà:Φ(Zêð ) = (1 − α)/2 = (1 − 0,01)/2 = 0,495.Ïî òàáëèöå ôóíêöèè Ëàïëàñà (Ïðèëîæåíèå 2) íàõîäèì Zêð = 2,58.Òàê êàê |Zíàáë | > Zêð , òî íóëåâóþ ãèïîòåçó îòâåðãàåì. Ýòî îçíà÷àåò,÷òî âûáîðî÷íûå ñðåäíèå îòëè÷àþòñÿ çíà÷èìî.97.3. Èç íîðìàëüíîé ãåíåðàëüíîé ñîâîêóïíîñòè ñ èçâåñòíîé äèñïåðñèåé σ 2 = 16 èçâëå÷åíà âûáîðêà îáú¼ìà n = 80, è ïî íåéíàéäåíà âûáîðî÷íàÿ ñðåäíÿÿ x̄ = 13,12. Òðåáóåòñÿ ïðè óðîâíå çíà÷èìîñòè 0,05 ïðîâåðèòü íóëåâóþ ãèïîòåçó H0 : a = a0 = 12 ïðèêîíêóðèðóþùåé ãèïîòåçå H1 : a ̸= 12.ÏðèìåðÐ å ø å í è å: Íàéäåì íàáëþäàåìîå çíà÷åíèå êðèòåðèÿ ïî ôîðìóëå(95.6):13,12 − 12 √Uíàáë =· 80 ≈ 2,504.4Òàê êàê êîíêóðèðóþùàÿ ãèïîòåçà a ̸= a0 , òî êðèòè÷åñêàÿ îáëàñòüäâóñòîðîííÿÿ.
Êðèòè÷åñêóþ òî÷êó íàéäåì èç ðàâåíñòâà:Φ(Zêð ) = (1 − α)/2 = (1 − 0,05)/2 = 0,475.Ïî òàáëèöå ôóíêöèè Ëàïëàñà (Ïðèëîæåíèå 2) íàõîäèì Zêð = 1,96.Òàê êàê Uíàáë > Zêð , òî íóëåâóþ ãèïîòåçó îòâåðãàåì. Äðóãèìè ñëîâàìè, âûáîðî÷íàÿ è ãèïîòåòè÷åñêàÿ ãåíåðàëüíûå ñðåäíèå ðàçëè÷àþòñÿçíà÷èìî.97.4. Ïàðòèÿ èçäåëèé ïðèíèìàåòñÿ, åñëè âåðîÿòíîñòüòîãî, ÷òî èçäåëèå îêàæåòñÿ áðàêîâàííûì, íå ïðåâûøàåò 0,03. Ñðåäèñëó÷àéíî îòîáðàííûõ 500 èçäåëèé îêàçàëîñü 20 áðàêîâàííûõ. Áóäåòëè ýòà ïàðòèÿ ïðèíÿòà?ÏðèìåðÐ å ø å í è å: Íóëåâàÿ ãèïîòåçà H0 : p = p0 = 0,03, à îòíîñèòåëüíàÿ÷àñòîòà áðàêà m/n = 20/500 = 0,04. Ïðèìåì â êà÷åñòâå êîíêóðèðóþùåé ãèïîòåçû H1 : p > 0,03 è óðîâåíü çíà÷èìîñòè α = 0,05. Ñ ó÷¼òîì94Ïðàêòè÷åñêîå çàíÿòèå 97. Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåçòîãî, ÷òî q0 = 1 − p0 = 0,97, ïî ôîðìóëå (95.8) íàéäåì íàáëþäàåìîåçíà÷åíèå êðèòåðèÿ:√(0,04 − 0,03) · 500√≈ 1,311.Uíàáë =0,03 · 0,97Òàê êàê êîíêóðèðóþùàÿ ãèïîòåçà ñîñòîèò â òîì, ÷òî p > p0 , òîêðèòè÷åñêàÿ îáëàñòü ïðàâîñòîðîííÿÿ.
Êðèòè÷åñêóþ òî÷êó íàéäåì èçðàâåíñòâà:Φ(uêð ) = (1 − 2α)/2 = (1 − 2 · 0,05)/2 = 0,45.Ïî òàáëèöå ôóíêöèè Ëàïëàñà (Ïðèëîæåíèå 2) íàõîäèì uêð = 1,645.Òàê êàê Uíàáë < uêð , òî íåò îñíîâàíèé îòâåðãíóòü ãèïîòåçó î òîì, ÷òîâåðîÿòíîñòü áðàêà â ïàðòèè íå ïðåâûøàåò 0,03. Ñëåäîâàòåëüíî, ïàðòèÿ ìîæåò áûòü ïðèíÿòà.Ïðèìåð 97.5. Ïðîèçâåäåíî n = 100 èçìåðåíèé íåêîòîðîé ñëó÷àéíîé âåëè÷èíû. Âñÿ ñîâîêóïíîñòü ýëåìåíòîâ âûáîðêè ðàçáèòà íà èíòåðâàëû.  èòîãå èìååòñÿ ñëåäóþùèé ñòàòèñòè÷åñêèé ðÿä.N èíòåðâàëà 123456789èíòåðâàëû 69,2 69,8 70,4 71,0 71,6 72,2 72,8 73,4 74,069,8 70,4 71,0 71,6 72,2 72,8 73,4 74,0 74,6÷àñòîòû mj14112127221031Íàéòè çàêîí ðàñïðåäåëåíèÿ äàííîé ñëó÷àéíîé âåëè÷èíû ξ .Ð å ø å í è å: Êàê âèäèì èç òàáëèöû, äëèíà èíòåðâàëà ∆aj = 0,6.Äëÿ âûäâèæåíèÿ ãèïîòåçû î âèäå ðàñïðåäåëåíèÿ íà ðèñ.
14 ïîñòðîèìãèñòîãðàììó äàííîãî âàðèàöèîííîãî ðÿäà.Ïîñêîëüêó ðàñïðåäåëåíèå ñèììåòðè÷íî è èìååò ìàêñèìóì â ñåðåäèíå, òî ìîæíî âûäâèíóòü íóëåâóþ ãèïîòåçó H0 î òîì, ÷òî èçó÷àåìàÿâåëè÷èíà ïîä÷èíÿåòñÿ íîðìàëüíîìó çàêîíó. Âñëåäñòâèå ýòîãî íóæíîñäåëàòü îöåíêè äëÿ êàæäîãî èç íåèçâåñòíûõ ïàðàìåòðîâ íîðìàëüíîãîðàñïðåäåëåíèÿ.Äëÿ ðàñ÷¼òîâ âîçüì¼ì ñåðåäèíû èíòåðâàëîâ ui = (xj−1 +xj )/2, j == 1, 2, ..., s, ãäå s ÷èñëî èíòåðâàëîâ.
Òîãäà âûáîðî÷íîå ñðåäíåå èäèñïåðñèÿ îïðåäåëÿòñÿ ïî ôîðìóëàì:ss∑∑2x̄ =mi ui /n,S =mi u2i /n − x̄2 .i=1i=1∗2Èñïðàâëåííàÿ äèñïåðñèÿ S = nS /(n − 1).  ðåçóëüòàòå âû÷èñëåíèé ïîëó÷èì: x̄ = 71, 876, S ∗ = 0,8982.2Ïðàêòè÷åñêîå çàíÿòèå 97. Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåç95hi1/21/31/6069,271,072,874,6xjÐèñ. 14. Ãèñòîãðàììà îòíîñèòåëüíûõ ÷àñòîò äëÿ ïðèìåðà 97.5Âåðîÿòíîñòü ïîïàäàíèÿ ñëó÷àéíîé âåëè÷èíû ξ â îïðåäåë¼ííûé èíòåðâàë (xj−1 , xj ) ìîæíî íàéòè ñ ïîìîùüþ ôóíêöèè ðàñïðåäåëåíèÿ,êàê ýòî óêàçàíî â ëåêöèè, èëè äëÿ íîðìàëüíîé ñëó÷àéíîé âåëè÷èíûïî ôîðìóëå()()xj − x̄xj−1 − x̄Pj = P (xj−1 < ξ < xj ) = Φ−Φ.S∗S∗Çäåñü çíà÷åíèÿ ôóíêöèè Ëàïëàñà ìîæíî íàéòè ïî òàáëèöàì. Òîãäàòåîðåòè÷åñêèå ÷àñòîòû îïðåäåëÿòñÿ êàê mj = Pj ·n.  èòîãå âìåñòî ýìïèðè÷åñêèõ ÷àñòîò ïîëó÷èì ñëåäóþùèå ïðèáëèæ¼ííûå çíà÷åíèÿ òåîðåòè÷åñêèõ ÷àñòîò m′i :N12345678′mj 0,8961 3,9749 11,4545 21,4611 26,1543 20,7354 10,6927 3,5848N9′mj 0,7808Ïîñëå ýòîãî ïîñëåäîâàòåëüíî íàõîäèì mi − m′i , (mi − m′i )2 , è(mi − mi )2 /m′i , à çàòåì ñóììó ïîñëåäíèõ çíà÷åíèé.
Ñîãëàñíî (96.6),êðèòåðèé χ2íàáë = 0,349.96Ïðàêòè÷åñêîå çàíÿòèå 97. Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåçÓ÷èòûâàÿ, ÷òî çäåñü êîëè÷åñòâî èíòåðâàëîâ s = 9, îïðåäåëÿåì÷èñëî ñòåïåíåé ñâîáîäû k ïî ôîðìóëå (96.7): k = 9 − 1 − 2 = 6. Çàäàäèìñÿ óðîâíåì çíà÷èìîñòè α = 0,05. Òîãäà ñ ÷èñëîì ñòåïåíåé ñâîáîäû6 ïî òàáëèöå êðèòè÷åñêèõ òî÷åê ðàñïðåäåëåíèÿ χ2 (ïðèëîæåíèå 4) íàõîäèì çíà÷åíèå êðèòåðèÿχ2êð (α, k) = χ2êð (0,05; 6) = 12,6.Òàê êàê χ2íàáë < χ2êð , òî íåò îñíîâàíèé îòâåðãíóòü ãèïîòåçó î íîðìàëüíîì ðàñïðåäåëåíèè.
Äðóãèìè ñëîâàìè, ðàñõîæäåíèå ýìïèðè÷åñêèõ è òåîðåòè÷åñêèõ ÷àñòîò íåçíà÷èìî. Ñëåäîâàòåëüíî, îïûòíûå äàííûå ñîãëàñóþòñÿ ñ ãèïîòåçîé î íîðìàëüíîì ðàñïðåäåëåíèè èçó÷àåìîéñëó÷àéíîé âåëè÷èíû ξ .Ñàìîñòîÿòåëüíàÿ ðàáîòàÏðèìåð 97.6. Ïî âûáîðêå îáú¼ìà n = 30 íàéäåí âûáîðî÷íûé êî∗ýôôèöèåíò êîððåëÿöèè rxy= 0,35. Ïðè óðîâíå çíà÷èìîñòè 0,1 ïðîâåðèòü ãèïîòåçó î ðàâåíñòâå íóëþ òåîðåòè÷åñêîãî êîýôôèöèåíòàêîððåëÿöèè ïðè êîíêóðèðóþùåé ãèïîòåçå rξζ ̸= 0.97.7. Ïî âûáîðêå îáú¼ìà n = 100, èçâëå÷åííîé èç äâóìåðíîé íîðìàëüíîé ãåíåðàëüíîé ñîâîêóïíîñòè, ñîñòàâëåíà êîððåëÿöèîííàÿ òàáëèöà:ÏðèìåðÈñõîäíûåξ\ζ 4 65 4 210 - 615 - 20 - 2äàííûå ïðèìåðà 97.78 1012- 7 40 512 148Íàéòè âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè è ïðè óðîâíå çíà÷èìîñòè 0,1 ïðîâåðèòü íóëåâóþ ãèïîòåçó î ðàâåíñòâå íóëþ òåîðåòè÷åñêîãî êîýôôèöèåíòà êîððåëÿöèè ïðè êîíêóðèðóþùåé ãèïîòåçårξζ ̸= 0.Ïðàêòè÷åñêîå çàíÿòèå 97.
Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåç97Òàáëèöà 97.3Nâàðèàíòà123456789101112131415161718192021222324252627282930n1 n213121110987651013121110913121110987651013121110917161514131211109817161514131011121314151617891314151617x151.481.6635.4025.654.4080.5314.3116.6232.127.208.815.034.6413.3316.0828.4380.3445.7825.3123.4616.3817.645.324.3819.238.3212.4823.4520.4413.25x2σ12σ22α55.04.381.29 0.0185.00 29.96 12.97 0.0530.30 9.843.90 0.0123.55 2.080.98 0.054.00 0.0295 0.008 0.0182.40 21.34 7.34 0.0512.21 17.82 3.70 0.0113.34 13.32 4.47 0.0530.10 18.92 3.45 0.015.15 10.52 3.18 0.055.85 11.68 4.62 0.016.215.222.90 0.054.026.732.39 0.0116.22 8.944.52 0.0513.11 7.352.02 0.0130.50 11.22 2.38 0.0178.10 24.35 11.71 0.0540.32 18.43 7.84 0.0522.84 8.513.04 0.0125.81 12.38 5.87 0.0518.21 11.64 3.66 0.0115.32 10.52 4.52 0.057.554.321.38 0.014.012.350.75 0.0517.34 7.481.82 0.016.294.358.25 0.0110.31 19.38 8.25 0.0120.81 17.25 8.50 0.0523.00 13.11 4.54 0.0111.49 10.12 6.98 0.0598Ïðàêòè÷åñêîå çàíÿòèå 97.
Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåçÏðèìåð 97.8. Ïî âûáîðêå îáú¼ìà n = 100 íàéäåí ñðåäíèé âåñ äåòàëåé x̄ = 210 ã, èçãîòîâëåííûõ íà ïåðâîì ñòàíêå; ïî âûáîðêå îáú¼ìà m = 90 íàéäåí ñðåäíèé âåñ ȳ = 208 ã äåòàëåé, èçãîòîâëåííûõ íàâòîðîì ñòàíêå. Ãåíåðàëüíûå äèñïåðñèè èçâåñòíû: σ12 = 80, σ22 = 70.Ïðåäïîëàãàåòñÿ, ÷òî ñëó÷àéíûå âåëè÷èíû ξ è ζ ðàñïðåäåëåíû íîðìàëüíî è âûáîðêè íåçàâèñèìû. Ïðè óðîâíå çíà÷èìîñòè 0,05 ïðîâåðèòü ãèïîòåçó H0 : M (ξ) = M (ζ) ïðè êîíêóðèðóþùåé ãèïîòåçåM (ξ) ̸= M (ζ).97.9.  òàáëèöå 97.3 äàíû âàðèàíòû çàäàíèé. Äëÿ êàæäîãî âàðèàíòà ïðèâåäåíû äâå íåçàâèñèìûå âûáîðêè îáú¼ìàìè n1 èn2 , íàéäåíû âûáîðî÷íûå ñðåäíèå x̄1 , x̄2 è èçâåñòíû äèñïåðñèè σ12 , σ22 .Íóæíî ïðîâåðèòü ïðè çàäàííîì óðîâíå çíà÷èìîñòè α ðàâåíñòâî ìàòåìàòè÷åñêèõ îæèäàíèé ïðè êîíêóðèðóþùåé ãèïîòåçå îá èõ íåðàâåíñòâå.Ïðèìåð97.10.
Çàäàíà âûáîðêà îáú¼ìà n = 120 èç íîðìàëüíîéñîâîêóïíîñòè ñ èçâåñòíûì ñðåäíèì êâàäðàòè÷åñêèì îòêëîíåíèåìσ = 5 è âûáîðî÷íîé ñðåäíåé x̄ = 23,54. Íåîáõîäèìî ïðè óðîâíå çíà÷èìîñòè 0,01 ïðîâåðèòü íóëåâóþ ãèïîòåçó H0 : a = a0 = 23 ïðèêîíêóðèðóþùåé ãèïîòåçå H1 : a ̸= 23.Ïðèìåð97.11. Ôèðìà ðàññûëàåò ðåêëàìíûå êàòàëîãè òîðãîâûìîðãàíèçàöèÿì. Âåðîÿòíîñòü òîãî, ÷òî îðãàíèçàöèÿ, ïîëó÷èâøàÿ êàòàëîã, çàêàæåò ðåêëàìèðóåìîå èçäåëèå, ðàâíà 0,12.
Ôèðìà ðàçîñëàëà400 íîâûõ óëó÷øåííûõ êàòàëîãîâ è ïîëó÷èëà 60 çàêàçîâ. Ìîæíî ëèñ÷èòàòü, ÷òî íîâûå êàòàëîãè çíà÷èìî ëó÷øå ñòàðûõ?Óêàçàíèå. Ïðèíÿòü íóëåâóþ ãèïîòåçó H0 : p = p0 = 0,12; êîíêóðèðóþùóþ H1 : p > 0,12; óðîâåíü çíà÷èìîñòè α = 0,05.ÏðèìåðÏðèìåð 97.12. Ïðîâåäåíî n èçìåðåíèé íåêîòîðîé íåïðåðûâíîéñëó÷àéíîé âåëè÷èíû. Äèàïàçîí ïîëó÷åííûõ èçìåðåíèé ðàçáèò íà èíòåðâàëû ïîñòîÿííîé äëèíû è êîëè÷åñòâî íàáëþäåíèé, ïîïàâøèõ âñîîòâåòñòâóþùèé èíòåðâàë, çàïèñàíî â òàáëèöó 97.5.  òàáëèöå97.4 ïðèâåäåíû ëåâàÿ è ïðàâàÿ ãðàíèöû íàáëþäåíèé è ÷èñëî èíòåðâàëîâ. Ïî ðåçóëüòàòàì èçìåðåíèé âû÷èñëèòü âûáîðî÷íîå ñðåäíåå èèñïðàâëåííóþ äèñïåðñèþ. Èñïîëüçóÿ êðèòåðèé Ïèðñîíà, ïðè óðîâíåçíà÷èìîñòè α = 0,05 ïðîâåðèòü, ñîãëàñóåòñÿ ëè ãèïîòåçà î íîðìàëüíîì ðàñïðåäåëåíèè ãåíåðàëüíîé ñîâîêóïíîñòè ξ ñ çàäàííûì ýìïèðè÷åñêèì ðàñïðåäåëåíèåì.Ïðèëîæåíèå 199Âàðèàíòû156101115152021252530Òàáëèöà 97.4Íàèìåíüøåå Íàèáîëüøåå×èñëîçíà÷åíèåçíà÷åíèåèíòåðâàëîâ9,189,761034,562,01020161066,5102,5120,210,45124,5044,54812ÏÐÈËÎÆÅÍÈßÏÐÈËÎÆÅÍÈE11Òàáëèöà çíà÷åíèé ôóíêöèè φ(x) = √2πe−x2 /201234567890,00,39893989398939883986398439823980397739730,139703965396139563951394539393932392539180,239103902389438853876386738573847383638250,338143802379037783765375237393726371236970,436833668365236373621360535893572355535380,535213503348534673448342934103391337233520,633323312329232713251323032093187316631440,731233101307930563034301129892966294329200,828972874285028272803780227562732270926850,926612637261325892565254125162492246824441,00,24202396237123472323229922752251222722031,121792155213121072083205920362012198919651,219421919189518721849182618041781175817361,317141691166916471626160415821561153915181,414971476145614351415139413741354133413151,512951276125712381219120011821163114511271,611091092107410571040102310060989097309571,709400925090908930878086308480833081808041,807900775076107480734072107070694068106691,90656064406320620060805960584057305620551100Ïðèëîæåíèå 1Ïðîäîëæåíèå òàáëèöû ïðèëîæåíèÿ 1N Âàðèàíòà123456789101112131415161718192021222324252627282930522414141224554324313612111301935638263588101066812448125642481218891610203371091616201618819396619206101056211234034206821431875211933303022661731581093739812151010353010×àñòîòû49 43 2640 38 1924 22 1380 76 3825 23 1249 40 2438 40 3480 69 1726 22 1124 21 1353 60 6450 65 6050 75 4080 70 3782 87 3239 48 4249 64 6074 63 5016 20 1818 28 1553 60 6458 74 6317 20 1830 40 3235 45 3019 42 4815 18 2843 50 4440 32 1244 76 78Òàáëèöà 97.5834641096443024151412195342131049501510183993910642214113412111010551037168531421047175206100102012211335115 2 117 7 312 8 26 5 13 1 017 7 216 12 89 7 42 1 12 4 18 6 33 1 09 5 32 1 16 4 2Ïðèëîæåíèå 110101234567892,00,05400529051905080498048804780468045904492,104400431042204130404039603870379037103632,203550347033903320325021703100303029702902,302830277027002640258025202460241023502292,402240219021302080203019801940189018401802,501750171016701630158015401510147014301392,601360132012901260122011901160113011001072,701040101009900960093009100880086008400812,800790077007500730071006900670065006300612,900600058005600550053005100500048004700453,00,00440043004200400039003800370036003500343,100330032003100300029002800270026002500253,200240023002200220021002000200019001800183,300170017001600160015001500140014001300133,400120012001200110011001000100010000900093,500090008000800080008000700070007000700063,600060006000600050005000500050005000500043,700040004000400040004000400030003000300033,800030003000300030003000200020002000200023,90002000200020002000200020002000200010001102Ïðèëîæåíèå 2Òàáëèöà çíà÷åíèé ôóíêöèèx0,000,010,020,030,040,050,060,070,080,090,100,110,120,130,140,150,160,170,180,190,200,210,220,230,240,250,260,270,280,290,300,31Ô(x)0,00000,00400,00800,01200,01600,01990,02390,02790,03190,03590,03980,04380,04780,05170,05570,05960,06360,06750,07140,07530,07930,08320,08710,09100,09480,09870,10260,10640,11030,11410,11790,1217x0,320,330,340,350,360,370,380,390,400,410,420,430,440,450,460,470,480,490,500,510,520,530,540,550,560,570,580,590,600,610,620,63Ô(x)0,12550,12930,13310,13680,14060,14430,14800,15170,15540,15910,16280,16640,17000,17360,17720,18080,18440,18790,19150,19500,19850,20190,20540,20880,21230,21570,21900,22240,22570,22910,23240,2357ÏÐÈËÎÆÅÍÈÅ2∫x1Ô(x) = √2πx0,640,650,660,670,680,690,700,710,720,730,740,750,760,770,780,790,800,810,820,830,840,850,860,870,880,890,900,910,920,930,940,95Ô(x)0,23890,24220,24540,24860,25170,25490,25800,26110,26420,26730,27030,27340,27640,27940,28230,28520,28810,29100,29390,29670,29950,30230,30510,30780,31060,31330,31590,31860,32120,32380,32640,3289e−z0x0,960,970,980,991,001,011,021,031,041,051,061,071,081,091,101,111,121,131,141,151,161,171,181,191,201,211,221,231,241,252 /2dzÔ(x)0,33150,33400,33650,33890,34130,34380,34610,34850,35080,35310,35540,35770,35990,36210,36430,36650,36860,37080,37290,37490,37700,37900,38100,38300,38490,38690,38830,39070,39250,3944Ïðèëîæåíèå 2x1,261,271,281,291,301,311,321,331,341,351,361,371,381,391,401,411,421,431,441,451,461,471,481,491,501,511,521,531,541,551,561,571,58Ô(x)0,39620,39800,39970,40150,40320,40490,40660,40820,40990,41150,41310,41470,41620,41770,41920,42070,42220,42360,42510,42650,42790,42920,43060,43190,43320,43450,43570,43700,43820,43940,44060,44180,4429103x1,591,601,611,621,631,641,651,661,671,681,691,701,711,721,731,741,751,761,771,781,791,801,811,821,831,841,851,861,871,881,891,901,91Ïðîäîëæåíèå òàáëèöû ïðèëîæåíèÿ 2Ô(x)xÔ(x)xÔ(x)0,4441 1,92 0,4726 2,50 0,49380,4452 1,93 0,4732 2,52 0,49410,4463 1,94 0,4738 2,54 0,49450,4474 1,95 0,4744 2,56 0,49480,4484 1,96 0,4750 2,58 0,49510,4495 1,97 0,4756 2,60 0,49530,4505 1,98 0,4761 2,62 0,49560,4515 1,99 0,4767 2,64 0,49590,4525 2,00 0,4772 2,66 0,49610,4535 2,02 0,4783 2,68 0,49630,4545 2,04 0,4793 2,70 0,49650,4554 2,06 0,4803 2,72 0,49670,4564 2,08 0,4812 2,74 0,49690,4573 2,10 0,4821 2,76 0,49710,4582 2,12 0,4830 2,78 0,49730,4591 2,14 0,4838 2,80 0,49740,4599 2,16 0,4846 2,82 0,49760,4608 2,18 0,4854 2,84 0,49770,4616 2,20 0,4861 2,86 0,49790,4625 2,22 0,4868 2,88 0,49800,4633 2,24 0,4875 2,90 0,49810,4641 2,26 0,4881 2,92 0,49820,4649 2,28 0,4887 2,94 0,49840,4656 2,30 0,4893 2,96 0,49850,4664 2,32 0,4898 2,98 0,49860,4671 2,34 0,4904 3,00 0,498650,4678 2,36 0,4909 3,20 0,499310,4686 2,38 0,4913 3,40 0,499660,4693 2,40 0,4918 3,60 0,4998410,4699 2,42 0,4922 3,80 0,4999280,4706 2,44 0,4927 4,00 0,4999680,4713 2,46 0,4931 4,50 0,4999970,4719 2,48 0,4934 5,00 0,499997104Ïðèëîæåíèå 3ÏÐÈËÎÆÅÍÈÅ 3Êðèòè÷åñêèå òî÷êè ðàñïðåäåëåíèÿ Ñòüþäåíòà×èñëîñòåïåíåéñâîáîäû k1234567891011121314151617181920212223242526272829304060120∞0,106,312,922,352,132,011,941,891,861,831,811,801,781,771,761,751,751,741,731,731,731,721,721,711,711,711,711,711,701,701,701,681,671,661,64Óðîâåíü çíà÷èìîñòè α(äâóñòîðîííÿÿ êðèòè÷åñêàÿ îáëàñòü)0,050,020,010,00212,731,8263,7318,34,306,979,9222,333,184,545,8410,222,783,754,607,172,573,374,035,892,453,143,715,212,363,003,504,792,312,903,364,502,262,823,254,302,232,763,174,142,202,723,114,032,182,683,053,932,162,653,013,852,142,622,983,792,132,602,953,732,122,582,923,692,112,572,903,652,102,552,883,612,092,542,863,582,092,532,853,552,082,522,833,532,072,512,823,512,072,502,813,492,062,492,803,472,062,492,793,452,062,482,783,442,052,472,773,422,052,462,763,402,052,462,763,402,042,462,753,392,022,422,703,312,002,392,663,231,982,362,623,171,962,332,583,090,001637,031,612,98,616,865,965,405,044,784,594,444,324,224,144,074,013,963,923,883,853,823,793,773,743,723,713,693,663,663,653,553,463,373,29Ïðèëîæåíèå 4105ÏÐÈËÎÆÅÍÈÅ 4Êðèòè÷åñêèå òî÷êè ðàñïðåäåëåíèÿ χ2×èñëîñòåïåíåéñâîáîäû k123456789101112131415161718192021222324252627282930Óðîâåíü çíà÷èìîñòè α0,016,69,211,313,315,116,818,520,121,723,224,726,227,729,130,632,033,434,836,237,638,940,341,643,044,345,647,048,349,650,90,0255,07,49,411,112,814,416,017,519,020,521,923,324,726,127,528,830,231,532,934,235,536,838,139,440,641,943,244,545,747,00,053,86,07,89,511,112,614,115,516,918,319,721,022,423,725,026,327,628,930,131,432,733,935,236,437,738,940,141,342,643,80,950,00390,1030,3520,7111,151,642,172,733,333,944,575,235,896,577,267,968,679,3910,110,911,612,313,113,814,615,416,216,917,718,50,9750,990,00098 0,000160,0510,0200,2160,1150,4840,2970,8310,5541,240,8721,691,242,181,652,702,093,252,563,823,054,403,575,014,115,634,666,265,236,915,817,566,418,237,018,917,639,598,2610,38,9011,09,5411,710,212,410,913,111,513,812,214,612,915,313,616,014,316,815,0k2 \k112345678910111213141516171405298,4934,1221,2016,2613,7412,2511,2610,5610,049,869,339,078,868,688,538,402499999,0130,8118,0013,2710,929,558,658,027,567,206,936,706,516,366,236,113540390,1729,4616,6912,069,788,457,596,996,556,225,955,745,565,425,295,18Óðîâåíü çíà÷èìîñòè α = 0,01456785625 5764 5889 5928 598199,25 99,33 99,30 99,34 99,3628,71 28,24 27,91 27,67 27,4915,98 15,52 15,21 14,98 14,8011,39 10,97 10,67 10,45 10,279,15 8,75 8,47 8,26 8,107,85 7,46 7,19 7,00 6,847,01 6,63 6,37 6,19 6,036,42 6,06 5,80 5,62 5,475,99 5,64 5,39 5,21 5,065,67 5,32 5,07 4,88 4,745,41 5,06 4,82 4,65 4,505,20 4,86 4,62 4,44 4,305,03 4,69 4,46 4,28 4,144,89 4,56 4,32 4,14 4,004,77 4,44 4,20 4,03 3,894,67 4,34 4,10 3,93 3,799602299,3627,3414,6610,157,986,715,915,354,954,634,394,194,033,893,783,6810605699,4027,2314,5410,057,876,625,825,264,854,544,304,103,943,803,693,5911608299,4127,1314,459,967,796,545,745,184,784,464,224,023,863,733,613,5212610699,4227,0514,379,897,726,475,675,114,714,404,163,963,803,673,553,45106Ïðèëîæåíèå 5ÏÐÈËÎÆÅÍÈÅ 5Êðèòè÷åñêèå òî÷êè ðàñïðåäåëåíèÿ F Ôèøåðà Ñíåäåêîðàk2 \k11234567891011121314151617Ïðîäîëæåíèå òàáëèöû ïðèëîæåíèÿ 5Óðîâåíü çíà÷èìîñòè α = 0,0512345678910111216120021622523023423723924124224324418,51 19,00 19,16 19,25 19,30 19,33 19,36 19,37 19,38 19,39 19,40 19,4110,13 9,55 9,28 9,12 9,01 8,94 8,88 8,84 8,81 8,78 8,76 8,747,71 6,94 6,59 6,39 6,26 6,16 6,09 6,04 6,00 5,96 5,93 5,916,61 5,79 5,41 5,19 5,05 4,95 4,88 4,82 4,78 4,74 4,70 4,685,99 5,14 4,76 4,53 4,39 4,28 4,21 4,15 4,10 4,06 4,03 4,005,59 4,74 4,35 4,12 3,97 3,87 3,79 3,73 3,68 3,63 3,60 3,575,32 4,46 4,07 3,84 3,69 3,58 3,50 3,44 3,39 3,34 3,31 3,285,12 4,26 3,86 3,63 3,48 3,37 3,29 3,23 3,18 3,13 3,10 3,074,96 4,10 3,71 3,48 3,33 3,22 3,14 3,07 3,02 2,97 2,94 2,914,84 3,98 3,59 3,36 3,20 3,09 3,01 2,95 2,90 2,86 2,82 2,794,75 3,88 3,49 3,26 3,11 3,00 2,92 2,85 2,80 2,76 2,72 2,694,67 3,80 3,41 3,18 3,02 2,92 2,84 2,77 2,72 2,67 2,63 2,604,60 3,74 3,34 3,11 2,96 2,85 2,77 2,70 2,65 2,60 2,56 2,534,54 3,68 3,29 3,06 2,90 2,79 2,70 2,64 2,59 2,55 2,51 2,484,49 3,63 3,24 3,01 2,85 2,74 2,66 2,59 2,54 2,49 2,45 2,424,45 3,59 3,20 2,96 2,81 2,70 2,62 2,55 2,50 2,45 2,41 2,38k1 - ÷èñëî ñòåïåíåé ñâîáîäû áîëüøåé äèñïåðñèè,k2 - ÷èñëî ñòåïåíåé ñâîáîäû ìåíüøåé äèñïåðñèèÏðèëîæåíèå 5107.
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