Design And Analysis Of Experiments 8th Edition Douglas C. Montgomery - Solutions

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(a) Show that the numerator sum of squares for testing $H_0 : (β_1 = 0)$ against the alternative hypothesis $H_A : (β_1 ≠ 0)$ is the same as the sum of squares ssc that would be obtained for testing that the linear trend is zero in the analysis of variance model (the relevant calculations were
(b) Obtain a 95% confidence band for the regression line.
(c) Calculate a 99% prediction interval for the fluid flow rate at 100 revolutions per minute.
(d) Estimate the intercept $β_0$. This is not zero, which suggests that the fluid flow rate is not zero at 0 rpm. Since this should not be the case, explain what is happening.
(a) Define the objectives of the experiment.
(b) Identify all sources of variation, including: (i) treatment factors and their levels, (ii) experimental units, (iii) blocking factors, noise factors, and covariates.
(c) Choose a rule for assigning the experimental units to the treatments.
(d) Specify the measurements to be made, the experimental procedure, and the anticipated difficulties.
(e) Run a pilot experiment.
f) Specify the model.
(g) Outline the analysis.
(h) Calculate the number of observations that need to be taken.
(i) Review the above decisions. Revise, if necessary.
A short description of the decisions that need to be made at each stage of the checklist is given below. Only after all of these decisions have been made should the data be collected.
1. Table 2.10 gives a list of experiments that can be run as class projects. Select a simple experiment of interest to you, but preferably not on the list. Complete steps (a)-(c) of the checklist with the intention of actually running the experiment when the checklist is complete.
2. For experiments 1 and 7 in Table 2.10, complete steps (a) and (b) of the checklist. There may be more than one treatment factor. Give precise definitions of their levels.
3. For experiment 2, complete steps (a)-(c) of the checklist.
4. For experiment 3, complete steps (a)-(c) of the checklist.
5. For experiment 4, list sources of variation. Decide which sources can be controlled by limiting the scope of the experiment or by specifying the exact experimental procedure to be followed. Of the remaining sources of variation, decide which are minor and which are major. Are there any blocking
(a) Find the section of Table A.7 for the numerator degrees of freedom v₁ = v - 1 and the specified a (only a = 0.01 and a = 0.05 are shown).
(b) Calculate the denominator degrees of freedom using v2 = 1000 in the first iteration and v2 = n - v = v(r-1) in the following iterations, and locate the appropriate row of the table.
(c) For the required power π(Δ), we use Table A.7 to determine ϕ and take the larger listed value if necessary.
(d) Calculate r = 2υσ²ϕ²/Δ², rounding up to the nearest integer.
(e) Repeat steps (b)-(d) until the value of r is unchanged or alternates between two values.Select the larger of alternating values.
1. Suppose that you are planning to run an experiment with one treatment factor having four levels and no blocking factors. Suppose that the calculation of the required number of observations has given r₁ = r₂ = r₃ = r₄ = 5. Assign at random 20 experimental units to the v = 4 levels of the
2. Suppose that you are planning to run an experiment with one treatment factor having three levels and no blocking factors. It has been determined that r₁ = 3, r₂ = r₃ = 5.Assign at random 13 experimental units to the v = 3 treatments, so that the first treatment is assigned 3 units and the
3. Suppose that you are planning to run an experiment with three treatment factors, where the first factor has two levels and the other two factors have three levels each. Write out the coded form of the 18 treatment combinations. Assign 36 experimental units at random to the treatment combinations
4. For the one-way analysis of variance model (3.3.1), page 36, the solution to the normal equations used by the SAS software is $$t_i = Y_{i.} - \bar{Y}_{..}$$ (i = 1,..., v) and $$û = \bar{Y}_{..}$$.(a) Is τ₁ estimable? Explain.
(b) Calculate the expected value of the least squares estimator for τ₁ - τ₂ corresponding to the above solution. Is τ₁ - τ₂ estimable? Explain.
5. Consider a completely randomized design with observations on three treatments (coded 1, 2, 3). For the one-way analysis of variance model (3.3.1), page 36, determine which of the following are estimable. For those that are estimable, state the least squares estimator.(a) τ₁ + τ₂ -
6. (requires calculus) Show that the normal equations for estimating μ, τ₁, ..., τν are those given in equation (3.4.3) on page 38.
7. (requires calculus) Show that the least squares estimator of μ + τ is Y.., for the linear model Yᵢⱼ = μ + τᵢ + εᵢⱼ (i = 1, ..., r; i = 1, 2, ..., v), where the εᵢⱼ's are independent random variables with mean zero and variance σ². (This is the reduced model for the one-way
8. For the model in the previous exercise, find an unbiased estimator for σ². (Hint: first calculate E[ssEo] in (3.5.11), page 45.)
9. (requires calculus) Find the least squares estimates of µ₁, µ₂, ..., µν for the linear model Yᵢⱼ = µᵢ + εᵢⱼ (i = 1,.r; i = 1, 2,..., v), where the εᵢⱼ's are independent random variables with mean zero and variance σ². Compare these estimates with the least squares
10. For the model in the previous exercise, find an unbiased estimator for σ². Compare the estimator with that in (3.4.8), page 43.
11. Verify, for the one-way analysis of variance model (3.3.1), page 36, that each treatment sample variance Sᵢ² is an unbiased estimator of the error variance σ², so that$$E(SSE) = \sum_{i}(r_i - 1)E(S_i²) = (n - v)σ^2.$$
12. Balloon experiment (Meily Lin, 1985)Prior to 1985, the experimenter had observed that some colors of birthday balloons seem to be harder to inflate than others. She ran this experiment to determine whether balloons of different colors are similar in terms of the time taken for inflation to a
(a) Plot inflation time versus color and comment on the results.
(b) Estimate the mean inflation time for each balloon color, and add these estimates to the plot from part (a).
(c) Construct an analysis of variance table and test the hypothesis that color has no effect on inflation time.
(d) Plot the data for each color in the order that it was collected. Are you concerned that the assumptions on the model are not satisfied? If so, why? If not, why not?
(e) Is the analysis conducted in part (c) satisfactory?
13. Heart-lung pump experiment, continued The heart-lung pump experiment was described in Example 3.4.4, page 40, and the data were shown in Table 3.2, page 41.
(a) Calculate an analysis of variance table and test the null hypothesis that the number of revolutions per minute has no effect on the fluid flow rate.
(b) Are you happy with your conclusion? Why or why not?
(c) Calculate a 90% upper confidence limit for the error variance σ².
14. Pedestrian light experiment (Larry Lesher, 1985)Larry Lesher ran this experiment in order to determine whether pushing a certain pedes-trian light button had an effect on how long he had to wait before the pedestrian light showed "walk." The treatment factor of interest was the number of pushes
(a) Plot the waiting times against the number of pushes of the button. What does the plot show?
(b) Construct an analysis of variance table and test the null hypothesis that the number of pushes of the pedestrian button has no effect on the waiting time for the "walk"sign.
(c) Estimate the mean waiting time for each number of pushes and the corresponding variance.
(d) Estimate the contrast τ0 - (τ1 + τ2 + τ3)/3, which compares the effect of no pushes of the button with the average effect of pushing the button once, twice, or three times.
Consider the battery experiment introduced in Section 2.5.2, for which a = b = 2 and r = 4. Suppose it is of interest to calculate confidence intervals for the four simple effects (11 - [12, [21 -T22, "11 - [21, [12 - 22, with an overall confidence level of 95%.
(a) Determine whether the Tukey or Bonferroni method of multiple comparisons would provide shorter confidence intervals.
(b) Apply the better of the methods from part (a) and comment on the results. (The data give y11. = 570.75, J12. = 860.50. F21. = 433.00, and y22. = 496.25 minutes per unit cost and msE = 2, 367.71.)
(c) Discuss the practical meaning of the contrasts estimated in (b) and explain what you have learned from the confidence intervals.
7. Weld strength experiment The data shown in Table 6.20 are a subset of the data given by Anderson and McLean(1974) and show the strength of a weld in a steel bar. Two factors of interest were gage bar setting (the distance the weld die travels during the automatic weld cycle) and time of welding
(a) Using the cell-means model (6.2.1) for these data, test the hypothesis that there is no difference in the effects of the treatment combinations on weld strength against the alternative hypothesis that at least two treatment combinations have different effects.
(b) Suppose the experimenters had planned to calculate confidence intervals for all pairwise comparisons between the treatment combinations, and also to look at the confidence interval for the difference between gage bar setting 3 and the average of the other two. Show what these contrasts look
8. Weld strength experiment, continued For the experiment described in Exercise 7, use the two-way complete model instead of the equivalent cell means model.
(a) Test the hypothesis of no interaction between gage bar setting and time of weld and state your conclusion.
(b) Draw an interaction plot for the two factors Gage bar setting and Time of welding.Does your interaction plot support the conclusion of your hypothesis test? Explain.
(c) In view of your answer to part (b), is it sensible to investigate the differences between the effects of gage bar setting? Why or why not? Indicate on your plot what would be compared.
(d) Regardless of your answer to (c), suppose the experimenters had decided to look at the linear trend in the effect of gage bar settings. Test the hypothesis that the linear trend in gage setting is negligible (against the alternative hypothesis that it is not negligible).
An experiment is to be run to examine three levels of factor A and four levels of factor B, using the two-way complete model (6.2.2). Determine the required sample size if the error variance o2 is expected to be less than 15 and simultaneous 99% confidence intervals for pairwise comparisons between
10. Bleach experiment, continued Use the data of the bleach experiment of Example 6.4.3, on page 151.
(a) Evaluate the effectiveness of a variance-equalizing transformation.
(b) Apply Satterthwaite's approximation to obtain 99% confidence intervals for the pairwise comparisons of the main effects of factor A using Tukey's method of multiple comparisons.
18. Nail varnish experiment, continued The nail varnish experiment, introduced in Example 6.5.1, page 159, concerned the dissolving time of three brands of nail varnish in two brands of solvent. There were r = 5 observations on each treatment combination, and the two-way main-effects model (6.2.3)
(a) Verify that the assumptions on the two-way main effects model hold for this experiment.
(b) Verify that $(\bar{Y}_{.1} + \bar{Y}_{.2}) - \bar{Y}_{.3}$, is an unbiased estimator of the difference of averages contrast $(\beta_1 + \beta_2) - \beta_3$.
(c) Obtain a 95% confidence interval for $(\beta_1 + \beta_2) - \beta_3$, as though this were preplanned and the only contrast of interest.
19. For the two-way main-effects model (6.2.3) with equal sample sizes,
(a) verify the computational formulae for ssE given in (6.5.39),
(b) and, if SSE is the corresponding random variable, show that E[SSE] is (n - a -b + 1)². [Hint: E[X²] = Var(X) + E[X]².]
20. An experiment is to be run to compare the two levels of factor A and to examine the pairwise differences between the four levels of factor B, with a simultaneous confidence level of 90%. The experimenter is confident that the two factors will not interact. Find the required sample size if the
21. Water boiling experiment (Kate Ellis, 1986)The experiment was run in order to examine the amount of time taken to boil a given amount of water on the four different burners of her stove, and with 0, 2, 4, or 6 teaspoons of salt added to the water. Thus the experiment had two treatment factors
(a) Check the assumptions on the two-way main-effects model.
(b) Calculate a 99% set of Tukey confidence intervals for pairwise differences between the levels of salt, and calculate separately a 99% set of intervals for pairwise differences between the levels of burner.
22. For v = 5 and r = 4, show that the first three "orthogonal polynomial contrasts" listed in Table A.2 are mutually orthogonal. (In fact all four are.) Find a pair of orthogonal contrasts that are not orthogonal polynomial contrasts. Can you find a third contrast that is orthogonal to each of
23. Air velocity experiment, continued (a) For the air velocity experiment introduced in Section 6.7.4 (page 173), calculate the sum of squares for each of the three interaction contrasts assumed to be negligible, and verify that these add to the value ssE = 175.739, as in Table 6.12.
(b) Check the assumptions on the model by plotting the standardized residuals against the predicted responses, the treatment factor levels, and the normal scores. State your conclusions.
15.21. Three different Pinot Noir wines were evaluated by a panel of eight judges. The judges are considered a random panel of all possible judges. The wines are evaluated on a 100-point scale. The wines were presented in random order to each judge, and the following results obtained.Analyze the
15.20. Discuss how the operating characteristic curves for the analysis of variance can be used in the analysis of covariance.
15.19. Show that in a single-factor analysis of covariance with a single covariate, the standard error of the difference between any two adjusted treatment means is 2 (x-x) 1/2 Exx Sadiy-Adivi MSE n
15.18. Show that in a single-factor analysis of covariance with a single covariate a 100(1 ! () percent confidence interval on the ith adjusted treatment mean is Yi-B(xx) fa/2,a(n-1)-1
15.16. Compute the adjusted treatment means and their standard errors using the data in Problem 15.15.
15.14. Find the standard errors of the adjusted treatment means in Example 15.5.
15.13. The sums of squares and products for a single-factor analysis of covariance follow. Complete the analysis and draw appropriate conclusions. Use ( $ 0.05. Source of Variation Degrees of Freedom Sums of Squares and Products x xy y Treatment 3 1500 1000 650 Error 12 Total 26 6000 1200 550 15
15.12. Compute the adjusted treatment means and the standard errors of the adjusted treatment means for the data in Problem 15.11.
15.10. Myers, Montgomery, Vining and Robinson (2010)describe an experiment to study spermatozoa survival. The design factors are the amount of sodium citrate, the amount of glycerol, and equilibrium time, each at two levels. The response variable is the number of spermatozoa that survive out of 50
15.9. Problem 12.10 suggests using ln (s2) as the response[refer to part (b)]. Does the Box–Cox method indicate that a transformation is appropriate?
15.8. In the 33 factorial design of Problem 12.12 one of the responses is a standard deviation. Use the Box–Cox method to investigate the usefulness of transformations for this response.Would your answer change if we used the variance as the response?
15.7. In the central composite design of Problem 12.11, two responses were obtained, the mean and variance of an oxide thickness. Use the Box–Cox method to investigate the potential usefulness of transformation for both of these responses.Is the log transformation suggested in part (c) of that
15.6. In the grill defects experiment described in Problem 8.30, a variation of the square root transformation was employed in the analysis of the data. Use the Box–Cox method to determine whether this is the appropriate transformation.
15.5. Reconsider the photoresist experiment in Problem 8.26. Use the variance of the resist thickness at each test combination as the response variable. Is there any indication that a transformation is required?
15.4. In Problem 8.25 a replicated fractional factorial design was used to study substrate camber in semiconductor manufacturing.Both the mean and standard deviation of the camber measurements were used as response variables. Is there any indication that a transformation is required for either
15.2. In Example 6.3 we selected a log transformation for the drill advance rate response. Use the Box–Cox procedure to demonstrate that this is an appropriate data transformation.
15.1. Reconsider the experiment in Problem 5.24. Use the Box–Cox procedure to determine whether a transformation on the response is appropriate (or useful) in the analysis of the data from this experiment.
14.27. Reconsider the experiment in Problem 14.26. This is a rather large experiment, so suppose that the experimenter had used a 25!1 design instead. Set up the 25!1 design in a split-plot, using the principal fraction. Then select the response data using the information from the full
14.26. An article in Quality Engineering (“Quality Quandries:Two-Level Factorials Run as Split-Plot Experiments,” Bisgaard et al., Vol. 8, No. 4, pp. 705–708, 1996) describes a 25 factorial experiment in a plasma process focused on making paper more susceptible to ink. Four of the factors

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