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Miller And Freunds Probability And Statistics For Engineers 9th Edition Richard A. Johnson - Solutions

12.35 (a) Using q0.10 = 3.921 for the Tukey HSD method, compare the strength of the 5 linen threads in Exercise 12.23. (b) Use the Bonferroni confidence interval approach on page 411, with α = 0.10, to compare the mean linen thread strengths in Exercise 12.20.
12.34 An approximation to Tukey HSD confidence intervals for mildly unequal sample sizes When the there are only small differences in the sample sizes, an approximation is available. The MSE and its degrees of freedom still come from the ANOVA table but √ 2 MSE / n is replaced by MSE 1 ni + 1
12.33 Referring to Exercise 12.3, use Bonferroni simultaneous confidence intervals with α = 0.06 to compare the mean number of electrodes coated by the experiment under the 3 different alternatives.
12.32 Referring to Example 1, use q0.05 = 3.776 for the Tukey HSD method to examine differences in mean coating weights.
12.31 Using q0.01 = 5.499 for the Tukey HSD method, examine differences among the ball bearings in Exercise 12.19.
12.30 Verify that the computing formulas for SST, SS(Tr). SS(BI), and SSE, given on page 409, are equiva- lent to the corresponding terms of the identity of Theorem 12.2.
12.29 Show that if μi j = μ + αi + βj, the mean of the μi j (summed on j) is equal to μ + αi, and the mean of μi j (summed on i and j) is equal to μ, it follows that a i=1 αi = b j=1 βj = 0
12.28 As was pointed out on page 308, two ways of increasing the size of a two-way classification experiment are (a) to double the number of blocks, and (b) to replicate the entire experiment. Discuss and compare the gain in degrees of freedom for the error sum of squares by the two methods.
12.27 The following are the number of defectives produced by the 4 workers operating, in turn, 3 different machines. In each case, the first figure represents the number of defectives produced on a Friday and the second figure represents the number of defectives produced on the following Monday:
12.26 If, in a two-way classification, the entire experiment is repeated r times, the model becomes Yijk = μ + αi + βj + εijk for i = 1, 2,...,a, j = 1, 2,...,b, and k = 1, 2,...,r, where the sum of the α’s the sum and the β’s are equal to zero. The εijk are independent normally
12.25 To emphasize the importance of blocking, reanalyze the cleanness data Example 7 as a one-way classification with the 4 detergents being the different treatments.
12.24 An industrial engineer tests 4 different shop-floor layouts by having each of 6 work crews construct a subassembly and measuring the construction times (minutes) as follows: Layout 1 Layout 2 Layout 3 Layout 4 Crew A Crew B Crew C Crew D Crew E Crew F 48.2 53.1 51.2 58.6 49.5 52.9 50.0 60.1
12.23 A laboratory technician measures the breaking strength of each of 5 kinds of linen thread by means of 4 different instruments and obtains the following results (in ounces): Measuring Instrument I1 I2 I3 I4 Thread 1 Thread 2 Thread 3 Thread 4 Thread 5 20.6 20.7 20.0 21.4 24.7 26.5 27.1 24.3
12.22 Four different, though supposedly equivalent, forms of a standardized reading achievement test were given to each of 5 students, and the following are the scores which they obtained: Student 1 Student 2 Student 3 Student 4 Student 5 Form A Form B Form C Form D 75 73 59 69 84 83 72 56 70 92 86
12.21 Looking at the days (rows) as blocks, rework Exercise 12.5 by the method of Section 12.3.
12.20 The analysis of variance for a randomized-block design is conveniently implemented using MINITAB. With reference to Example 7, first open C12Ex7.MTW in the MINITAB data bank. Dialog Box: Stat > ANOVA > Balanced ANOVA. Enter Cleanness in Responses. Enter Engine and Detergent in Model. Click
12.19 Concerns about the increasing friction between some machine parts prompted an investigation of four dif- ferent types of ball bearings. Five different machines were available and each type of ball bearing was tried in each machine. Given the observations on tempera- ture, coded by subtracting
12.18 A randomized-block experiment is run with three treatments and four blocks. The three treatment means are y1• = 6, y2• = 7, and y3• = 11. The total (corrected) sum of squares is 220 = 3 i=1 b j=1 ( yi j − y• • ) 2 The analysis of variance (ANOVA) table takes the form Source of
12.17 Samples of peanut butter produced by 2 different manufacturers are tested for aflatoxin content, with the following results: Aflatoxin Content (ppb) Brand A Brand B 0.5 4.7 0.0 6.2 3.2 0.0 1.4 10.5 0.0 2.1 1.0 0.8 8.6 2.9(a) Use analysis of variance to test whether the two brands differ in
12.16 With reference to Exercise 12.9, determine individual 95% confidence intervals for the differences of mean reaction times.
12.15 Verify the alternative formulas for computing SST and SS(Tr) given on page 399.
12.14 Show that if μi = μ + αi and μ is the mean of the μi, it follows that k i=1 ni αi = 0
12.13 Referring to the discussion on page 386, assume that the standard deviations of the tin-coating weights determined by any one of the 4 laboratories have the common value σ = 0.012, and that it is desired to be 95% confident of detecting a difference in means between any 2 of the
12.12 Corrosion rates (percent) were measured for 4 different metals that were immersed in a highly corrosive solution: Aluminum: 75 77 76 79 74 77 75 Stainless Steel: 74 76 75 78 74 77 75 77 Alloy I: 73 74 72 74 70 73 74 71 Alloy II: 71 74 74 73 74 73 71 (a) Perform an the analysis of variance
12.11 Two tests are made of the compressive strength of each of 6 samples of poured concrete. The force required to crumble each of 12 cylindrical specimens, measured in kilograms, is as follows: Sample A B CDE F Test 1 110 125 98 95 104 115 Test 2 105 130 107 91 96 121 Test at the 0.05 level of
12.10 Several different aluminum alloys are under consideration for use in heavy-duty circuit-wiring applications. Among the desired properties is low electrical resistance, and specimens of each wire are tested by applying a fixed voltage to a given length of wire and measuring the current
12.9 To find the best arrangement of equipment at the rear of a fire truck, 3 different arrangements were tested by simulating a fire condition and observing the reaction time required to extinguish the fire. The reaction time (in seconds) of 24 firefighters (randomly assigned to the different
12.8 The one-way analysis of variance is conveniently implemented using MINITAB. With reference to the example on page 389, we first set the observations in columns: DATA: C1: 13 10 8 11 8 C2: 13 11 14 14 C3: 4 1 3 4 2 4 Dialog Box: Stat > ANOVA > One-way. Pull down Response data are in
12.7 Given the following observations collected according to the one-way analysis of variance design, Setting 1 18 17 15 18 Setting 2 2 4 5 86 Setting 3 16 10 7 Setting 4 14 18 (a) decompose each observation yi j as
12.6 With reference to the example on page 389, suppose one additional observation y25 = 8 is available using formula B. Construct the analysis of variance table and test the equality of the mean curing times using α = 0.05.
12.5 The following are the numbers of mistakes made in 5 successive days for 4 technicians working for a photographic laboratory: Technician Technician Technician Technician I II III IV 5 17 9 9 12 12 11 13 9 15 6 7 8 14 14 10 11 17 10 11 Test at the level of significance α = 0.01 whether the
12.4 Using the sum of squares obtained in Exercise 12.3, test at the level of significance α = 0.01 whether the differences among the means obtained for the 3 samples are significant.
12.3 Three alternatives are suggested for electroplating to reduce dissolved metal cations so that they form a coherent metal coating on an electrode. In an experiment conducted to compare the manufacturing yields using the three alternatives, experimenters record the number of electrodes coated
12.2 A certain bon vivant, wishing to ascertain the cause of his frequent hangovers, conducted the following experiment. On the first night, he drank nothing but whiskey and water; on the second night, he drank vodka and water; on the third night, he drank gin and water; and on the fourth night, he
12.1 An experiment is performed to compare the rotational speed of two conveyers, Conveyer X and Conveyer Y. 30 belts are loaded with an optimal weight, each is put on one of the conveyers, and the speed of the conveyer is measured. Criticize the following aspects of the experiment. (a) To
11.86 Robert Boyle (1627–1691) established the law that (pressure × volume) = constant for a gas at a constant temperature. By pouring mercury into the open top of the long side of a J-shaped tube, he increased the pressure on the air trapped in the short leg. The volume of trapped air = h ×
11.85 Robert A. Millikan (1865–1953) produced the first accurate measurements on the charge e of an electron. He devised a method to observe a single drop of water or oil under the influence of both electric and gravitational fields. Usually, a droplet carried multiple electrons, and direct
11.84 With reference to Exercise 11.78, use the theory of Exercise 11.61 to calculate the multiple correlation coefficient (which measures how strongly the damage is related to both weight and distance).
11.83 Assuming that the necessary assumptions are met, construct a 95% confidence interval for ρ when (a) r = 0.78 and n = 15; (b) r = −0.62 and n = 32; (c) r = 0.17 and n = 35.
11.82 If for certain paired data n = 18 and r = 0.44, test the null hypothesis ρ = 0.30 against the alternative hypothesis ρ > 0.30 at the 0.01 level of significance.
11.81 If r = 0.41 for one set of paired data and r = 0.29 for another, compare the strengths of the two relationships.
11.80 Use the expression on page 367, involving deviations from the mean, to calculate r for the following data: x y 3 8 1 6 2 3 9 6 5 7
11.79 With reference to Exercise 11.9, (a) find a 95% confidence interval for the mean current density when the strain is x = 0.50; (b) find 95% limits of prediction for the current density when a new diode has stress x = 0.50.
11.78 The following are sample data provided by a moving company on the weights of six shipments, the distances they are moved, and the damage that was incurred: Weight Distance Damage (1,000 pounds) (1,000 miles) (dollars) x1 x2 y 4.0 1.5 160 3.0 2.2 112 1.6 1.0 69 1.2 2.0 90 3.4 0.8 123 4.8
11.77 The rise of current in an inductive circuit having the time constant τ is given by I = 1 − e−t/τ where t is the time measured from the instant the switch is closed, and I is the ratio of the current at time t to the full value of the current given by Ohm’s law. Given the measurements
11.76 With reference to Exercise 11.75, use the method of Section 11.2 to construct a 95% confidence interval for γ . State what assumptions will have to be made.
11.75 In an experiment designed to determine the specific heat ratio γ for a certain gas, measurements of the volume and corresponding pressure p produced the data: p (lb/in. 2) 16.6 39.7 78.5 115.5 195.3 546.1 V(in. 3 ) 50 30 20 15 10 5 Assuming the ideal gas law p ·Vγ = C, use these data to
11.74 With reference to Example 15, (a) find the least squares line for predicting the chromium in the effluent from that in the influent after taking natural logarithms of each variable;(b) predict the mean ln (effluent) when the influent has 500 μg/l chromium.
11.73 With reference to the preceding exercise, construct a 95% confidence interval for α.
11.72 To determine how well existing chemical analyses can detect lead in test specimens in water, a civil engineer submits specimens spiked with known concentrations of lead to a laboratory. The chemists are told only that all samples are from a study about measurements on “low”
11.71 With reference to Exercise 11.69, find the proportion of variance in the amount of NOx explained by the amount of additive.
11.70 With reference to Exercise 11.69, find the 95% limits of prediction when the amount of additive is 4.5.
11.69 A chemical engineer found that by adding different amounts of an additive to gasoline, she could reduce the amount of nitrous oxides (NOx) coming from an automobile engine. A specified amount was added to a gallon of gas and the total amount of NOx in the exhaust collected. Suppose, in
11.68 With reference to Exercise 11.65, (a) find a 99% confidence interval for the mean bat- tery backup at x = 1.25; (b) find 95% limits of prediction for the battery backup provided by a laptop charged for 1.25 hours.
11.67 With reference to Exercise 11.65, test the null hypothe- sis =1.5 against the alternative hypothesis > 1.5 at the 0.01 level of significance.
11.66 With reference to Exercise 11.65, construct a 99% con- fidence interval for a.
11.65 The data below pertains to the number of hours a laptop has been charged for and the number of hours of backup provided by the battery. (a) Use the first set of expressions on page 330, involving deviations from the mean, to fit a least squares line to the observations. (b) Use the equation
11.64 To calculate r using MINITAB when the x values are in C1 and the y values are in C2, use Dialog box: Stat>Basic Statistics>Correlation Type C1 and C2 in Variables. Click OK. Also, you can make a scatter plot using the plot procedure in Exercise 11.22. Use the computer to do Exercise 11.50.
11.63 Referring to the nano twisting data in Exercise 11.24, calculate the correlation coefficient.
11.61 Instead of using the computing formula on page 367, we can obtain the correlation coefficient r with the formula r = ± 1 − (y −y)2 (y − y)2 which is analogous to the formula used to define ρ. Although the computations required by the use of this formula are tedious, the formula
11.60 Show that for the bivariate normal distribution (a) independence implies zero correlation; (b) zero correlation implies independence.
11.59 (a) Evaluating the necessary integrals, verify the identities μ2 = α + β μ1 and σ2 2 = σ2 + β2 σ2 1 on page 374. (b) Substitute μ2 = α + βμ1 and σ2 2 = σ2 + β2σ2 1 into the formula for the bivariate density given on page 374, and show that this gives the final form shown on
11.58 Assuming that the necessary assumptions are met, construct a 95% confidence interval for ρ when (a) r = 0.72 and n = 19; (b) r = 0.35 and n = 25; (c) r = 0.57 and n = 40.
11.57 If data on the ages and prices of 25 pieces of equipment yielded r = −0.58, test the null hypothesis ρ = −0.40 against the alternative hypothesis ρ < −0.40 at the 0.05 level of significance. Assume bivariate normality.
11.56 If r = 0.83 for one set of paired data and r = 0.60 for another, compare the strengths of the two relationships.
11.55 Referring to Example 3 concerning nanopillars, calculate the correlation coefficient between height and width.
11.54 The following are measurements of the total dissolved salts (TDS) and hardness index of 22 samples of water.TDS (ppm) Hardness Index 200 15 325 24 110 8 465 34 580 43 925 69 680 50 290 21 775 57 375 28 850 63 625 46 430 32 275 20 170 13 555 41 595 44 850 63 245 18 195 14 650 48 775 57 (a)
11.53 Calculate r for the changes in the flow of vehicles and the level of pollution in Exercise 11.12. Assuming that the necessary assumptions can be met, construct a 99% confidence interval for the population correlation coefficient ρ.
11.52 Calculate r for the temperatures and tearing strengths of Exercise 11.3. Assuming that the necessary assumptions can be met, test the null hypothesis ρ = 0.60 against the alternative hypothesis ρ > 0.60 at the 0.10 level of significance.
11.51 With reference to Exercise 11.50, test ρ = 0 against ρ = 0 at α = 0.05.
11.50 The following data pertain to the processing speed (GHz) of a computer and the time (minutes) it takes to boot up:Processing Speed Boot Time 1.2 5 1.0 1 1.3 3 1.6 2 1.8 4 1.1 8 0.8 1 0.9 7 1.1 1 1.4 2 1.3 4 1.2 3 1.1 8 1.7 1 0.5 6 0.8 2 1.5 9 1.6 1 1.3 10 1.7 5 2.0 1 1.4 2 1.2 3 1.5 6
significance.
11.49 Calculate r for the air velocities and evaporation coefficients of Example 2. Also, assuming that the necessary assumptions can be met, test the null hypothesis ρ = 0 against the alternative hypothesis ρ = 0 at the 0.05 level of
11.48 Use the expressions on page 367, involving the deviations from the mean, to calculate r for the following data: x y 6 6 9 2 10 4 2 8 8 10
11.47 Data, collected from cities of widely varying sizes, revealed a high positive correlation between the amount of beer consumed and the number of weddings in the past year. Will consuming lots of beer increase the number of weddings? Explain your answer.
11.46 Data, collected over seven years, reveals a positive correlation between the annual starting salary of engineers and the annual sales of diet soft drinks. Will buying more diet drinks increase starting salaries? Explain your answer and suggest a possible lurking variable.
11.45 The following residuals and predicted values were obtained from an experiment that related yield of a chemical process (y) to the initial concentration (x) of a component (the time order of the experiments is given in parentheses): Predicted Residual Predicted Residual 4.1 (5) −2 3.5 (3) 0
11.44 With reference to Exercise 11.39, analyze the residuals from the regression plane.
11.43 With reference to Exercise 11.40, in order to plot residuals, before clicking OK, you must select Dialog box: Click Storage. Check Residuals and Fits. Click OK twice. The additional steps, before clicking the second OK. Dialog box: Click Graphs. Check Residuals versus fits. Click OK twice.
11.42 To fit the quadratic regression model using MINITAB, when the x values are in C1 and the y values in C2, you must select Dialog box: Stat >Regression > Fitted Line Plot Enter C2 in Response (Y) and enter C1 in Predictor (X). Under Type of Regression Model choose Quadratic. Click OK. Use the
11.41 Using MINITAB we can transform the x values in C1 and/or the y values in C2. For instance, to obtain the logarithm to the base 10 of y, select Dialog box: Calc > Calculator Type C3 in Store, LOGT(C2) in Expression. Click OK. Use the computer to repeat the analysis of Exercise 11.27.
11.40 Multiple regression is best implemented on a computer. The following MINITAB commands fits the y values in C1 to predictor values in C2 and C3. Dialog box: Stat > Regression > Regression > Fit Regression Model. Type C1 in Response. Type C2 and C3 in Continuous predictors. Click OK. It
11.39 The following sample data were collected to determine the relationship between processing variables and the current gain of a transistor in the integrated circuit: Diffusion time Sheet resistance (hours) (2-cm) Current gain x1 x2 y 1.5 2.5 0.5 338 66 5.3 87 7.8 69 7.4. 1.2 141 9.8 2.6 93
11.38 A compound is produced for a coating process. It is added to an otherwise fixed recipe and the coating process is completed. Adhesion is then measured. The following data concern the amount of adhesion and its relation to the amount of an additive and temperature of a reaction. Additive
11.37 With reference to Exercise 11.36, estimate the hardness of a sheet of steel with a copper content of 0.05% and an annealing temperature of 1,150 degrees Fahrenheit.
11.36 Twelve specimens of cold-reduced sheet steel, having different copper contents and annealing temperatures, are measured for hardness with the following results: Copper Annealing Hardness content temperature (Rockwell 30-T) (%) (degrees F) 78.9 0.02 1,000 65.1 0.02 1,100 55.2 0.02 1,200 56.4
11.35 Verify that the system of normal equations on page 357 corresponds to the minimization of the sum of squares.
11.34 With reference to Example 11, verify that the predicted drying time is minimum when the amount of additive used is 5.1 grams.
11.33 When fitting a polynomial to a set of paired data, we usually begin by fitting a straight line and using the method on page 339 to test the null hypothesis β1 = 0. Then we fit a second-degree polynomial and test whether it is worthwhile to carry the quadratic term by comparing σ2 1 , the
11.32 The following data pertain to the amount of hydrogen present, y, in parts per million in core drillings made at 1-foot intervals along the length of a vacuum-cast ingot, x, core location in feet from x 1 2 3 4 5 6 7 8 9 10 y 1.28 1.53 1.03 0.81 0.74 0.65 0.87 0.81 1.10 1.03 (a) Draw a scatter
11.31 The number of inches which a newly built structure is settling into the ground is given by y = 3 − 3 e−α x where x is its age in months. x 2 4 6 12 18 24 y 1.07 1.88 2.26 2.78 2.97 2.99 Use the method of least squares to estimate α. [Hint: Note that the relationship between ln (3 − y
11.30 Plot the curve obtained in the preceding exercise and the one obtained in Exercise 11.26 on one diagram and compare the fit of these two curves.
11.29 Fit a Gompertz curve of the form y = eeαx + β to the data of Exercise 11.26.
11.28 Refer to Example 10. Two new observations are available. Rate of discharge(A) Capacity(Ah) ln(Capacity) 5 149.4 5.0066 18 108.2 6.5840 Add these observations to the data set in Example 10 and rework the example.
11.27 With reference to the preceding exercise, change the equation obtained in part (a) to the form y = a · e−cx, and use the result to rework part (b).
11.26 The following data pertain to water pressure at various depths below sea level: Depth (m) Pressure (psi) x y 10 140 50 74 150 218 1,600 2,400 2,110 3,060 3,580 5,150 4,800 7,000 (a) Fit an exponential curve. (b) Use the result obtained in part (a) to estimate the mean pressure at a depth of
11.25 The following data pertain to the growth of a colony of bacteria in a culture medium: Days since inoculation Count x y 3 115,000 6 147,000 9 239,000 12 356,000 15 579,000 18 864,000 (a) Plot log yi versus xi to verify that it is reasonable to fit an exponential curve. (b) Fit an exponential
11.24 Nanowires, tiny wires just a few millionths of a centimeter thick, which spiral and have a pine tree-like appearance, have recently been created.2 The investigators’ ultimate goal is to make better nanowires for high-performance integrated circuits, lasers, and biosensors. The twist, in
11.23 Referring to Example 3, the nanopillar data on height (nm) and width (nm) are Width Height 62 221 68 234 69 245 80 266 68 265 79 253 83 274 70 278 74 290 73 276 74 272 75 276 80 276 Width Height 77 290 80 292 83 289 73 284 79 271 100 292 93 308 92 303 101 308 87 315 96 309 99 300 94 305 Width
11.22 It is tedious to perform a least squares analysis without using a computer. We illustrate here a computer-based analysis using the MINITAB package. The observations on page 328 are entered in C1 and C2 of the worksheet. DATA 11-22.DAT x:01224456 y: 25 20 30 40 45 50 60 50 We first obtain the

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