Exam+solution - Háskóli Íslands Raunvísindadeild...

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Unformatted text preview: Háskóli Íslands Raunvísindadeild Staerðfraeðiskor 09.10.25 Líkindareikningur og tölfraeði Fimmtudagurinn 29. apríl 2004 Lausnir Kennarar: Dr. Birgir Hrafnkelsson lektor Verkfraeðideild Dr. Hermann Þórisson prófessor Raunvísindadeild Yfirfarið af Atla Norðmann Sigurðssyni 20.janúar 2007 Framsett í L A T E Xaf Páli Jens Reynissyni Verkfraeðideild 27. janúar 2007 1 1 daemi (16.67%) Eftirfarandi gögn eru um þyngd (í tonnum) nokkurra bifreiða og eldsneytisnotkun þeirra í lítrum á 100 kílóme- tra. Bíll núm. Tegund Þyngd bíls Eldsneytisnotkun (tonn) ( x ) (lítrar/100 km) ( y ) 1 Buick Estate Wagon 4,360 13,979 2 Ford Country Squire Wagon 4,054 15,242 3 Chrysler LeBaron Wagon 3,940 11,468 . . . . . . . . . . . . 27 Datsun 810 2,815 10,739 28 BMW 320i 2,600 10,988 29 VW Rabbit 1,925 7,406 Svo virðist sem líkan á forminu Y i = γx θ i e φ i , φ i ∼ N (0 ,η 2 ) , i = 1 ,..., 29 , lýsi gögnunum viðunandi vel. Nokkrar reiknistaerðir byggðar á gögnunum eru: n = 29 , ¯ x = 2 , 753 , ¯ y = 9 , 869 , S xx = n X i =1 ( x i- ¯ x ) 2 = 11 , 936 , S yy = n X i =1 ( y i- ¯ y ) 2 = 202 , 11 , S xy = n X i =1 ( x i- ¯ x )( y i- ¯ y ) = 45 , 679 , SS R = ( S xx S yy- S 2 xy ) /S xx = 27 , 306 . Nokkrar reiknistaerðir byggðar á logranum af gögnunum, u i = ln( x i ) , v i = ln( y i ) , i = 1 ,..., 29 , eru: n = 29 , ¯ u = 1 , 0530 , ¯ v = 2 , 3248 , S uu = n X i =1 ( u i- ¯ u ) 2 = 1 , 6052 , S vv = n X i =1 ( v i- ¯ v ) 2 = 2 , 1399 , S uv = n X i =1 ( u i- ¯ u )( v i- ¯ v ) = 1 , 7036 , SS R,uv = ( S uu S vv- S 2 uv ) /S uu = 0 , 3320 . 1. Reiknið 95% öryggisbil fyrir γx θ þegar x = 2 , 2 tonn. Ábending : Reiknið fyrst öryggisbil fyrir ln( γx θ ) . 2. Prófið núlltilgátuna um að θ = 1 , á móti gagntilgátunni um að θ 6 = 1 . Notið marktektarkröfu α = 0 , 01 . 2 1.( 16.67%) (English version) The following data are on the weight(in tons) of several cars and their mileage in liters per 100 kilometers. Car no. Type of car Weight of car Mileage (tons) ( x ) (liters/100 km) ( y ) 1 Buick Estate Wagon 4.360 13.979 2 Ford Country Squire Wagon 4.054 15.242 3 Chrysler LeBaron Wagon 3,940 11,468 . . . . . . . . . . . . 27 Datsun 810 2.815 10.739 28 BMW 320i 2.600 10.988 29 VW Rabbit 1.925 7.406 It appears that a model of the form Y i = γx θ i e φ i , φ i ∼ N (0 ,η 2 ) , i = 1 ,..., 29 , fits the data adequately well. A few summary statistics are: n = 29 , ¯ x = 2 . 753 , ¯ y = 9 . 869 , S xx = n X i =1 ( x i- ¯ x ) 2 = 11 . 936 , S yy = n X i =1 ( y i- ¯ y ) 2 = 202 . 11 , S xy = n X i =1 ( x i- ¯ x )( y i- ¯ y ) = 45 . 679 , SS R = ( S xx S yy- S 2 xy ) /S xx = 27 . 306 . A few summary statistics based on the logarithm of the data, u i = ln( x i ) , v i = ln( y i ) , i = 1 ,..., 29 , are as follows: n = 29 , ¯ u = 1 . 0530 , ¯ v = 2 . 3248 , S uu = n X i =1 ( u i- ¯ u ) 2 = 1 . 6052 , S vv = n X i =1 ( v i- ¯ v ) 2 = 2 . 1399 , S uv = n X i =1 ( u i- ¯ u )( v i- ¯ v ) = 1...
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This homework help was uploaded on 04/19/2008 for the course VERKHI 09.10.25 taught by Professor Hþ during the Spring '08 term at Uni. Iceland.

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Exam+solution - Háskóli Íslands Raunvísindadeild...

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