Exam+solution - Háskóli Íslands Raunvísindadeild...

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Unformatted text preview: Háskóli Íslands Raunvísindadeild Staerðfraeðiskor 09.10.25 Líkindareikningur og tölfraeði Mánudagurinn 17. ágúst 2005 Lausnir Kennarar: Dr. Birgir Hrafnkelsson lektor Verkfraeðideild Dr. Hermann Þórisson prófessor Raunvísindadeild Yfirfarið af Þorsteini Þorsteinssyni 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 gufuþrýsting vatns (mm Hg) og hitastig ( ◦ K). Maeling Hitastig Gufuþrýstingur i x (K) y (mm Hg) 1 273 4,6 2 283 9,2 3 293 17,5 4 303 31,8 5 313 55,3 6 323 92,5 7 333 149,4 8 343 233,7 9 353 355,1 10 363 525,8 11 373 760,0 Svo virðist sem líkan á forminu Y i = κe λx- 1 i + φ i , φ i ∼ N (0 ,τ 2 ) , i = 1 ,..., 11 , þar sem κ > , λ ∈ < og τ > , lýsi gögnunum viðunandi vel. Nokkrar reiknistaerðir byggðar á v i = ln( y i ) í stað y i og u i = x- 1 i í stað x i , i = 1 ,..., 11 , eru: n = 11 , ¯ u = 0 , 003126 , ¯ v = 4 , 3489 , S uu = n X i =1 ( u i- ¯ u ) 2 = 1 , 0541 * 10- 6 , S vv = n X i =1 ( v i- ¯ v ) 2 = 28 , 5149 , S uv = n X i =1 ( u i- ¯ u )( v i- ¯ v ) =- , 005482 , SS R,uv = ( S uu S vv- S 2 uv ) /S uu = 0 , 003846 . 1. Reiknið 95% öryggisbil fyrir gufuþrýstinginn, κe λx- 1 , þegar hitastigið er x = 328 ◦ K. Ábending : Reiknið fyrst öryggisbil fyrir logrann af gufuþrýstinginum. 2. Prófið núlltilgátuna um að λ =- 5200 á móti gagntilgátunni um að λ 6 =- 5200 . Notið marktektarkröfu α = 0 , 01 . 2 1. ( 16 , 67 %) (English version) The following data are on vapor pressure (mm Hg) and temperature ( ◦ K). Obs. Temperature Vapor pressure i x (K) y (mm Hg) 1 273 4.6 2 283 9.2 3 293 17.5 4 303 31.8 5 313 55.3 6 323 92.5 7 333 149.4 8 343 233.7 9 353 355.1 10 363 525.8 11 373 760.0 We can assume that the model Y i = κe λx- 1 i + φ i , φ i ∼ N (0 ,τ 2 ) , i = 1 ,..., 11 , where κ > , λ ∈ < and τ > , describes the data adequately well. A few summary statistics based on v i = ln( y i ) instead of y i and u i = x- 1 i instead of x i , i = 1 ,..., 11 , are: n = 11 , ¯ u = 0 . 003126 , ¯ v = 4 . 3489 , S uu = n X i =1 ( u i- ¯ u ) 2 = 1 . 0541 * 10- 6 , S vv = n X i =1 ( v i- ¯ v ) 2 = 28 . 5149 , S uv = n X i =1 ( u i- ¯ u )( v i- ¯ v ) =- . 005482 , SS R,uv = ( S uu S vv- S 2 uv ) /S uu = 0 . 003846 . 1. Compute a 95% confidence interval for vapor pressure, κe λx- 1 , when the temperature is x = 328 ◦ K. Hint : Find first a confidence interval for the logarithm of the vapor pressure. 2. Test the null hypothesis that λ =- 5200 , versus the alternative hypothesis that λ 6 =- 5200 . Use a significance level α = 0 . 01 . 3 a) 100(1- α )% öryggisbil fyrir ln ( κ * e λ * x- 1 ) = ln ( κ ) + λ * x- 1 = ln ( κ ) + λ * u er: ln (ˆ κ ) + ˆ λ * u ± t α/ 2 ,n- 2 * S r,uv s 1 n + ( u- ¯ u ) 2 S uu ln (ˆ κ ) = ˆ κ = ¯ v- ˆ λ * ¯ u = F ˆ λ = S uv S uu =- . 005482 1 . 0541 * 10- 6 =- 5200 , 6451 F = 4 , 3489 + 5200 , 6451 * , 003126 = 20...
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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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