exactly the same as the error sum of squares for the standard ANOVA. (a) True (b) False 3.66 Fisher's LSD procedure is an extremely conservative method for comparing pairs of treatment means following an ANOVA. (a) True (b) False 3.67 The REML method of estimating variance components is a technique based on maximum likelihood, while the ANOVA method is a method-of-moments procedure. (a) True (b) False 3.68 One advantage of the REML method of estimating variance components is that it automatically produces confidence intervals on the variance components. (a) True (b) False 3.63 The normality assumption is extremely important in the analysis of variance. (a) True (b) False 3.64 The analysis of variance treats both quantitative and qualitative factors alike so far as the basic computations for sums of squares are concerned. (a) True (b) False 3.65 If a single-factor experiment has a levels of the factor and a polynomial of degree a1 is fit to the experimental data, the error sum of squares for the polynomial model will be 3.69 The Tukey method is used to compare all treatment means to a control. (a) True (b) False 3.70 An experiment with a single factor has been conducted as a completely randomized design and analyzed using computer software. A portion of the output is shown below. (a) Fill in the missing information. (b) How many levels of the factor were used in this experiment? (c) How many replicates were used in this experiment? (d) Find bounds on the P-value. 3.71 The estimate of the standard deviation of any observation in the experiment in Problem 3.70 is (a) 7.03 (b) 2.65 (c) 5.91 (d) 1.95 (e) none of the above 3.5 The mean square for error in the ANOVA provides an estimate of (a) The variance of the random error (b) The variance of an individual treatment average (c) The standard deviation of an individual observation (d) None of the above 3.6 It is always a good idea to check the normality assumption in the ANOVA by applying a test for normality such as the Anderson-Darling test to the residuals. (a) True (b) False els. What conclusions can you draw from this simple example? 6.53 In two-level design, the expected value of a nonsignificant factor effect is zero. (a) True (b) False 6.54 A half-normal plot of factor effects plots the expected normal percentile versus the effect estimate. (a) True (b) False 6.10 Problems 307 6.55 In an unreplicated design, the degrees of freedom associated with the "pure error" component of error are zero. (a) True (b) False 6.56 In a replicated 23 design (16 runs), the estimate of the model intercept is equal to one-half of the total of all 16 runs. (a) True (b) False 6.57 Adding center runs to a 2k design affects the estimate of the intercept term but not the estimates of any other factor effects. (a) True (b) False 6.58 The mean square for pure error in a replicated factorial design can get smaller if nonsignificant terms are added to a model. (a) True (b) False 6.59 A 2k factorial design is a D-optimal design for fitting a first-order model. (a) True (b) False 6.60 If a D-optimal design algorithm is used to create a 12-run design for fitting a first-order model in three variables with all three two-factor interactions, the algorithm will construct a 23 factorial with four center runs. (a) True (b) False 6.61 Suppose that you want to replicate 2 of the 8 runs in a 23 factorial design. How many ways are there to choose the 2 runs to replicate? Suppose that you decide to replicate the run with all three factors at the high level and the run with all three factors at the low level. (a) Is the resulting design orthogonal? (b) What are the relative variances of the model coefficients if the main effects plus two-factor interaction model are fit to the data from this design? (c) What is the power for detecting effects of two standard deviations in magnitude? 6.62 The display below summarizes the results of analyzing a 24 factorial design. (a) Fill in the missing information in this table. (b) Construct a normal probability plot of the effects. Which factors seem to be active? 6.1 In a 24 factorial design, the number of degrees of freedom for the model, assuming the complete factorial model, is (a) 7 (b) 5 (c) 6 (d) 11 (e) 12 (f) none of the above 6.2 A 23 factorial is replicated twice. The number of pure error or residual degrees of freedom are (a) 4 (b) 12 (c) 15 (d) 2 (e) 8 (f) none of the above 6.3 A 23 factorial is replicated twice. The ANOVA indicates that all main effects are significant but the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a) 12 (b) 8 (c) 16 (d) 14 (e) 10 (f) none of the above 6.4 A 23 factorial is replicated three times. The ANOVA indicates that all main effects are significant but two of the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a) 12 (b) 14 (c) 6 (d) 10 (e) 8 (f) none of the above 5.1 An interaction effect in the model from a factorial experiment involving quantitative factors is a way of incorporating curvature into the response surface model representation of the results. (a) True (b) False 5.2 A factorial experiment may be conducted as a RCBD by running each replicate of the experiment in a unique block. (a) True (b) False 5.3 If an interaction effect in a factorial experiment is significant, the main effects of the factors involved in that interaction are difficult to interpret individually. (a) True (b) False 5.4 A biomedical researcher has conducted a two-factor factorial experiment as part of the research to develop a 4.1 Suppose that a single-factor experiment with four levels of the factor has been conducted. There are six replicates and the experiment has been conducted in blocks. The error sum of squares is 500 and the block sum of squares is 250 . If the experiment had been conducted as a completely randomized design the estimate of the error variance 2 would be. (a) 25.0 (b) 25.5 (c) 35.0 (d) 37.5 (e) None of the above 4.2 Suppose that a single-factor experiment with five levels of the factor has been conducted. There are three replicates and the experiment has been conducted as a complete randomized design. If the experiment had been conducted in blocks, the pure error degrees of freedom would be reduced by (a) 3 (b) 5 (c) 2 (d) 4 (e) None of the above 4.3 Blocking is a technique that can be used to control the variability transmitted by uncontrolled nuisance factors in an experiment. (a) True (b) False 172 Chapter 4 Randomized Blocks, Latin Squares, and Related Designs 4.4 The number of blocks in the RCBD must always equal the number of treatments or factor levels. (a) True (b) False 4.5 The key concept of the phrase "Block if you can, randomize if you can't." is that: (a) It is usually better to not randomize within blocks. (b) Blocking violates the assumption of constant variance. (c) Create blocks by using each level of the nuisance factor as a block and randomize within blocks. (d) Randomizing the runs is preferable to randomizing blocks. 4.57 Suppose that a single-factor experiment with five levels of the factor has been conducted. There are three replicates and the experiment has been conducted as a complete randomized design. If the experiment had been conducted in blocks, the pure error degrees of freedom would be reduced by (choose the correct answer): (a) 3 (b) 5 (c) 2 (d) 4 (e) none of the above (b) How many blocks were used in this experiment? (c) What conclusions can you draw? 4.9 Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in 5-gallon milk containers. The analysis is done in a laboratory, and only three trials can be run on any day. Because days could represent a potential source of variability, the experimenter decides to use a randomized block design. Observations are taken for four days, and the data are shown here. Analyze the data from this experiment (use =0.05 ) and draw conclusions. 4.10 Plot the mean tensile strengths observed for each chemical type in Problem 4.8 and compare them to an appropriately scaled t distribution. What conclusions would you draw from this display? 4.11 Plot the average bacteria counts for each solution in Problem 4.9 and compare them to a scaled t distribution. What conclusions can you draw? 4.12 Consider the hardness testing experiment described in Section 4.1. Suppose that the experiment was conducted as exactly the same as the error sum of squares for the standard ANOVA. (a) True (b) False 3.66 Fisher's LSD procedure is an extremely conservative method for comparing pairs of treatment means following an ANOVA. (a) True (b) False 3.67 The REML method of estimating variance components is a technique based on maximum likelihood, while the ANOVA method is a method-of-moments procedure. (a) True (b) False 3.68 One advantage of the REML method of estimating variance components is that it automatically produces confidence intervals on the variance components. (a) True (b) False 3.63 The normality assumption is extremely important in the analysis of variance. (a) True (b) False 3.64 The analysis of variance treats both quantitative and qualitative factors alike so far as the basic computations for sums of squares are concerned. (a) True (b) False 3.65 If a single-factor experiment has a levels of the factor and a polynomial of degree a1 is fit to the experimental data, the error sum of squares for the polynomial model will be 3.69 The Tukey method is used to compare all treatment means to a control. (a) True (b) False 3.70 An experiment with a single factor has been conducted as a completely randomized design and analyzed using computer software. A portion of the output is shown below. (a) Fill in the missing information. (b) How many levels of the factor were used in this experiment? (c) How many replicates were used in this experiment? (d) Find bounds on the P-value. 3.71 The estimate of the standard deviation of any observation in the experiment in Problem 3.70 is (a) 7.03 (b) 2.65 (c) 5.91 (d) 1.95 (e) none of the above 3.5 The mean square for error in the ANOVA provides an estimate of (a) The variance of the random error (b) The variance of an individual treatment average (c) The standard deviation of an individual observation (d) None of the above 3.6 It is always a good idea to check the normality assumption in the ANOVA by applying a test for normality such as the Anderson-Darling test to the residuals. (a) True (b) False els. What conclusions can you draw from this simple example? 6.53 In two-level design, the expected value of a nonsignificant factor effect is zero. (a) True (b) False 6.54 A half-normal plot of factor effects plots the expected normal percentile versus the effect estimate. (a) True (b) False 6.10 Problems 307 6.55 In an unreplicated design, the degrees of freedom associated with the "pure error" component of error are zero. (a) True (b) False 6.56 In a replicated 23 design (16 runs), the estimate of the model intercept is equal to one-half of the total of all 16 runs. (a) True (b) False 6.57 Adding center runs to a 2k design affects the estimate of the intercept term but not the estimates of any other factor effects. (a) True (b) False 6.58 The mean square for pure error in a replicated factorial design can get smaller if nonsignificant terms are added to a model. (a) True (b) False 6.59 A 2k factorial design is a D-optimal design for fitting a first-order model. (a) True (b) False 6.60 If a D-optimal design algorithm is used to create a 12-run design for fitting a first-order model in three variables with all three two-factor interactions, the algorithm will construct a 23 factorial with four center runs. (a) True (b) False 6.61 Suppose that you want to replicate 2 of the 8 runs in a 23 factorial design. How many ways are there to choose the 2 runs to replicate? Suppose that you decide to replicate the run with all three factors at the high level and the run with all three factors at the low level. (a) Is the resulting design orthogonal? (b) What are the relative variances of the model coefficients if the main effects plus two-factor interaction model are fit to the data from this design? (c) What is the power for detecting effects of two standard deviations in magnitude? 6.62 The display below summarizes the results of analyzing a 24 factorial design. (a) Fill in the missing information in this table. (b) Construct a normal probability plot of the effects. Which factors seem to be active? 6.1 In a 24 factorial design, the number of degrees of freedom for the model, assuming the complete factorial model, is (a) 7 (b) 5 (c) 6 (d) 11 (e) 12 (f) none of the above 6.2 A 23 factorial is replicated twice. The number of pure error or residual degrees of freedom are (a) 4 (b) 12 (c) 15 (d) 2 (e) 8 (f) none of the above 6.3 A 23 factorial is replicated twice. The ANOVA indicates that all main effects are significant but the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a) 12 (b) 8 (c) 16 (d) 14 (e) 10 (f) none of the above 6.4 A 23 factorial is replicated three times. The ANOVA indicates that all main effects are significant but two of the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a) 12 (b) 14 (c) 6 (d) 10 (e) 8 (f) none of the above 5.1 An interaction effect in the model from a factorial experiment involving quantitative factors is a way of incorporating curvature into the response surface model representation of the results. (a) True (b) False 5.2 A factorial experiment may be conducted as a RCBD by running each replicate of the experiment in a unique block. (a) True (b) False 5.3 If an interaction effect in a factorial experiment is significant, the main effects of the factors involved in that interaction are difficult to interpret individually. (a) True (b) False 5.4 A biomedical researcher has conducted a two-factor factorial experiment as part of the research to develop a 4.1 Suppose that a single-factor experiment with four levels of the factor has been conducted. There are six replicates and the experiment has been conducted in blocks. The error sum of squares is 500 and the block sum of squares is 250 . If the experiment had been conducted as a completely randomized design the estimate of the error variance 2 would be. (a) 25.0 (b) 25.5 (c) 35.0 (d) 37.5 (e) None of the above 4.2 Suppose that a single-factor experiment with five levels of the factor has been conducted. There are three replicates and the experiment has been conducted as a complete randomized design. If the experiment had been conducted in blocks, the pure error degrees of freedom would be reduced by (a) 3 (b) 5 (c) 2 (d) 4 (e) None of the above 4.3 Blocking is a technique that can be used to control the variability transmitted by uncontrolled nuisance factors in an experiment. (a) True (b) False 172 Chapter 4 Randomized Blocks, Latin Squares, and Related Designs 4.4 The number of blocks in the RCBD must always equal the number of treatments or factor levels. (a) True (b) False 4.5 The key concept of the phrase "Block if you can, randomize if you can't." is that: (a) It is usually better to not randomize within blocks. (b) Blocking violates the assumption of constant variance. (c) Create blocks by using each level of the nuisance factor as a block and randomize within blocks. (d) Randomizing the runs is preferable to randomizing blocks. 4.57 Suppose that a single-factor experiment with five levels of the factor has been conducted. There are three replicates and the experiment has been conducted as a complete randomized design. If the experiment had been conducted in blocks, the pure error degrees of freedom would be reduced by (choose the correct answer): (a) 3 (b) 5 (c) 2 (d) 4 (e) none of the above (b) How many blocks were used in this experiment? (c) What conclusions can you draw? 4.9 Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in 5-gallon milk containers. The analysis is done in a laboratory, and only three trials can be run on any day. Because days could represent a potential source of variability, the experimenter decides to use a randomized block design. Observations are taken for four days, and the data are shown here. Analyze the data from this experiment (use =0.05 ) and draw conclusions. 4.10 Plot the mean tensile strengths observed for each chemical type in Problem 4.8 and compare them to an appropriately scaled t distribution. What conclusions would you draw from this display? 4.11 Plot the average bacteria counts for each solution in Problem 4.9 and compare them to a scaled t distribution. What conclusions can you draw? 4.12 Consider the hardness testing experiment described in Section 4.1. Suppose that the experiment was conducted as