Problem Set 5 (tutorial 5: s2/02) — page 1 620-270 Applied Statistics Problem Set 5 (tutorial 5: s2/02) If you have time after ﬁnishing lab 5, the random allocations in Question 5.2 could be done using minitab. Answers to this Problem Set are on the Web. 5.1 Four varieties of pine trees were to be compared and 5 observations of each variety (volume of wood produced) would be taken. Each of the situations below could have given 20 values as desired. For each situation, say if it is a completely randomized one-factor design, a randomized block design or neither. For those that are neither, describe in one sentence what is wrong with them, using some of the terms blocking, randomization, fake replication, confounding. (a) Four ﬁelds were each divided into 5 plots and variety A was planted in the 5 plots in the ﬁrst ﬁeld, variety B in the 5 plots in the second ﬁeld, and so on. (b) 20 plots of land were roped oﬀ in a given area and 5 were randomly selected to plant with A, 5 others to plant with B, and so on. (c) 20 plots of land were roped oﬀ in a given area and the plots were grouped so that the four in each group were similar to each other in terms of soil type. The northernmost plot of each group of four was planted with B (B was randomly selected from the four possible varieties), the next most northern plot with D (similarly randomly selected), and so on. (d) Five ﬁelds were each divided into four plots and, in each ﬁeld, one of the four was randomly chosen to be planted with A, another with B and so on. 5.2 In an experiment to study the eﬀect of three tray types (A, B and C) on the drying time of sultanas, the setup below was used. The 9 individual drying trays are stacked in three racks of three trays each as shown. ---------------------Rack 1 ---------------------Rack 2 ---------------------Rack 3 Use the random numbers below to assign the three tray types to positions to achieve (a) a completely randomized design; (b) a randomized block design, blocking for rack; (c) a randomized block design, blocking for vertical position; (d) a Latin square design. Start a new row of random numbers for each design (this is not necessary for a good design, but will enable you to compare your designs with those in the answers). Random numbers: 9 0 8 0 6 4 9 8 2 4 4 7 9 7 2 0 8 7 6 2 0 6 1 4 3 8 8 2 6 1 2 5 3 2 3 9 0 3 3 6 1 4 9 6 5 3 2 1 8 4 6 7 5 6 8 9 1 9 1 5 0 5 9 6 7 1 6 3 8 2 0 4 5 3 7 6 0 3 8 7 3 5 1 1 1 9 9 2 7 5 2 3 7 8 8 9 5.3 An experiment is conducted to compare four treatments (1, 2, 3 and 4) for hardening the hooves of sheep. You have available 12 pens in a paddock which have been fenced oﬀ from one another. Four sheep are placed in each pen. For each situation below: i. State what the experimental unit is, and how many there are; ii. State what type of design is being used; iii. Comment on any design ﬂaws. Problem Set 5 (tutorial 5: s2/02) — page 2 (a) The hardening treatments are given by injection. In each pen, one sheep is randomly chosen to receive injection 1, another to receive injection 2, etc. The average hardness of the four hooves is recorded and analysed (one number per sheep). (b) The hardening treatments are given as in (a). The hardness of the front left hoof of each sheep is recorded and analysed. (c) The hardening treatments are given as in (a). The four hardness values (one for each hoof) are recorded and analysed (four numbers per sheep). (d) The hardening treatments are given by injection, with treatment 1 applied to the ﬁrst sheep the experimenter sees upon entering the pen, treatment 2 to the second, etc. The average hardness of the four hooves is recorded and analysed. (e) The hardening treatments are given by applying salve to the hoof. For each sheep, one hoof is randomly chosen to get salve 1, another to get salve 2, etc. The hardness of each hoof is recorded and analysed. (f) The hardening treatments are applied in the feed put in a pen. Three of the 12 pens are randomly chosen to get feed 1, three to get feed 2, etc. The average hardness of all 16 hooves in the pen is recorded (one number for each pen) and analysed. (g) The hardening treatments are applied in the feed put in a pen. The three pens closest to the door are given feed 1, the next three feed 2, etc. The average hardness of all 16 hooves in the pen is recorded (one number for each pen) and analysed. 5.4 Thirty-two plots are available for an experiment examining how well fungicides can control the disease “powdery mildew” on tomato plants. The plots, each of which comprises 6 tomato plants, are laid out in a 4 × 8 rectangle, as shown in the diagram. The land slopes down steeply from left to right, with the plots at the bottom (i.e. to the right in the diagram) more likely to accumulate water and encourage disease in the plants. Four chemical treatments are being compared in this experiment: (1) Flosul; (2) Cosavet; (3) Flosul and Cosavet together; (4) no chemical (i.e. a control). (a) How would you design the experiment? Give an example of how the treatments might be allocated (you don’t need to actually perform the randomisation). (b) Suppose that during the experiment, the eight plots closest to the bottom of the slope become waterlogged and all the tomato plants in them die. Explain how your design copes with this diﬃculty. (c) Suppose that the plants at the top of the diagram are more in the sun, and those at the bottom are more shaded. How would you design the experiment? Give an example of how the treatments might be allocated. (d) Instead of expressing the treatments as four separate treatments, express them as a factorial treatment structure. (e) Now suppose that there is another treatment factor to test, which is a growth stimulant applied at either a low or high rate. The low and high rates are to be applied for each of the four chemical treatments. Express the treatments as a factorial treatment structure, and list all the treatment combinations. (f) Ignoring the sun factor mentioned in (c), how would you design the experiment?