Solutions to the Spring 2001 Stat 402 Final Exam 1. a) SOURCE DF garden 1 rate 2 <--- These three lines would add to garden*rate 2 model d.f. of 5 The error would have 18-6=12 d.f. b) yhat=-1.11+3.69+-.58*rate+1.33*rate=2.58+0.75*rate c) yhat=-1.11+1.33*rate d) t=-2.38 or F=5.69 p-value=0.0318 e) garden 1: 2.58+0.75*7=7.83 garden 2: -1.11+1.33*7=8.2 Garden 2 would have the higher expected yield. 2 a) F=11.08 p-value=0.0076 There was a significant difference between soil types. b) Soil 1: (8+3.5+10)/3=7.1667 Soil 2: (6+2+7.5)/3=5.1667 c) Potting soil 2 because it had the smallest lsmean and that mean was significantly less than the lsmean for potting soil 1 (p-value=0.0076). d) (8+6)/2 - (3.5+2)/2 + or - 2.228*sqrt[(0.5^2)/2+(0.5^2)/4+(0.5^2)/4+(0.5^2)/1] 2.67 to 5.83 e) Variety 1 took significantly longer to germinate than variety 2. 3. a) Latin square 12345 23451 34512 45123 41234 b) Because of the breaks over the weekend, Mondays are likely to be more similar than days in general. The same goes for Tuesdays, Wednesdays, etc. Thus it is important to construct blocks according to day of the week. (For example, the 5 Mondays are one block, the 5 Tuesdays are another block, etc.) Performance could also vary with week. For example, the professor could get a cold that would reduce performance for a whole week. The professor might get tired by week 5 or perhaps stronger. Thus it is also important to consider the days in one week a block. 4. a) Use a randomized complete block design (RCBD) with the feed lots as blocks. For example... FEED PEN LOT 1 2 3 4 5 1 A C D E B 2 D B A E C 3 A D E C B 4 E D C B A b) pens c) steers d) SOURCE DF lot 3 diet 4 lot*diet 12 steer(lot diet) 100 c.total 119 e) y_ijk=mu+L_i+d_j+(LD)_ij+e_ijk y_ijk is ribeye area of the kth steer at feed lot i that received diet j. mu is overall mean L_i are random effects for feed lots (i=1 to 4) d_j are fixed effects for diets (j=1 to 5) (LD)_ij are random effects for lot*diet interaction they correspond to pens e_ijk are random effects for steers in pens (k=1 to 6) All random effects are normally distributed and independent with mean 0. The L_i have one common variance component. The (LD)_ij have one common variance component. The e_ijk have one common variance component. proc glm; class lot diet; model y=lot diet lot*diet; random lot lot*diet; run; proc mixed method=type3; class lot diet; model y=diet; random lot lot*diet; run; 5. a) b) SOURCE DF time 1 bank(time) bank(time) 8 random grass 3 grass*bank(time) time*grass 3 grass*bank(time) grass*bank(time) 24 random c.total 39 c) (i) numerator=3, denominator=24 (ii) There was no significant difference among grasses at one month, but there are significant differences among grasses at three months. This makes sense as it probably takes awhile for the grasses to appear.