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engineering
design and analysis of experiments

Design And Analysis Of Experiments 5th Edition Douglas C., Montgomery - Solutions

One of the variables in the experiment described in Problem 6-15, heat treatment method (C), is a categorical variable. Assume that the remaining factors are continuous.(a) Write two regression models for predicting crack length, one for each level of the heat treatment method variable. What
How would your answer to Problem 6-13 change if arsenic flow rate was more difficult to control in the process than the deposition time?Problem 6-13An article in the AT&T Technical Journal (March/April 1986, Vol. 65, pp. 39-50) describes the application of two-level factorial designs to
Use the regression model in part (c) of Problem 6-12 to generate a response surface contour plot for epitaxial layer thickness. Suppose it is critically important to obtain layer thickness of 14.5 Am. What settings of arsenic flow rate and decomposition time would you recommend?Problem 6-12An
An article in the AT&T Technical Journal (March/April 1986, Vol. 65, pp. 39-50) describes the application of two-level factorial designs to integrated circuit manufacturing. A basic processing step is to grow an epitaxial layer on polished silicon wafers. The wafers mounted on a susceptor are
Calculate approximate 95 percent confidence limits for the factor effects in Problem 6-10. Do the results of this analysis agree with the analysis of variance performed in Problem 6-10?Problem 6-10In Problem 6-9, the engineer was also interested in potential fatigue differences resulting from the
In Problem 6-9, the engineer was also interested in potential fatigue differences resulting from the two types of bottles. As a measure of the amount of effort required, he measured the elevation of the heart rate (pulse) induced by the task. The results follow. Analyze the data and draw
An industrial engineer employed by a beverage bottler is interested in the effects of two different types of 32-ounce bottles on the time to deliver 12-bottle cases of the product. The two bottle types are glass and plastic. Two workers are used to perform a task consisting of moving 40 cases of
A bacteriologist is interested in the effects of two different culture media and two different times on the growth of a particular virus. She performs six replicates of a 22 design, making the runs in random order. Analyze the bacterial growth data that follow and draw appro-priate conclusions.
An experiment was performed to improve the yield of a chemical process. Four factors were selected, and two replicates of a completely randomized experiment were run. The results are shown in the following table: (a) Estimate the factor effects.(b) Prepare an analysis of variance table, and
Reconsider the experiment described in Problem 6-1. Suppose that the experimenter only performed the eight trials from replicate I. In addition, he ran four center points and obtained the following response values: 36, 40, 43, 45.(a) Estimate the factor effects. Which effects are large?(b) Perform
A router is used to cut locating notches on a printed circuit board. The vibration level at the surface of the board as it is cut is considered to be a major source of dimensional variation in the notches. Two factors are thought to influence vibration: bit size (A) and cutting speed (B). Two bit
Plot the factor effects from Problem 6-1 on a graph relative to an appropriately scaled t distribution. Does this graphical display adequately identify the important factors? Com-pare the conclusions from this plot with the results from the analysis of variance. Problem 6-1.An engineer is
Find the standard error of the factor effects and approximate 95 percent confidence limits for the factor effects in Problem 6-1. Do the results of this analysis agree with the conclusions from the analysis of variance?Problem 6-1.An engineer is interested in the effects of cutting speed (A), tool
Reconsider part (c) of Problem 6-1. Use the regression model to generate response surface and contour plots of the tool life response. Interpret these plots. Do they provide insight regarding the desirable operating conditions for this process?Problem 6-1.An engineer is interested in the effects of
An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results follow: (a) Estimate the factor
An article in the IEEE Transactions on Electron Devices (Nov. 1986, pp. 1754) describes a study on polysilicon doping. The experiment shown below is a variation of their study. The response variable is base current.(a) Is there evidence (with α = 0.05) indicating that either polysilicon doping
An article in the Journal of Testing and Evaluation (Vol. 16, no. 2, pp. 508-515) investigated the effects of cyclic loading and environmental conditions on fatigue crack growth at a constant 22 MPa stress for a particular material. The data from this experiment are shown below (the response is
Consider the data in Problem 5-6. Analyze the data, assuming that replicates are blocks.Problem 5-6.The factors that influence the breaking strength of a synthetic fiber are being studied. Four production machines and three operators are chosen and a factorial experiment is run using fiber from the
Consider the data in Problem 5-5. Analyze the data, assuming that replicates are blocks.Problem 5-5Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, Wiley, 1977) describe an experiment to investigate warping of copper plates. The two factors studied
The yield of a chemical process is being studied. The two factors of interest are temperature and pressure. Three levels of each factor are selected; however, only nine runs can be made in one day. The experimenter runs a complete replicate of the design on each day. The data are shown in the
In Problem 5-1, suppose that we wish to reject the null hypothesis with a high probability if the difference in the true mean yield at any two pressures is as great as 0.5. If a reasonable prior estimate of the standard deviation of yield is 0.1, how many replicates should be run?Problem 5-1The
The quality control department of a fabric finishing plant is studying the effect of several factors on the dyeing of cotton—synthetic cloth used to manufacture men's shirts. Three operators, three cycle times, and two temperatures were selected, and three small speci-mens of cloth were dyed
The percentage of hardwood concentration in raw pulp, the vat pressure, and the cooking time of the pulp are being investigated for their effects on the strength of paper. Three levels of hardwood concentration, three levels of pressure, and two cooking times are selected. A factorial experiment
Consider the three-factor model Notice that there is only one replicate. Assuming all the factors are fixed, write down the analysis of variance table, including the expected mean squares. What would you use as the "experimental error" to test hypotheses? i = 1, 2, ... , a j = 1, 2, 1, 2, ...
The shear strength of an adhesive is thought to be affected by the application pressure and temperature. A factorial experiment is performed in which both factors are assumed to be fixed. Analyze the data and draw conclusions. Perform a test for nonadditivity. Temperature (°F) Pressure (Ib/in?)
Consider the following data from a two-factor factorial experiment. Analyze the data and draw conclusions. Perform a test for nonadditivity. Use α = 0.05. Column Factor Row Factor 1 3 4 36 39 36 32 18 20 22 20 3 30 37 33 34 2. 1,
Derive the expected mean squares for a two-factor analysis of variance with one observation per cell, assuming that both factors are fixed.
An experiment was conducted to determine if either firing temperature or furnace position affects the baked density of a carbon anode. The data are shown below: Suppose we assume that no interaction exists. Write down the statistical model. Conduct the analysis of variance and test
Use Tukey's test to determine which levels of the pressure factor are significantly different for the data in Problem 5-1.Problem 5-1.The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor
Consider the experiment in Problem 5-1. Fit an appropriate model to the response data. Use this model to provide guidance concerning operating conditions for the process.Problem 5-1.The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and
An experiment is conducted to study the influence of operating temperature and three types of face-plate glass in the light output of an oscilloscope tube. The following data are collected: (a) Use α = 0.05 in the analysis. Is there a significant interaction effect? Does glass type or
A mechanical engineer is studying the thrust force developed by a drill press. He suspects that the drilling speed and the feed rate of the material are the most important factors. He selects four feed rates and uses a high and low drill speed chosen to represent the extreme operating conditions.
The factors that influence the breaking strength of a synthetic fiber are being studied. Four production machines and three operators are chosen and a factorial experiment is run using fiber from the same production batch. The results are as follows:(a) Analyze the data and draw conclusions. Use α
Johnson and Leone (Statistics and Experimental Design in Engineering and the Physical Sciences, Wiley, 1977) describe an experiment to investigate warping of copper plates. The two factors studied were the temperature and the copper content of the plates. The response variable was a measure of the
An article in Industrial Quality Control (1956, pp. 5-8) describes an experiment to investigate the effect of the type of glass and the type of phosphor on the brightness of a television tube. The response variable is the current necessary (in microamps) to obtain a specified brightness level. The
For the data in Problem 5-2, compute a 95 percent confidence interval estimate of the mean difference in response for feed rates of 0.20 and 0.25 in/min.Problem 5-2An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. He selects three feed
An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. He selects three feed rates and four depths of cut. He then conducts a factorial experiment and obtains the following data: (a) Analyze the data and draw conclusions. Use α
The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data follow: (a) Analyze the data and
Occasionally the block size obeys the relationship a < k < 2a. An extended incomplete block design consists of a single replicate of each treatment in each block along with an incomplete block design with k* = k - a. In the balanced case, the incomplete block design will have parameters k* =
Show that the variance of the intrablock estimators {τ̂i} is k(a - 1)σ2/(λa2).
Verify that a BIRD with the parameters a = 8, r = 8, k = 4, and b = 16 does not exist.
Perform the interblock analysis for the design in Problem 4-29.Problem 4-29.Seven different hardwood concentrations are being studied to determine their effect on the strength of the paper produced. However, the pilot plant can only produce three runs each day. As days may differ, the analyst uses
Perform the interblock analysis for the design in Problem 4-27.Problem 4-27.An engineer is studying the mileage performance characteristics of five types of gasoline additives. In the road test he wishes to use cars as blocks; however, because of a time constraint, he must use an incomplete block
An experimenter wishes to compare eight treatments in blocks of four runs. Find a BIRD with 14 blocks and λ = 3.
An experimenter wishes to compare four treatments in blocks of two runs. Find a BIBD for this experiment with six blocks.
Prove that is the adjusted sum of squares for treatments in a BIBD. 15 (Ла)
Analyze the data in Example 4-6 using the general regression significance test.
Seven different hardwood concentrations are being studied to determine their effect on the strength of the paper produced. However, the pilot plant can only produce three runs each day. As days may differ, the analyst uses the balanced incomplete block design that follows. Analyze the data from
Construct a set of orthogonal contrasts for the data in Problem 4-27. Compute the sum of squares for each contrast.Problem 4-27. An engineer is studying the mileage performance characteristics of five types of gasoline additives. In the road test he wishes to use cars as blocks; however,
An engineer is studying the mileage performance characteristics of five types of gasoline additives. In the road test he wishes to use cars as blocks; however, because of a time constraint, he must use an incomplete block design. He runs the balanced design with the five blocks that follow. Analyze
Consider the randomized block design with one missing value in Table 4-7. Analyze this data by using the exact analysis of the missing value problem discussed in Section 4-1.4. Compare your results to the approximate analysis of these data given in Table 4-8.Table 4-7.Table 4-8. Table 4-7
Consider the data in Problems 4-15 and 4-23. Suppressing the Greek letters in 4-23, analyze the data using the method developed in Problem 4-19.Problems 4-15An industrial engineer is investigating the effect of four assembly methods (A, B, C, D) on the assembly time for a color television
Construct a 5 × 5 hypersquare for studying the effects of five factors. Exhibit the analysis of variance table for this design.
Suppose that in Problem 4-15 the engineer suspects that the workplaces used by the four operators may represent an additional source of variation. A fourth factor, workplace (α, β, γ, δ) may be introduced and another experiment conducted, yielding the Graeco—Latin square that follows. Analyze
The yield of a chemical process was measured using five batches of raw material, five acid concentrations, five standing times (A, B, C, D, E), and five catalyst concentrations (α, β, γ, δ, ∈). The Graeco-Latin square that follows was used. Analyze the data from this experiment (use α =
Suppose that in Problem 4-14 the data taken on day 5 were incorrectly analyzed and had to be discarded. Develop an appropriate analysis for the remaining data.Problem 4-14A chemist wishes to test the effect of four chemical agents on the strength of a particular type of cloth. Because there might
Discuss how the operating characteristics curves in the Appendix may be used with the Latin square design.
Designs involving several Latin squares. [See Cochran and Cox (1957), John (1971).] The p × p Latin square contains only p observations for each treatment. To obtain more replications the experimenter may use several squares, say n. It is immaterial whether the squares used are the same or
Derive the missing value formula (Equation 4-24) for the Latin square design.Equation 4-24 p(y. + y. + y)- 2y Yijk (р - 2)(р — 1)
Consider a p × p Latin square with rows (αi), columns (βk), and treatments (τj) fixed. Obtain least squares estimates of the model parameters αi βk, and τj.
Suppose that in Problem 4-14 the observation from batch 3 on day 4 is missing. Estimate the missing value from Equation 4-24, and perform the analysis using the value.Equation 4-24Problem 4-14The effect of five different ingredients (A, B, C, D, E) on the reaction time of a chemical process is
An industrial engineer is investigating the effect of four assembly methods (A, B, C, D) on the assembly time for a color television component. Four operators are selected for the study. Furthermore, the engineer knows that each assembly method produces such fatigue that the time required for the
The effect of five different ingredients (A, B, C, D, E) on the reaction time of a chemical process is being studied. Each batch of new material is only large enough to permit five runs to be made. Furthermore, each run requires approximately 1 1/2 hours, so only five runs can be made in one day.
An industrial engineer is conducting an experiment on eye focus time. He is interested in the effect of the distance of the object from the eye on the focus time. Four different distances are of interest. He has five subjects available for the experiment. Because there may be differences among
Two missing values in a randomized block. Suppose that in Problem 4-1 the observations for chemical type 2 and bolt 3 and chemical type 4 and bolt 4 are missing.(a) Analyze the design by iteratively estimating the missing values, as described in Section 4-1.3. (b) Differentiate SSE with
Suppose that the observation for chemical type 2 and bolt 3 is missing in Problem 4-1. Analyze the problem by estimating the missing value. Perform the exact analysis and compare the results.Problem 4-1.A chemist wishes to test the effect of four chemical agents on the strength of a particular type
Draw an operating characteristic curve for the design in Problem 4-2. Does the test seem to be sensitive to small differences in the treatment effects?Problem 4-2Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in 5-gallon milk
Assuming that chemical types and bolts are fixed, estimate the model parameters τi and βj in Problem 4-1.Problem 4-1.A chemist wishes to test the effect of four chemical agents on the strength of a particular type of cloth. Because there might be variability from one bolt to another, the chemist
Analyze the data in Problem 4-2 using the general regression significance test.Problem 4-2Three 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
An aluminum master alloy manufacturer produces grain refiners in ingot form. The company produces the product in four furnaces. Each furnace is known to have its own unique operating characteristics, so any experiment run in the foundry that involves more than one furnace will consider furnaces as
Consider the ratio control algorithm experiment described in Chapter 3, Section 3.8. The experiment was actually conducted as a randomized block design, where six time periods were selected as the blocks, and all four ratio control algorithms were tested in each time period. The average cell
An article in the Fire Safety Journal ("The Effect of Nozzle Design on the Stability and Performance of Turbulent Water Jets," Vol. 4, August 1981) describes an experiment in which a shape factor was detennined for several different nozzle designs at six levels of jet efflux velocity. Interest
Plot the average bacteria counts for each solution in Problem 4-2 and compare them to a scaled t distribution. What conclusions can you draw?Problem 4-2Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in 5-gallon milk containers. The
Plot the mean tensile strengths observed for each chemical type in Problem 4-1 and compare them to an appropriately scaled t distribution. What conclusions would you draw from this display?Problem 4-1A chemist wishes to test the effect of four chemical agents on the strength of a particular type of
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
A chemist wishes to test the effect of four chemical agents on the strength of a particular type of cloth. Because there might be variability from one bolt to another, the chemist decides to use a randomized block design, with the bolts of cloth considered as blocks. She selects five bolts and
Consider the experiment in Example 3-1. Suppose that the largest observation on tensile strength is incorrectly recorded as 50. What effect does this have on the usual analysis of variance? What effect does it have on the Kruskal-Wallis test?
Use the Kruskal-Wallis test for the experiment in Problem 3-12. Are the results comparable to those found by the usual analysis of variance?Problem 3-12.Four different designs for a digital computer circuit are being studied to compare the amount of noise present. The following data have been
Use the Kruskal-Wallis test for the experiment in Problem 3-11. Compare the conclusions obtained with those from the usual analysis of variance.Problem 3-11.The effective life of insulating fluids at an accelerated load of 35 kV is being studied. Test data have been obtained for four types of
Consider the data shown in Problem 3-10.(a) Write out the least squares normal equations for this problem, and solve them for µ̂ and τ̂i, using the usual constraint (Σ3i =1 τ̂i = 0). Estimate τ1 – τ2.(b) Solve the equations in (a) using the constraint τ3 = 0. Are the estimators τ̂i,
Four different feed rates were investigated in an experiment on a CNC machine producing a component part used in an aircraft auxiliary power unit. The manufacturing engineer in charge of the experiment knows that a critical part dimension of interest may be affected by the feed rate. However, prior
Refer to the aluminum smelting experiment in Section 3-8. Verify the analysis of variance for pot noise summarized in Table 3-13. Examine the usual residual plots and comment on the experimental validity.Table 3-13. Degrees of Freedom Sum of Mean Source of Variation Squares Square Fo P-Value Ratio
Refer to the aluminum smelting experiment described in Section 3-8. Verify that ratio control methods do not affect average cell voltage. Construct a normal probability plot of the residuals. Plot the residuals versus the predicted values. Is there an indication that any underlying assumptions are
Refer to Problem 3-26.(a) How would your answer change if a reasonable estimate of the experimental error variance were σ2 = 36?(b) How would your answer change if a reasonable estimate of the experimental error variance were σ2 = 49?(c) Can you draw any conclusions about the sensitivity of your
Suppose that four normal populations have means of µ1 = 50, µ2 = 60, µ3 = 50, and µ4 = 60. How many observations should be taken from each population so that the probability of rejecting the null hypothesis of equal population means is at least 0.90? Assume that α = 0.05 and that a reasonable
Consider the experiment in Problem 3-14. If we wish to construct a 95 percent confidence interval on the difference in two mean battery lives that has an accuracy of ±2 weeks, how many batteries of each brand must be tested?Problem 3-14.Three brands of batteries are under study. It is suspected
Refer to Problem 3-14.(a) If we wish to detect a maximum difference in battery life of 10 hours with a probability of at least 0.90, what sample size should be used? Discuss how you would obtain a preliminary estimate of σ2 for answering this question.(b) If the difference between brands is great
Refer to Problem 3-10. If we wish to detect a maximum difference in mean response times of 10 milliseconds with a probability of at least 0.90, what sample size should be used? How would you obtain a preliminary estimate of σ2?Problem 3-10.The response time in milliseconds was determined for three
Use the modified Levene test to determine if the assumption of equal variances is satisfied in Problem 3-14. Use α = 0.05. Did you reach the same conclusion regarding the equality of variances by examining residual plots?Problem 3-14.Three brands of batteries are under study. It is suspected that
Use Bartlett's test to determine if the assumption of equal variances is satisfied in Problem 3-14. Use α = 0.05. Did you reach the same conclusion regarding equality of variances by examining residual plots?Problem 3-14.Three brands of batteries are under study. It is suspected that the lives (in
In a fixed effects experiment, suppose that there are n observations for each of four treatments. Let Q21, Q22, Q23 be single-degree-of-freedom components for the orthogonal contrasts. Prove that SSTreatmens = Q21 + Q22 + Q23.
Show that the variance of the linear combination E1c,yi, is o2n,c.
Consider testing the equality of the means of two normal populations, where the variances are unknown but are assumed to be equal. The appropriate test procedure is the pooled t test. Show that the pooled t test is equivalent to the single-factor analysis of variance.
A semiconductor manufacturer has developed three different methods for reducing particle counts on wafers. All three methods are tested on five wafers and the after-treatment particle count obtained. The data are shown below:(a) Do all methods have the same effect on mean particle count?(b) Plot
An experiment was performed to investigate the effectiveness of five insulating materials. Four samples of each material were tested at an elevated voltage level to accelerate the time to failure. The failure times (in minutes) are shown below:(a) Do all five materials have the same effect on mean
Four catalysts that may affect the concentration of one component in a three-component liquid mixture are being investigated. The following concentrations are obtained: (a) Do the four catalysts have the same effect on the concentration?(b) Analyze the residuals from this experiment.(c)
Three brands of batteries are under study. It is suspected that the lives (in weeks) of the three brands are different. Five batteries of each brand are tested with the following results:(a) Are the lives of these brands of batteries different?(b) Analyze the residuals from this experiment.(c)
Four chemists are asked to determine the percentage of methyl alcohol in a certain chemical compound. Each chemist makes three determinations, and the results are the following:(a) Do chemists differ significantly? Use α = 0.05.(b) Analyze the residuals from this experiment.(c) If chemist 2 is a
The effective life of insulating fluids at an accelerated load of 35 kV is being studied. Test data have been obtained for four types of fluids. The results were as follows: (a) Is there any indication that the fluids differ? Use α = 0.05.(b) Which fluid would you select, given that the
The response time in milliseconds was determined for three different types of circuits that could be used in an automatic valve shutoff mechanism. The results are shown in the following table:(a) Test the hypothesis that the three circuit types have the same response time. Use α = 0.01.(b) Use

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