STA 6166 UNIT 5 Section 2 Exercises
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# Unit 5 Section 2 Exercises

Note: After the general questions I have provided practice analyses for the majority of designs covered in this chapter. Note that each subject area has a different design. If you are interested in a particular design, work that particular problem. If I had provided all possible combinations of subject areas (4) by design types (3) you would be working 12 examples. Answers to the questions can be found here(answers).

 General Questions. What are the advantages of a Completly Randomized Design (CRD)? What are the disadvantages of a CRD? What are the advantages of using a Randomized Complete Block Design (RCBD)? What are the disadvantages of using a RCBD? If multiple replicates of each treatment can be accommodated in each block of a blocked design which disadvantages of a RCBD are removed? What are the advantages of a Latin Square Design (LSD)? What are the disadvantages of a LSD? An experiment is to be designed to explore the impact of two factors (call them A and B) on a response. Suppose one factor (A) is to be examined at 5 levels and the other factor (B) is to be examined at 5 levels, how many treatments must the experiment accommodate? For question 8, how many experimental units must we have if we wish 4 replicates of the complete set of treatments? For question 8, how many experimental units must each block contain if you wish to use a RCBD for our experiment? For question 8, how many experimental units must we have if we wish to implement a Latin Square Design using factor A for the row factor and another factor, factor C for the column factor? For question 8, how many experimental units must we have if we wish to implement a Latin Square Design for our treatments but now the row factor is a fourth factor, factor D, and the column factor is still factor C, and the treatment is the factorial in factors A and B? For students in agriculture and environmental fields. A Randomized Block Design Six greenhouse benches were set up as blocks. Within each block one of each of four varieties of house plants was planned. The plant heights (in cm) obtained are tabulated as follows. Using the ANOVA for a RCBD, test the hypothesis that all four varieties of plants reach the same maximum height at measurement time. ```Block Var_1 Var_2 Var_3 Var_4 1 19.8 21.9 16.4 14.7 2 16.7 19.8 15.4 13.5 3 17.7 21.0 14.8 12.8 4 18.2 21.4 15.6 13.7 5 20.3 22.1 16.4 14.6 6 15.5 20.8 14.6 12.9 ``` Note that to get any statistics package to perform the analysis, these data will have to be stacked. For students in engineering fields. Randomized Block Design with Replication An experiment was designed to examine the effectiveness of three types of gasoline additives on boosting the miles per gallon on a specific type of car. Three randomly selected cars (Block factor at levels Car_1, Car_2 and Car_3) of the same model/year were purchased direct from the factory. Each additive (Treatment factor at levels A, B and C) was tested four times on each of the three cars. The assignment of additive by replicates were randomly assigned within each car. The same driver performed all tests on the same track under controlled conditions of temperature and humidity. The data follow. Determine whether there is a significant difference in the average gas milage for the three additives. ``` Car Additive Rep MPG CAR_1 A 1 18.24 CAR_1 A 2 18.73 CAR_1 A 3 19.17 CAR_1 A 4 19.25 CAR_1 B 1 17.95 CAR_1 B 2 18.07 CAR_1 B 3 18.07 CAR_1 B 4 17.64 CAR_1 C 1 16.92 CAR_1 C 2 16.91 CAR_1 C 3 16.39 CAR_1 C 4 16.25 CAR_2 A 1 21.97 CAR_2 A 2 21.78 CAR_2 A 3 21.93 CAR_2 A 4 21.80 CAR_2 B 1 21.35 CAR_2 B 2 21.28 CAR_2 B 3 20.72 CAR_2 B 4 20.99 CAR_2 C 1 19.75 CAR_2 C 2 19.43 CAR_2 C 3 19.18 CAR_2 C 4 20.00 CAR_3 A 1 21.32 CAR_3 A 2 21.96 CAR_3 A 3 22.14 CAR_3 A 4 22.02 CAR_3 B 1 20.54 CAR_3 B 2 20.64 CAR_3 B 3 21.57 CAR_3 B 4 21.15 CAR_3 C 1 19.38 CAR_3 C 2 19.20 CAR_3 C 3 19.81 CAR_3 C 4 19.14 ``` For students in toxicology and health science fields. Two Factor Factorial in a CRD You are interested in the effect on biomass reductions in lettuce shoots after exposure to a pesticide. You further suspect that the temperature at time of application may also have an impact. Starting with 42 lettuce plants you randomly assign them to treatments constructed as the combination of Pesticide concentration (%) at 7 levels and Temperature applied at three levels, with two replicates for each treatment combination. Biomass is reported as a natural log of biomass to account for suspected heterogeneity of variances. Using the data below, determine whether there are main effects due to pesticide concentration or temperature and whether there is an interaction between the two. [This is an example of stacked data ready for running in most statistics packages that will perform a two way analysis of variance.] ``` Conc Temp Rep Ln_Biomass 0 10 1 0.343 0 10 2 1.511 0 15 1 0.140 0 15 2 1.456 0 20 1 0.530 0 20 2 1.099 0.33 10 1 -0.246 0.33 10 2 1.049 0.33 15 1 1.140 0.33 15 2 0.617 0.33 20 1 0.436 0.33 20 2 0.462 0.5 10 1 -0.333 0.5 10 2 0.356 0.5 15 1 0.294 0.5 15 2 0.008 0.5 20 1 0.518 0.5 20 2 0.628 1 10 1 -0.843 1 10 2 -1.680 1 15 1 0.352 1 15 2 0.043 1 20 1 -0.296 1 20 2 -0.206 2 10 1 -0.561 2 10 2 -0.629 2 15 1 0.285 2 15 2 -0.284 2 20 1 -0.102 2 20 2 -0.421 5 10 1 -1.348 5 10 2 -0.232 5 15 1 -0.550 5 15 2 -1.057 5 20 1 -0.079 5 20 2 0.234 10 10 1 -1.809 10 10 2 -1.617 10 15 1 -1.266 10 15 2 -0.066 10 20 1 -0.378 10 20 2 -0.618``` For students in community development, education and social services fields. Latin Square Design Below is a hypothetical experiment on the effect of various types of background music on the productivity of workers on an electronics assembly line. It is well known that productivity differs among various days of the week as well as during certain times of the day. Hence we design the experiment with the column factor being day of the week and the row factor being times of the day. The treatment are five "types" of music. A: Rock and Roll B: Country/Western C: Easy listening D: Classical E: No music Productivity is measured by the average number of correctly working components per 15 minutes that exit the line during the test period. The data for the analysis are presented below in stacked format. ``` Times Day Music Components 9_10 Monday C 10.5 10_11 Monday B 10.3 11_12 Monday D 8.6 1_2 Monday A 7.3 2_3 Monday E 8.1 9_10 Tuesday B 11.4 10_11 Tuesday A 6.5 11_12 Tuesday C 12.1 1_2 Tuesday E 11.7 2_3 Tuesday D 9.6 9_10 Wednesday E 10.4 10_11 Wednesday D 14.0 11_12 Wednesday A 11.3 1_2 Wednesday C 16.0 2_3 Wednesday B 12.0 9_10 Thursday A 8.0 10_11 Thursday E 11.4 11_12 Thursday B 12.6 1_2 Thursday D 16.0 2_3 Thursday C 12.7 9_10 Friday D 9.3 10_11 Friday C 12.5 11_12 Friday E 10.6 1_2 Friday B 13.0 2_3 Friday A 8.0 ``` ```  ```