STAT 511 FINAL EXAM NAME Spring 2001

Transcription

1 STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte your answers on addtonal sheets of paper. There s not enough room to wrte your answers on ths exam. Please wrte your name on every sheet of paper that you submt. Submt ths copy of the exam wth your solutons.. In a study of the amno acd phenylalanne 7 young male rats weghng 5 to 55 grams were fed a det lackng phenylalanne for two days. Then the rats were randomly assgned to new dets wth three rats assgned to each level of phenylalanne. The weght gan (or weght loss) after 5 days on the new det was recorded for each rat. The results are shown n the followng table. Negatve values for weght gan ndcate a weght loss. Percent Phenylalanne n the new det Weght Gan (grams) Least squares estmaton was used to ft the followng model to the data β β exp( β - β X ) 3 ε where s the observed 5 day weght gan for the j-th rat fed a new det contanng the percent phenylalanne gven by X and the ε s are ndependent and dentcally dstrbuted random errors wth mean zero and varance σ.

2 The estmated coeffcents are gven below along wth an estmate of the varance covarance matrx for the estmated coeffcents. b ~ Øb b b º b 3 ß Ø º 4.79ß V Ø º ß A graph showng the data and the estmated curve s shown on the next page along wth some resdual plots. A. Wth respect to the relatonshp between mean weght gan n rats and the level of phenylalanne n ther det gve an nterpretaton of the parameters β β and β n ths model. ou are not asked to gve a precse nterpretaton of β 3 whch s a coeffcent related to the rate at whch mean weght gan ncreases as the level of phenylalnne ncreases. B. Assumng that the proposed model s correct show how to construct an approxmate 95% confdence nterval for β. C. Assumng that the proposed model s correct show how to use b V and the delta method construct an approxmate 95% confdence nterval for the level of phenylalanne n the det correspondng to a 5 day mean weght gan of zero. Ths s the level of phenylalnne at whch an average rat wll nether gan nor lose weght. (ou are not requred to complete the numercal evaluaton of the confdence lmts.) D. Assumng that the proposed model s correct explan how you would use a bootstrap procedure to construct a confdence nterval for the mean 5 day weght gan for rats wth. percent phenylalanne n ther det. Gve only one answer correspondng to what you thnk s the best way to use the bootstrap. (If you gve more than one answer you wll receve credt only for your weakest response.) E. Examne the resdual plots. What do they suggest? F. Show how to construct a test statstc to test the null hypothess that the condtonal mean for the 5 day weght change corresponds to the proposed β formula E( X) β +. Ether present a formula for a + exp( β - β3 X) test statstc and show how to use t or descrbe what you would do. ou do not have to provde any theoretcal justfcaton for your answer.

3 3. A certan bonsectcde s used to control nsects that lve n sol and damage the roots of plants. Ths bonsectcde conssts of spores of a certan fungus that are suspended n a soluton. After they are ntroduced nto the sol the spores develop nto fungal colones that nfect and destroy the nsects. The object of one study was to examne how dfferent levels of rrgaton affect the vablty of spores n the sol. Ten dfferent felds were used n ths study. These felds were randomly selected from a large set of felds that could have been used n the study. Each feld was subdvded nto three plots of equal sze. Each plot was sprayed wth a concentraton of the bonsectcde that deposted about 5 x 8 spores per cm of surface area. One plot n each feld was rrgated wth cm of water per day. Another plot n each feld was rrgated wth. cm of water per day. The remanng plot n each feld receved no rrgaton. The three levels of rrgaton were randomly assgned to the plots wth a dfferent randomzaton wthn each feld. Sol samples were taken from each plot at 5 and days after the plot was sprayed wth the bonsectcde and the number fungal colones () was measured for each sol sample. Consder the model k µ + β + α + δ + τ + γ + j k jk ε k where k s an observaton on the number of fungal colones n a sol sample α j corresponds to the j-th level of rrgaton τ k corresponds to the k-th samplng date γ jk corresponds to the j-th level of rrgaton and the k-th samplng date β s a random feld (block) effect wth β ~ NID( ) δ s a random effect wth δ ~ NID( ) σ p σ f ε s a random effect wth ε ~ NID( ) k k σ e and β δ and ε k are mutually ndependent.

4 4 A. An analyss of varance table for ths model ncludes the followng sources of varaton: Source of Varaton Degrees of Freedom Expected Mean Square Felds σ e + 4σp + σf Plots wthn felds σ + 4σ Error σ e e p Report the approprate degrees of freedom and use these results to obtan formulas for estmates of the varance components. B. For the proposed model show how you would test the null hypothess that there s no nteracton between the levels of the rrgaton factor and tme (sample dates). Gve a formula for your test statstc and explan how you would use t to make an nference about the potental nteracton between levels of rrgaton and tme. C. Suppose the researchers want to compare the results at days for the rrgaton levels correspond to cm of water per day and cm of water per day. Averagng across felds let 4 4 denote the average of the observatons at days usng cm of water per day and let denote the average of the observatons at days usng cm of water per day. Fnd the varance of 4-34 and show how to construct a 95% confdence nterval for the dfference n the mean results at days for rrgaton levels of cm and cm of water per day. D. Whch of the followng are true statements about the proposed model? Crcle any statement that s true.. Each observaton has the same varance.. Any two observatons taken n the same feld have the same level of correlaton.. Two observatons taken from two dfferent felds are uncorrelated. v. The REML estmates of the varance components are equal to the estmates of the varance components that you were asked to derve n part (A). α + γ - α + s estmable. v. ( ) ( ) 4 3 γ 34 E. Suppose that one of the objectves of the study was to show that there s a tme effect when plots are rrgated wth cm of water per day. Consequently the researchers want to test the alternatve that the mean results dffer for at least two tme ponts. Show how to construct an F-test for ths null hypothess.

5 5 3. A second model was proposed for the data n problem. Usng the notaton from problem ths model s k µ + β + α + δ + τ + j k k where β s a random feld (block) effect wth β ~ NID( σ f ) and the k s are random effects that are ndependent of the β s and Ø º 3 4 ß ~ Ø Ø σ ρ σσ ρ σσ3 ρ σσ NID ρ σσ σ ρ σσ3 ρ σσ4 9 5 ρ σ σ3 ρ σσ3 σ3 ρ σ3σ4 Ł º ß 9 5 º ρ σσ4 ρ σσ4 ρ σ3σ4 σ4 ßł A. Explan how ths model dffers from the model proposed n problem. B. Explan how you would perform tests to determne f ether ths model or the model from problem was approprate for the data. Report the degrees of freedom for any test you propose and explan how you would use t to make a decson. Fnal Exam Score Course Grade