Примеры решения задач (1094491), страница 7
Текст из файла (страница 7)
Òîãäà ïî ôîðìóëå (93.2) íàéäåì:1x̄ = (10 · 0,9 + 25 · 1 + 20 · 1,2 + 15 · 1,4 + 10 · 1,5) = 1,175.80Äëÿ íàõîæäåíèÿ äèñïåðñèè âîñïîëüçóåìñÿ ôîðìóëîé (93.7):1S 2 = (10 · 0,92 + 25 · 12 + 20 · 1,22 + 15 · 1,42 + 10 · 1,52 ) − (1,175)2 ≈80≈ 1,4225 − 1,3806 ≈ 0,042.50Ïðàêòè÷åñêîå çàíÿòèå 94. Òî÷å÷íûå è èíòåðâàëüíûå îöåíêèÇàìåòèì, ÷òî îòëè÷íàÿ îò íóëÿ äèñïåðñèÿ ÿâëÿåòñÿ âñåãäà ïîëîæèòåëüíîé√ âåëè÷èíîé. Âûáîðî÷íîå ñðåäíåå êâàäðàòè÷åñêîå îòêëîíåíèåS = 0,042 ≈ 0,205.94.3. Ïî âûáîðêå îáú¼ìà n = 50 íàéäåíà ñìåù¼ííàÿ îöåíêà S = 9,8 ãåíåðàëüíîé äèñïåðñèè. Íàéòè íåñìåù¼ííóþ îöåíêó äèñïåðñèè ãåíåðàëüíîé ñîâîêóïíîñòè.Ïðèìåð2Ð å ø å í è å: Ñîãëàñíî (93.9), èñïðàâëåííàÿ âûáîðî÷íàÿ äèñïåðñèÿ, ÿâëÿþùàÿñÿ â òî æå âðåìÿ íåñìåù¼ííîé îöåíêîén50S ∗2 =· S2 =· 9,8 = 10.n−149Ïðèìåð 94.4. Íàéòè äîâåðèòåëüíûé èíòåðâàë äëÿ îöåíêè ñ íàäåæíîñòüþ 0,99 íåèçâåñòíîãî ìàòåìàòè÷åñêîãî îæèäàíèÿ a íîðìàëüíîãî ðàñïðåäåëåíèÿ, åñëè ñðåäíåå êâàäðàòè÷åñêîå îòêëîíåíèåσ = 3, âûáîðî÷íîå ñðåäíåå x̄ = 32 è îáú¼ì âûáîðêè n = 36.Ð å ø å í è å:  äàííîì ïðèìåðå âîñïîëüçóåìñÿ âûðàæåíèåì(93.11).
Ïîñêîëüêó çäåñü γ = 0,99, òî ïàðàìåòð τ γ2 íàéäåì ñ ïîìîùüþðàâåíñòâà:γ0,99Φ(τ γ2 ) = == 0,495.22Îòñþäà ïî òàáëèöàì ôóíêöèè Ëàïëàñà îïðåäåëèì τ γ2 = 2,57. Çäåñüëåâàÿ ãðàíèöà èíòåðâàëàσ3x̄ − τ γ2 · √ = 32 − 2,57 · = 32 − 1,285 = 30,715.6nÏðàâàÿ ãðàíèöà îïðåäåëèòñÿ êàê 32 + 1,285 = 33,285. Òàêèì îáðàçîì, èñêîìûé äîâåðèòåëüíûé èíòåðâàë äëÿ ìàòåìàòè÷åñêîãî îæèäàíèÿ a áóäåò30,715 < a < 33,285.Ïðèìåð 94.5. Èç ãåíåðàëüíîé ñîâîêóïíîñòè èçâëå÷åíà âûáîðêàîáú¼ìà n = 16:xi 3,5 4,1 4,7 5,4 5,6 6,2ni 232432Îöåíèòü ñ íàäåæíîñòüþ 0,95 ìàòåìàòè÷åñêîå îæèäàíèå a íîðìàëüíî ðàñïðåäåë¼ííîé ñëó÷àéíîé âåëè÷èíû ïî âûáîðî÷íîìó ñðåäíåìó ñïîìîùüþ äîâåðèòåëüíîãî èíòåðâàëà.Ïðàêòè÷åñêîå çàíÿòèå 94.
Òî÷å÷íûå è èíòåðâàëüíûå îöåíêè51Ð å ø å í è å:  äàííîì ñëó÷àå äèñïåðñèÿ íåèçâåñòíà è äîâåðèòåëüíûé èíòåðâàë îïðåäåëÿåòñÿ ïî ôîðìóëå (93.15). Âûáîðî÷íîå ñðåäíååâû÷èñëèì ïî ôîðìóëå (93.2):1(2 · 3, 5 + 3 · 4,1 + 2 · 4,7 + 4 · 5,4 + 3 · 5,6 + 2 · 6,2) ≈ 4,969.16Âûáîðî÷íóþ äèñïåðñèþ óäîáíåå èñêàòü ñ ïîìîùüþ âûðàæåíèÿ(93.7):61 ∑2ni x2i − 4,9692 ≈ 2,426.S =15 i=1x̄ =Ñîãëàñíî (93.9), èñïðàâëåííàÿ âûáîðî÷íàÿ äèñïåðñèÿ16· 2,426 = 2,587.15Îòñþäà íàõîäèì èñïðàâëåííîå ÑÊÎ S ∗ ≈ 1,608.
Ïðè α = 1 − γ == 0,05 è ÷èñëå ñòåïåíåé ñâîáîäû n − 1 = 15 èç òàáëèöû ïðèëîæåíèÿ 3îïðåäåëèì tγ = 2,15 è ãðàíèöû èíòåðâàëàS ∗2 =S∗x̄ − tγ · √ ≈ 4,104,nS∗x̄ + tγ · √ ≈ 5,833.nÒàêèì îáðàçîì, èñêîìûé äîâåðèòåëüíûé èíòåðâàë áóäåò:4,1042 < a < 5,8333.Ïîêàæåì, êàê íàéòè äîâåðèòåëüíûé èíòåðâàë ñ èñïîëüçîâàíèåìïðîãðàììû íà Mathcad.Ïðèìåð 94.6. Íàéòè äîâåðèòåëüíûé èíòåðâàë äëÿ ìàòåìàòè÷åñêîãî îæèäàíèÿ, äëÿ ïðèâåä¼ííîé âûáîðêè èç íîðìàëüíîãî ðàñïðåäåëåíèÿ ñ íàäåæíîñòüþ 0,95.ORIGIN := 1 k := 12 i := 1 .
. . k/*Çíà÷åíèÿ âàðèàíò ïðèâåäåíû â ìàññèâå x:*/x := (904.3 910.2 916.6 928.8 935 941.2 947.4 953.6 959.8 966 972.2978.4 )T/*Çíà÷åíèÿ ÷àñòîò ïðèâåäåíû â ìàññèâå m:*/m := (3 1 2 7 8 10 4 2 4 1 1 1 )T/*Íàéäåìîáú¼ì âûáîðêè: */∑n :=min = 44i/*Íàéäåì âûáîðî÷íîå ñðåäíåå: */52Ïðàêòè÷åñêîå çàíÿòèå 94. Òî÷å÷íûå è èíòåðâàëüíûå îöåíêè1 ∑·mi · x iM x = 938.693n i/*è âûáîðî÷íóþ äèñïåðñèþ:*/∑1S2 :=·mi · (xi − M x)2 S2 = 282.988n−1 i/* äëÿ íàõîæäåíèÿ ãðàíèö äîâåðèòåëüíîãî èíòåðâàëà ñ íàäåæíîñòüþ 0.95,( íàéäåì tγ )èç óðàâíåíèÿ (93.16):*/0.05t := qt 1 −,nt = 2.015,2/*ãäå qt îáðàòíàÿ ê ôóíêöèè ðàñïðåäåëåíèÿ Ñòüþäåíòà.*//*Íàéäåì√ ðàäèóñ äîâåðèòåëüíîãî èíòåðâàëà:S2ε := t ·ε = 5.111n/*à òàêæå ëåâóþ xLef t è ïðàâóþ xRight ãðàíèöû èíòåðâàëà:*/xLef t := M x − εxRight := M x + εxLef t = 933.582xRight = 943.804Maxima-ïðîãðàììà:(%i1) kill(all)$ numer:true$ fpprintprec:6$ ratprint:true$ k:12$(%i5) x:[904.3,910.2, 916.6,928.8, 935, 941.2, 947.4, 953.6,959.8, 966, 972.2, 978.4];(%i6) m:[3, 1, 2, 7, 8, 10, 4, 2, 4, 1, 1, 1];(%i7) n:sum(m[i],i,1,k);(%o7) 44(%i8) Mx:sum(m[i]*x[i], i, 1, k)/n;(%o8) 938.693(%i9) S2:sum(m[i]*(x[i]-Mx)^2, i, 1, k)/(n-1);(%o9) 282.988/* Èç òàáëèöû Ïðèëîæåíèå 3 ïðè k = 43 è a = 0,05 ïîëó÷àåì çíà÷åíèå t*/(%i10) t:2.015;(%i11) eps:t*sqrt(S2/n);(%o11) 5.11014(%i12) xLeft:Mx-eps; xRight:Mx+eps;(%o12) 933.583(%o13) 943.803Îòâåò: (933,583; 943,803).M x :=Ñàìîñòîÿòåëüíàÿ ðàáîòàÏðàêòè÷åñêîå çàíÿòèå 94.
Òî÷å÷íûå è èíòåðâàëüíûå îöåíêè53 òàáëèöå 94.1 äàíû 30 âàðèàíòîâ çàäàíèé. Ïî äàííûì â òàáëèöåðåçóëüòàòàì èçìåðåíèé íàéòè äîâåðèòåëüíûå èíòåðâàëû äëÿ îöåíêèíåèçâåñòíîãî ìàòåìàòè÷åñêîãî îæèäàíèÿ a íîðìàëüíîãî ðàñïðåäåëåíèÿ ñ çàäàííîé íàäåæíîñòüþ γ .Óêàçàíèå. Âûáîðî÷íîå ñðåäíåå è èñïðàâëåííîå âûáîðî÷íîå ÑÊÎíàõîäèòü ñîîòâåòñòâåííî ïî ôîðìóëàì (93.1) è (93.9), äîâåðèòåëüíûéèíòåðâàë ïî ôîðìóëå (93.15).èçì.123456789101112131415âàð.
\4,504,4330,1080,205,4014,2820,1236,4114,4680,3015,3816,065,907,7115,2114,514,4130,4080,105,4114,2620,1136,4214,4580,3015,4016,205,887,7315,2824,524,3930,3080,305,4014,2720,1036,4414,4180,2015,4116,165,977,7015,2534,534,4530,0079,705,4214,3020,1036,4514,4080,3015,4216,005,957,7215,2444,544,4029,4579,805,3914,3119,9836,4814,4280,2015,4316,155,937,6815,1754,494,3529,6579,805,3814,3219,9736,4914,4879,8015,3716,055,857,6715,1864,544,4230,0580,105,3814,3120,0236,4614,5079,8015,3616,015,887,6515,2074,474,4030,1580,005,3714,2920,0336,4514,4279,7015,3616,035,877,6315,19894,494,3729,9079,705,3514,3020,0236,4214,4579,9015,3416,025,877,7015,26Èñõîäíûå äàííûå äëÿ ñàìîñòîÿòåëüíîé ðàáîòû4,464,3830,0080,305,4014,2620,1036,3814,4480,5015,3316,025,907,7115,2210íàä¼æíîñòü γ0,950,990,9990,950,990,9990,9990,990,950,950,990,9990,950,990,999Òàáëèöà 94.154Ïðàêòè÷åñêîå çàíÿòèå 94.
Òî÷å÷íûå è èíòåðâàëüíûå îöåíêèèçì.161718192021222324252627282930âàð. \2,584,3814,4110,255,9716,0242,803,4436,408,3515,2827,907,715,4128,7012,504,4014,4510,235,8816,0342,723,4736,708,4015,2527,307,725,4028,3022,574,3714,4610,245,9016,0242,753,3836,908,3815,2126,907,705,4228,8032,494,4214,4010,185,9316,0542,903,3936,808,4415,2427,307,735,4028,8042,494,3514,4810,175,9516,0142,983,4636,308,4515,1827,407,675,3928,0052,514,4014,4210,185,8816,1542,853,4936,708,4415,1727,507,685,3828,1062,554,3914,5010,195,8516,0042,073,3936,908,3715,2027,007,655,3727,9072,534,4514,4510,205,8716,1642,933,4736,908,3915,1927,607,635,3828,70892,534,4314,4210,195,9016,2042,773,4636,408,3615,2627,707,715,3528,10Èñõîäíûå äàííûå äëÿ ñàìîñòîÿòåëüíîé ðàáîòû2,554,4114,4410,175,8716,0642,833,4536,008,4215,2227,407,705,4028,2010íàä¼æíîñòü γ0,950,990,9990,9990,990,950,950,990,9990,950,990,9990,950,990,999Òàáëèöà 94.2Ïðàêòè÷åñêîå çàíÿòèå 94.
Òî÷å÷íûå è èíòåðâàëüíûå îöåíêè5556Ëåêöèÿ 95. Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåçËåêöèÿ 95. Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåçÎñíîâíûå ïîíÿòèÿ. Ïðîâåðêà ãèïîòåçû î çíà÷èìîñòè âûáîðî÷íîãî êîýôôèöèåíòà êîððåëÿöèè. Ñðàâíåíèå äâóõ ìàòåìàòè÷åñêèõ îæèäàíèé. Ñðàâíåíèå ìàòåìàòè÷åñêîãî îæèäàíèÿ ñ çàäàííûì çíà÷åíèåì. Ñðàâíåíèå âåðîÿòíîñòè ñ çàäàííûì çíà÷åíèåì95.1. Îñíîâíûå ïîíÿòèÿ ïðîâåðêè ñòàòèñòè÷åñêèõ ãèïîòåç95.1. Ñòàòèñòè÷åñêîé íàçûâàåòñÿ ãèïîòåçà î âèäå ðàñïðåäåëåíèÿ èëè î çíà÷åíèÿõ åãî ïàðàìåòðîâ.ÎïðåäåëåíèåÃèïîòåçû áóäåì îáîçíà÷àòü H0 , H1 , H2 , . . .
.Ðàçëè÷àþò ïðîâåðÿåìóþ èëè îñíîâíóþ ãèïîòåçó H0 è àëüòåðíàòèâíóþ èëè êîíêóðèðóþùóþ H1 , êîòîðàÿ äîëæíà ïðîòèâîðå÷èòü îñíîâíîé.Ïðèìåð 95.1. Ïðîâåðÿåìàÿ ãèïîòåçà H0 ñîñòîèò â òîì, ÷òî ìàòåìàòè÷åñêîå îæèäàíèå ñëó÷àéíîé âåëè÷èíû ξ ðàâíî çàäàííîìó çíà÷åíèþ a0 . H0 : M (ξ) = a0 . Àëüòåðíàòèâíàÿ H1 : M (ξ) > a0 .Äëÿ ïðîâåðêè ñòàòèñòè÷åñêîé ãèïîòåçû íà îñíîâàíèè âûáîðêèx1 , x2 , . . .
, xn âû÷èñëÿþò çíà÷åíèå êðèòåðèÿ, çàâèñÿùåãî îò íàáëþäåíèé:T = T (x1 , x2 , . . . xn ).Âñ¼ ìíîæåñòâî çíà÷åíèé êðèòåðèÿ äåëèòñÿ íà òàê íàçûâàåìóþêðèòè÷åñêóþ îáëàñòü, ïðè ïîïàäàíèè â êîòîðóþ êðèòåðèÿ ïðîâåðÿåìàÿ ãèïîòåçà îòâåðãàåòñÿ, è îáëàñòü ïðèíÿòèÿ ãèïîòåçû.Ïðè ïðèíÿòèè ðåøåíèÿ î ñïðàâåäëèâîñòè ãèïîòåçû H0 âîçìîæíûñëåäóþùèå îøèáêè: ãèïîòåçà H0 îòâåðãàåòñÿ, õîòÿ íà ñàìîì äåëå îíà âåðíà (îøèáêà ïåðâîãî ðîäà) ; ãèïîòåçà H0 ïðèíèìàåòñÿ, õîòÿ íà ñàìîì äåëå îíà íå âåðíà, àñïðàâåäëèâà ãèïîòåçà H1 (îøèáêà âòîðîãî ðîäà) .Íàðÿäó ñ ýòèì âîçìîæíû ñëåäóþùèå ïðàâèëüíûå ðåøåíèÿ: ãèïîòåçà H0 ïðèíèìàåòñÿ è îíà äåéñòâèòåëüíî âåðíà; ãèïîòåçà H0 îòâåðãàåòñÿ è íà ñàìîì äåëå ñïðàâåäëèâà ãèïîòåçà H1 .Ëåêöèÿ 95.
Ïðîâåðêà ñòàòèñòè÷åñêèõ ãèïîòåç57Îïðåäåëåíèå 95.2. Âåðîÿòíîñòü îøèáêè ïåðâîãî ðîäà íàçûâàåòñÿ óðîâíåì çíà÷èìîñòè êðèòåðèÿ è îáû÷íî îáîçíà÷àåòñÿ α.Âåðîÿòíîñòü ïðàâèëüíî îòâåðãíóòü ïðîâåðÿåìóþ ãèïîòåçó íàçûâàåòñÿ ìîùíîñòüþ êðèòåðèÿ è îáû÷íî îáîçíà÷àåòñÿ β , òîãäà âåðîÿòíîñòüîøèáêè âòîðîãî ðîäà ðàâíà 1 − β .Îäíîâðåìåííî óìåíüøèòü âåðîÿòíîñòè îøèáîê ïåðâîãî è âòîðîãîðîäà ìîæíî òîëüêî óâåëè÷èâ îáú¼ì âûáîðêè n. Ïðè ôèêñèðîâàííîìn îáû÷íî çàäàþò äîïóñòèìûé óðîâåíü îøèáêè ïåðâîãî ðîäà α è ñòàðàþòñÿ ìèíèìèçèðîâàòü âåðîÿòíîñòü îøèáêè âòîðîãî ðîäà 1 − β , ò.å.ìàêñèìèçèðîâàòü ìîùíîñòü êðèòåðèÿ β .Íà ïðàêòèêå ïðè ïðîâåðêå ñòàòèñòè÷åñêîé ãèïîòåçû íà îñíîâàíèèíàáëþäåíèé âû÷èñëÿþò íàáëþäàåìîå çíà÷åíèå êðèòåðèÿ Tíàáë è ïîçàäàííîìó óðîâíþ çíà÷èìîñòè α îïðåäåëÿþò ãðàíèöû êðèòè÷åñêîéîáëàñòè êðèòè÷åñêèå òî÷êè.Åñëè êðèòè÷åñêàÿ îáëàñòü ïðàâîñòîðîííÿÿ, ò.å.
(têð2 ; +∞), ïðèâûïîëíåíèè óñëîâèÿ Tíàáë > têð2 äåëàþò âûâîä: ïðîâåðÿåìàÿ ãèïîòåçàH0 îòâåðãàåòñÿ ñ óðîâíåì çíà÷èìîñòè α â ïîëüçó ãèïîòåçû H1 ; åñëè ýòîóñëîâèå íå âûïîëíÿåòñÿ, ò.å. Tíàáë 6 têð2 , äåëàþò áîëåå îñòîðîæíûéâûâîä: íåò îñíîâàíèé äëÿ òîãî, ÷òîáû îòâåðãíóòü ãèïîòåçó H0 â ïîëüçóãèïîòåçû H1 ñ óðîâíåì çíà÷èìîñòè α.Åñëè êðèòè÷åñêàÿ îáëàñòü ëåâîñòîðîííÿÿ, ò.å. (−∞; têð1 ), ãèïîòåçàH0 îòâåðãàåòñÿ ïðè âûïîëíåíèè óñëîâèÿ Tíàáë < têð1 .  ñëó÷àå äâóñòîðîííåé êðèòè÷åñêîé îáëàñòè âèäà (−∞; têð1 )∪(têð2 ; +∞) ãèïîòåçàH0 îòâåðãàåòñÿ ïðè âûïîëíåíèè óñëîâèÿ Tíàáë < têð1 èëè Tíàáë > têð2 .95.2.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
