Design And Analysis Of Experiments 1st Edition Angela M. Dean, Daniel Voss - Solutions

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22. A total of 4217 eggs have to be packed in boxes that can hold 36 eggs each. By typing one line (command) in the Command Window, calculate how many eggs will remain unpacked if every box that is used has to be full. (Hint: Use MATLAB built-in function fix.)
21. The perimeter of an ellipse can be approximated by: P=z(a+b)/3- (3a+b)(a+36) a+b Calculate the perimeter of an ellipse with a 18 in. and b=7in.
20. The distance d from a point P (xp, yp, 2p) to the line that passes through the two points A (xy) and B (x, y, z) can be calculated by d=28/r where r is the distance between the B, given points A and by r = (x-x)+(y)+(2-2) and S is the area of the triangle defined by the three points cal-
19. For the triangle shown. =72. =43", and its perimeter is p=114 mm. Definea, , and p, as variables, and then: (a) Calculate the triangle sides (Use the law of sines). (b) Calculate the radius of the circle inscribed in the triangle using the formula: (s-as-bys-c) where s (a+b+c)/2.
18. In the triangle shown a 27 in., 6=43in., and c=57 in. Definea, b, and e as variables, and then: (a) Calculate the anglesa, , and y by substituting the variables in the law of cosines. (Law of cosines: ca+b-2abcos) (b) Verify the law of tangents by substituting the results into the right and
17. A frustum of cone is filled with ice cream such that the portion above the cone is a hemisphere. Define the variables d=1.25 in., do-2.25 in., h=2 in., and determine the volume of the ice cream.
16. The three shown circles, with radius 15 in., 10.5 in., and 4.5 in., are tangent to each other. (a) Calculate the angle y (in degrees) by using the law of cosines (Law of cosines: +b-2abcos) (b) Calculate the angles y and a (in degrees) using the law of sines. (c) Check that the sum of the
15. The arc length of a segment of a parabola ABC is given by: Lase = +47 ++(2)+1) Determine LABC if a-8 in. and h-13 in.
14. A rectangular box has the dimensions shown. (a) Determine the angle BAC to the nearest degree. (b) Determine the area of the triangle ABC to the nearest tenth of a centimeter. Law of cosines: =d+b-2abcos Heron's formula for triangular area: B 23 cm App-a)(p-b)(p-c), where p = (a+b+c)/2.
13. Given: following definite integral: cos xdx. 1/6
12. Define two variables: alpha =/8, and beta = /6. Using these variables, show that the following trigonometric identity is correct by calculating the values of the left and right sides of the equation. tana+lan tan(+9)= 1-tano tan cos xdx = 2 x cos x + (x-2)sin x. Use MATLAB to calculate the
11. Two trigonometric identities are given by: (a) (sin x + cos x)=1+2 sin xos x (b) 1-2 cos x-3 cos x x-3cos 1-3cos x sin x 1-cos x For each part, verify that the identity is correct by calculating the values of the left and right sides of the equation, substituting x=20.
10. Two trigonometric identities are given by: (a) posx-sinx=12sinx tan x (b) sin x-2 tan x For each part, verify that the identity is correct by calculating the values of the left and right sides of the equation, substituting x = x/10.
9. Define the variablesa, b,c, and d as: c=4.6 d 1.7 a=ca, and b=- x+y (b) +2x-xy (x-y)" c+a then evaluate: c-d (a) e+c+a-(ca) (6)+() --
8. Define the variables x and y as x=6.5 and y = 3.8; then evaluate: 12/3 (a) (x + y)+x
7. Define the variable as 13.2; then evaluate: (a) e-3.81 (b) 6+61-2 7-1
6. Define the variable as 4.5; then evaluate: cos 20 (a) 0.42 +3.12-162.3z-80.7 (b) (-23)/(+17.5)
5. Calculate: (a) sin(0.2) cos(2/6) +tan 72 (b) (tan 64cos 15)+ sin 37
4. Calculate: 3.8 (a) 275-41-25 5.2+1.8 (b) 2.110-15.210 3-61011
3. Calculate: 23+ 45 (a) +logo 589006 16 0.7 (b) (36.1-2.25x)+20)
2. Calculate: (a)8++ (b) +55x0.41 26
1. Calculate: (a) (5-1+253) (b) 73.1+1130-15%/x 5
2.15. interest 2.15. The shelf life of a carbonated beverage is of Ten bottles are randomly selected and tested, and the follow- ing results are obtained: Days 108 138 124 163 124 159 106 134 115 134
2.10. Consider the Minitab output shown below.Two-Sample T-Test and CI: Y1, Y2 Two-sample T for Y1 vs Y2 N Mean Std. Dev. SE Mean 11 20 50.19 1.71 0.38 12 20 52.52 2.48 0.55 Difference = (Y1) - (Y2)Estimate for difference: -2.33341 95% CI for difference: (-3.69547, -0.97135)T-Test of difference = 0
2.9. Consider the Minitab output shown below.One-Sample T: Y Test of mu = 25 vs. not = 25 Variable N Mean StDev SE Mean 95% CI T P Y 12 25.4818 2.3360 1 0.034(a) How many degrees of freedom are there on the t-test statistic?(b) Fill in the missing information.
2.8. Consider the Minitab output shown below.One-Sample T: Y Test of mu = 91 vs. not = 91 Variable N Mean StDev SE Mean 95% CI T P Y 25 92.5805 0.4673 191.6160, 113.38 0.002(a) Fill in the missing values in the output. Can the null hypothesis be rejected at the 0.05 level? Why?(b) Is this a
2.7. Suppose that we are testing Hp versus H, with a sample size of 10. Calculate bounds on the P-value for the following observed values of the test statistic (a) -248 (bi-3.95 (c) -2.69 (d) 4 1.88 (e)=-1.25
2.6. Suppose that we are testing H, a versus with a sample size of a 15. Calculate bounds H, on the P-value for the following observed values of the test statistic: - (a) -2.35 (b) 3.55 (c) 2.00 (d) -1.55 (a)-2.35 (b)-3.55 (d)4-1.55
2.5. Consider the Minitab output shown below.One-Sample Z Test of µ = 30 vs not = 30 The assumed standard deviation = 1.2 N Mean SE Mean 95% CI Z P 16 31.2000 0.3000 (30.6120, 31.7880) * *(a) Fill in the missing values in the output. What conclusion would you draw?(b) Is this a one-sided or
2.2. The Minitab output for a random sample of data is shown below. Some of the quantities are missing. Compute the values of the missing quantities.Variable N Mean St. Dev. Sum* 16 7 $$8.159$$ $$399.851$$
2.1. The Minitab output for a random sample of data is shown below. Some of the quantities are missing. Compute the values of the missing quantities.variable N Mean St. Dev. Variance Minimum Maximum* 8 16 3.12 $$15.94$$ $$21.14$$
1.3. Suppose that you want to compare the growth of flow- ers with different conditions of sunlight, water, fertilizer, and soil conditions. Complete steps 1-3 of the guidelines for designing experiments in Section 1.4.
6. What difference in average observed hardness between the two quenching media will be considered important?
5. What method of data analysis should be used?
4. How should the test coupons be assigned to the quenching solutions, and in what order should the data be collected?
3. How many coupons of alloy should be tested in each quenching solution?
2. Are there any other factors that might affect hardness that should be investigated or controlled in this experiment (such as, the temperature of the quenching media)?
1. Are these two solutions the only quenching media of potential interest?
(f) Using the results of (c) and (e), argue that μ + τi and β and all linear combinations of these are estimable.
(e) Verify that E[μ + τi] = μ + τi, where μ + τi = Yi. - β(xi. - x.).
(d) Verify that Var(β) = σ2/ss*xx and Cov(Yi., β) = 0.
(c) Show that E[β] = β, where β = sp*xy/ss*xx and ss*xx = ΣiΣr(xir - xi)2.
(b) Verify that sp*xy = ΣiΣr (xir - xi)Yir, given that sp*xy = ΣiΣr(xir - xi)(Yir -Yi).
(a) Compute E[Yir].
2. (optional) Assume that the analysis of covariance model (9.2.2) holds, so that Yir =μ + τi + β(xir - ), + εir.
(e) Construct a 95% confidence interval for the difference in treatment effects. Discuss the results.
(d) Test for equality of the treatment effects, using a significance level of α = 0.05.Discuss the results.
(c) Test for inequality of slopes, using a level of significance α = 0.05.
(b) Plot the residuals against the covariate, the predicted values, and normal scores.Use the plots to evaluate the model assumptions.
1. Consider the hypothetical data of Example 9.3.2, in which two treatments are to be compared.(a) Fit the analysis of covariance model (9.2.1) or (9.2.2) to the data of Table 9.1, page 282.
1. For the following hypothetical data sets of Section 7.2.2 reproduced below, draw interaction plots to evaluate the BC and ABC interaction effects, with levels of B on the horizontal axis and levels of C for labels. In each case, comment on the apparent presence or absence of BC and ABC
2. In planning a five-factor experiment, it is determined that the factors A, B, and C might interact and the factors D and E might interact but that no other interaction effects should be present. Draw a line graph for this experiment and give an appropriate model.
3. Consider an experiment with four treatment factors, A, B, C, and D, ata, b,c, and d levels, respectively, with r observations per treatment combination. Assume that the four-way complete model is a valid representation of the data. Use the rules of Section 7.3 to answer the following.
(a) Find the number of degrees of freedom associated with the AC interaction effect.
(b) Obtain an expression for the sum of squares for AC.
(c) Give a rule for testing the hypothesis that the AC interaction is negligible against the alternative hypothesis that it is not negligible. How should the results of the test be interpreted, given the other terms in the
model?
(d) Write down a contrast for measuring the AC interaction. Give an expression for
(a) How would you decide whether or not the error variables have approximately the same variance for each fabric?
(b) Using the cell-means model, test the hypothesis Ho: [τ₁ = ... = τ36] against the alternative hypothesis HA: [at least two τ's differ]. What can you conclude?
(c) Write down a contrast in the treatment combinations that compares the polyester fabric with the nylon fabric. Is your contrast estimable?
(d) If your contrast in (c) is estimable, give a formula for the least squares estimator and its variance. Otherwise, go to part (e).
(e) Assuming that you are likely to be interested in a very large number of contrasts and you want your overall confidence level to be 95%, calculate a confidence interval for any pairwise comparison of your choosing. What does the interval tell you?
(f) Calculate a 90% confidence bound for σ².
(g) If you were to repeat this experiment and you wanted your confidence interval in (d) to be of length at most 20%, how many observations would you take on each treatment combination?
6. **Weathering experiment, continued**Suppose you were to analyze the weathering experiment described in Exercise 5 using a four-way complete model.
(a) What conclusions can you draw from the analysis of variance table?
(b) Give an explicit formula for testing that the FA-interaction is negligible.
(c) Would confidence intervals for differences in fabrics be of interest? If not, why not?If so, how would they be interpreted? Give a formula for such confidence intervals assuming that these intervals are preplanned and are the only intervals envisaged, and the overall level is to be at least
Two observations were taken on each brand-oil combination for each of the pop-ping times. The observations were collected in a random order and are summarized in Table 7.25.
(a) Analyze the experiment as a mixed array, using a three-way complete model. Draw an ABT interaction plot, similar to that of Figure 7.8, page 220. If the goal of the experiment is to find brand-oil combinations that give a high percentage of edible kernels and that are not too sensitive to the
(b) Does the store brand of popcorn differ substantially in terms of percentage of edible kernels from the average of the name brands? Do the different types of oil differ?State your overall confidence levels or significance levels.
(c) Analyze the experiment as a product array, and calculate the sample average and the log sample variance percentage of popped kernels for each brand-oil combi-nation. Draw AB interaction plots similar to those of Figure 7.9, page 223. If the goal of the experiment is still to find brand-oil
(a) Explain why we could have regarded this as an experiment for the control of noise variability, with brand of paper towel as the design factor and the amount and type of liquid as noise factors.
(b) Use the complete model and construct an analysis of variance table. For each signif- icant interaction effect involving brand (if any), draw the corresponding interaction plot. Discuss the overall performance and relative robustness of the two brands of paper towel.
(c) For each level j of brand of paper towel, there are twelve combinations ikt (i = 1, 2, k = 1, 2,1 = 1, 2, 3) of levels of the noise factors "amount," "type of liquid," and observation number. For each j, calculate the sample variances and log sample variance In(s) of these twelve observations.
(d) The best brand is that which has the greatest average strength and the smallest variance in the strength (calculated over the noise factors). Which brand would you recommend? Does your recommendation agree with your discussion in part (b)?
3. Systolic blood pressure experiment A pilot experiment was run by John Spitak in 1987 to investigate the effect of jogging on systolic blood pressure. Only one subject was used in the pilot experiment, and a main experiment involving a random sample of subjects from a population of interest would
(a) Fit a simple linear regression model to the data and test for model lack of fit.
(b) Use residual plots to check the assumptions of the simple linear regression model.
(c) Give a 95% confidence interval for the slope of the regression line.
(d) Using the confidence interval in part (c), test at significance level α = 0.05 whether the linear term is needed in the model.
(e) Repeat the test in part (d) but using the formula for the orthogonal polynomial linear trend coefficients for unequally spaced levels and unequal sample sizes given in Section 4.2.4. Do these two tests give the same information?
(f) Estimate the mean systolic blood pressure of the subject after jogging in place for 35 seconds and calculate a 99% confidence interval.
(g) The current experiment was only a pilot experiment. Write out a checklist for the main experiment.
4. Trout experiment, continued The data in Table 8.13 show the measurements of hemoglobin (grams per 100 ml) in the blood of brown trout. (The same data were used in Exercise 15 of Chapter 3.) The trout were placed at random in four different troughs. The fish food added to the troughs contained,
(a) Fit a quadratic regression model to the data.
(b) Test the quadratic model for lack of fit.
(c) Use residual plots to check the assumptions of the quadratic model.
(d) Test whether the quadratic term is needed in the model.
(e) Use the fitted quadratic model to estimate the number of grams of sulfamerazine per 100 pounds of fish to maximize the mean amount of hemoglobin in the blood of the brown trout.
5. Bean-soaking experiment, continued Use residual plots to check the assumptions of the cubic regression model for the data of the bean-soaking experiment. (The data are in Table 8.6, page 264).
6. Bean-soaking experiment, continued Suppose the experimenter in the bean-soaking experiment of Section 8.8 had presumed that the quadratic regression model would be adequate for soaking times ranging from 12 to 30 hours.
(a) Figure 8.8, page 268, shows the fitted response curve and the standardized residuals each plotted against soaking time. Based on these plots, discuss model adequacy.
(b) Test the quadratic model for lack of fit.
8. Orthogonal polynomials Consider use of the quadratic orthogonal polynomial regression model (8.7.19), page 260, for the data at levels 18, 24, and 30 of the bean-soaking experiment—the data are in Table 8.6, page 264.
(a) Compute the least squares estimates of the parameters.
(b) Why is it not possible to test for lack of fit of the quadratic model?
(c) Give an analysis of variance table and test the hypothesis that a linear model would provide an adequate representation of the data.
9. Heart-lung pump experiment, continued In Example 8.5, page 254, we fitted a linear regression model to the data of the heart-lung pump experiment. We rejected the null hypothesis that the slope of the line is zero.

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Design And Analysis Of Experiments 1st Edition Angela M. Dean, Daniel Voss - Solutions

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