Questions and Answers of Design And Analysis Of Experiments

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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, ∈).
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
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
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
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 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.
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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.
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
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
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
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
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)

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