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Probability Statistics And Reliability For Engineers And Scientists 3rd Edition Bilal M. Ayyub, Richard H. McCuen - Solutions

Using the data from Problem 12.29, compute the two-sided, 95% confidence interval for the mean value of Y at X = 6. Also, compute a similar interval at X = 0. Compare the two intervals and explain the difference in their widths.
Compare correlation analysis and regression analysis as tools in decision making.
Using the data set given below, regress (a) Y on X, and (b) X on Y. (c) Compute the correlation coefficient for each regression; (d) Transform the regression of Y on X into an equation for computing X,and compare these coefficients with the coefficients for the regression of X on Y. (e) Why do the
Regress Y on X1, Y on X2, and Y on X3 with the following data:X1 X2 X3 Y 1 2 3 1 2 2 1 3 5 5 7 5 6 4 4 7 Then transform the three equations algebraically to solve for X. Regress X1 on Y, X2 on Y, and X3 on Y.Use the equations for regressing the X’s on Y and the transformed equation to predict X.
The data shown are the results of a tensile test of a steel specimen, where Y is elongation in thousands of an inch that resulted when the tensile force was X thousands of pounds. Fit the data with a linear model and make a complete assessment of the results (use α = 5% where necessary).X 0.8 1.6
The stream reaeration coefficient (Y) is related to the water temperature (T) in °C. Fit the following with a linear model and make a complete assessment of the results (use α = 10% where necessary):X 14 23 17 11 20 15 18 11 25 19 13 24 16 23 21 Y 2.89 4.20 4.17 3.69 3.78 3.56 3.35 2.69 3.58 3.06
Use simulation to verify the relationships for computing the standard errors of the two coefficients of the linear bivariate regression equation.
Use simulation to find the standard error and the critical values for 10 and 5% of the slope coefficient of the linear zero-intercept bivariate regression model (Yˆ = bX) for a sample size of 10. Assume X to be normally distributed, X ~ N(mean = 20, standard deviation = 4), and generate values of
Identify variables that would affect the rate of corrosion of steel bridges.
Identify variables that would affect the output of a solar panel.
Identify variables that would affect the accumulation of trash in a stream or river.
Show sketches of (X,Y) data that have correlation coefficients of approximately (a) 0.95; (b) 0.5; (c) 0;(d) – 0.8.
A set of data for predicting a student’s score on a test (Y) in a probability and statistics course includes the following variables: X1 is the student’s grade point average (GPA); X2 is the student’s age; X3 is the number of hours the student studied for the test; and X4 is the number of
A data set of 32 stream sections includes the following variables: X1 is the depth of flow; X2 is the roughness of the channel bottom; X3 is the slope of the stream bed; X4 is the mean soil particle diameter;and Y is the erosion rate of the channel bed. The data yield the following correlation
Using the principle of least squares, derive the normal equations for the following model:Yˆ = b X + b X + b X 1 1 2 1 3 3
For parts (a) and (b) use the data, which consist of a criterion variable and three predictor variables;plot all combinations of variables (six graphs); compute the correlation matrix; discuss the degree of intercorrelation and predictor–criterion correlation and the potential for accurate
The cost of producing power (P) in mills per kilowatt hour is a function of the load factor (L) in percent and the cost of coal (C) in cents per million Btu. Perform a regression analysis to develop a regression equation for predicting P using the following data:L C P 85 15 4.1 80 17 4.5 70 27 5.6
Visitation to lakes for recreation purposes varies directly with the population of the nearest city or town and inversely with the distance between the lake and city. Use the data below to develop a regression equation to predict visitation (V) in person-days per year, using as predictors the
Discuss the potential effects of using bivariate regression models when the criterion variable Y is in fact a function of several predictor variables.
What is a partial regression coefficient? Standardized partial regression coefficient? What is the relationship between the two coefficients?
For the data base that follows, derive the least-squares estimates of the regression coefficients for the model y = bo + b1X1 + b2X2. (Note: The computations can be simplified by subtracting the mean value from each observation; if such a transformation is made, it is necessary to adjust the
The peak discharge (Q) of a river is an important parameter for many engineering design problems, such as the design of dam spillways and levees. Accurate estimates can be obtained by relating Q to the watershed area (A) and the watershed slope (S). Perform a regression analysis using the following
Sediment is a major water-quality problem in many rivers. Estimates are necessary for the design of sediment control programs. The sediment load (Y) in a stream is a function of the size of the contributing drainage area (A) and the average stream discharge (Z). Perform a regression analysis using
Snowmelt runoff rate (M), which serve as a source of water for irrigation, water supply, and power, are a function of the precipitation (P) incident to the snowpack and the mean daily temperature (T).Using the following data, derive a relationship for predicting M from P and T.P (cm/day) T (°C) M
Find the standardized partial regression coefficients for the following correlation matrix: X X Y X, [1.0 -0.4 -0.6] X Y 1.0 +0.4 1.0
Develop the calculation of the standardized partial regression coefficients for the matrix X, X, Y X [1.0 0.6 0.4 Y 1.0 0.7 1.0
Derive the standardized partial regression coefficients for the matrix X X, X, Y X, [1.0 -0.5 -0.71 X Y 1.0 0.3 1.0
Find the standardized partial regression coefficient (t) for the following matrix: Y X, X X[1.0 0.3 0.37] X Y 1.0 0.71 1.00
The stream reaeration coefficient (r) is a necessary parameter for computing the oxygen deficit sag curve for a stream reach. The coefficient is a function of the mean stream velocity in feet per second (X1), water depth in feet (X2 ), and the water temperature in degrees Celsius (X3). Use the
Interception losses (I) vary with the rainfall rate (P) and the type of cover. For a small grain, the interception is also a function of the average crop height (h). The following data were measured during rainstorms from an experimental plot.h (ft) P (in.) I (in.) h (ft) P (in.) I (in.)0.83 0.68
Compute the values of the determinants for the following intercorrelation matrices, and discuss the implications for the rationality of regression coefficients (a) [1.0 0.6 -0.4] (b) [1.0 -0.2 1.0 -0.5 1.0 0.87 1.0 -0.5 1.0
Compute the determinants of matrices D and E and discuss the potential rationality of the regression coefficients. 1 0.2 0 1 0.2 0 (a) D=0.2 1 0.2 (b) E=0.2 1 0 0 0.2 1 0 0 1
Compute the determinant of matrix G and discuss the potential rationality of the regression coefficients: 1.00 0.22 0.58 0.82 0.22 1.00 0.26 0.30 G= 0.58 0.26 1.00 0.75 0.82 0.30 0.75 1.00
Compute the determinant of matrix H and discuss the implication for model rationality: 1.00 0.34 -0.17 -0.44 0.34 1.00 -0.05 -0.19 H = -0.17 -0.05 1.00 0.07 -0.44 -0.19 0.07 1.00
Determine the determinant for the following matrix of intercorrelations. Discuss the implications for model rationality. 1.0 -0.621 0.8 (a) A = (b) B= -0.62 1.0 0.8 1
Determine the determinant for the following matrix of intercorrelations. Discuss the implications for model rationality. 0.3 0.71 - 0.2 0.2 0.5 (b) C=0.2 1 0.2 (a) B- 1 1 0.2 0.2 1
Determine the determinant for the following matrix of intercorrelations. Discuss the implications for model rationality 1 -0.7 0.25] 1 0.8 0.8 (a) G= 1 -0.20 1.0 = (b) F 0.8 1 F- 0.8 0.8 0.8 1
Starting with Equation 13.8, derive Equation 13.2.
By creating a new predictor variable for the term involving the product X1X2, show the normal equations for the following model:Using the data base from Table 13.16, insert the values of the summations into the normal equations.(Note: It is not necessary to solve for the values of the unknowns.) Y
Discuss the advantages and disadvantages of both the multiple correlation coefficient and R2 as goodness of fit statistics.
Discuss the advantages and disadvantages of the standard error of estimate (Se) and (Se/Sy) as a goodness of fit statistics.
Use the principle of least squares to derive the normal equations for the following model: = bxas + 52 b
Using the method of least squares, derive the normal equations for the model Yˆ = b1X + b2X2. Evaluate the coefficients using the data of Problem 12.18.
Using the data of Problem 12.47, fit a linear model and a quadratic polynomial and make a complete assessment of each. Which model is best for prediction? Support your choice.
Using the data of Problem 12.29, fit a linear model and a quadratic polynomial and make a complete assessment of each. Which model is best for prediction? Support your choice.
Using the following measurements of y and x, perform a bivariate polynomial regression analysis.Use a partial F test with a 5% level of significance to determine whether the model should be linear, quadratic, or cubic. Compute the correlation coefficient, standard error of estimate, and total F
Using the following measurements y and x, perform a regression analysis for a bivariate polynomial.Use a partial F test with 1% level of significance to decide whether the model should be linear, quadratic, or cubic. Compute the correlation coefficient, standard error of estimate, and the total F
Given the values of y and x, determine the values of the coefficients for the model y = b0 + b1x + b2x2.Assess the goodness of fit (R, R2 Se/Sy, Se). y 5 DO 8 11 13 15 x 1 2 3 4 55 19 6
For a set of data based on a sample size of 24, the coefficients of determination (R) for linear, quadratic, and cubic polynomials are 0.53, 0.64, and 0.75. Perform an ANOVA. At a 5% level of significance, which model would be recommended?
For a set of data based on a sample size of 11, the coefficients of determination (R) for linear, quadratic, and cubic polynomials are 0.41, 0.46, and 0.59. Perform an ANOVA. At a 5% level of significance, which model would be recommended?
For a set of data based on a sample size of 18, the coefficients of determination (R) for linear, quadratic, cubic, and quartic polynomials are 0.47, 0.58, 0.61, and 0.74, respectively. Perform an ANOVA. At a 5% level of significance, which model would be recommended?
Using the method of least squares, derive the normal equations for the model Yˆ = axb. Evaluate the coefficients using the data of Problem 12.18. Solve the problem (a) directly (without transformation)and (b) by converting the equation to linear form by making a logarithmic transformation. Compare
Derive the normal equations for the following model: = axx
Using the data of Problem 12.71, fit a linear model and a power model. Make a complete assessment of each. Which model is best for prediction? Support your choice.
Using the data of Problem 12.72, fit a linear model and a power model. Make a complete assessment of each. Which model is best for prediction? Support your choice.
Using the data of Problem 13.9, fit a multiple power model. Make a complete assessment of the model.
Using the method of least squares, derive the normal equations for the model Yˆ = b0 + b1 X .Evaluate the coefficients using the data of Problem 12.18.
Place the equation y b x x b b = + −0 12 1 2 1 ( ) 1 in a form that can be calibrated using linear multiple regression.Show the equations for estimating the coefficients.
Use the principle of least squares to derive formulas for computing the partial regression coefficients for the nonlinear equation y = b1x + b2x2.
Develop a table with column headings: (1) criteria, (2) indication of, (3) advantages, (4) disadvantages,(5) decision mode, and (6) rank. Under the column “criteria” list each of the criteria that are used in selecting a final model. In the column “rank” specify the order of importance of
Transform the predictor variable for the modelso that the coefficients can be evaluated using bivariate regression. Obtain estimates of the unknowns a and b using the data base.Compute the correlation coefficient and standard error of estimate, and compare the goodness of fit statistics with those
Transform the following model to linear form so that the coefficients can be evaluated using the principle of least squares. Obtain estimates of the unknowns a and b using the data baseCompute the correlation coefficient and the standard error of estimate, and comment on the goodness of fit. y= 10+
Transform the modelto a linear form a so that the coefficients can be evaluated using least squares. Obtain estimates for the regression coefficients a and b using the data baseCompute the correlation coefficient and standard error of estimate, and comment on the goodness of fit. y = e+ bx
For the following performance function, determine the safety index (β) using (a) the FORM and(b) the ASM method:The noncorrelated random variables are assumed to have the following probabilistic characteristics: Z=X,X-X,
For the following performance function, determine the safety index (β) using (a) the FORM and(b) the ASM method:The noncorrelated random variables are assumed to have the following probabilistic characteristics: Z=X,X-x
Redo Problem 14.1 using (a) the direct MCS method, and (b) the CE method. Use 100 and 2000 simulation cycles to estimate the failure probability and its COV.
Redo Problem 14.2 using (a) the direct MCS method, and (b) the CE method. Use 100 and 2000 simulation cycles to estimate the failure probability and its COV.
For the following performance function, determine the safety index (β) using (a) the FORM and(b) the ASM method:The uncorrelated random variables are assumed to have the following probabilistic characteristics:Random Variable Mean Value COV Distribution Type X1 10 0.25 Normal X2 5 0.05 Normal X3 3
For the following performance function, determine the safety index (β) using (a) the FORM, and(b) the ASM method:Z = X1 − X2 − X3 The noncorrelated random variables are assumed to have the following probabilistic characteristics:Random Variable Mean Value COV Distribution Type X1 10 0.25
For the following performance function, determine an estimated failure probability and its COV using (a) the direct simulation method, and (b) the CE method with 100 and 2000 cycles:The last term in this equation introduces noise to the performance function. Assume that the constants I1 = 1 and I2
In Problem 14.7, study the effect of changing the level of noise in the performance function.Investigate the effect of using I1 = 1, 5, and 10 given that I2 = 1 or 10 using the CE method, with N = 10, 100, 1000, and 2000 cycles. Discuss your results.
Redo Problem 14.7 using (a) IS, and (b) CE with AV.
Redo Problem 14.8 using (a) IS, and (b) CE with AV.
The stability of a vehicle, such as a truck, can be measured using a simplified static model as follows:Stability =T 2H where T is the nonrandom truck width (i.e., center of the right front tire to the center of the left front tire), and H is the random location of the center of gravity of the
Use the stability model for a truck as provided in Problem 14.11 to compute the mean H that you would permit for loading a truck by stacking loads vertically so that the failure (rollover) probability does not exceed 0.01 for mean values of T of 5 to 10 ft. Use lognormal probability distributions.
Catchment basins are used to protect residential areas and highways from mud slides. A mudslide basin has a capacity of C, which is a random variable with a COV of 0.2. The expected 25-year maximum inflow is denoted F with a COV of 1.0. Develop an expression for computing the reliability index and
Use the reliability model for a catchment basin as provided in Problem 14.13 to compute the mean C that you would permit for a flow F so that the failure probability does not exceed 0.01. Provide your answer in the form of a mean C/F ratio. Perform parametric analysis by varying the target failure
The change in the length of a rod due to axial force P is given byL PL AE=where L is the length of the rod, P is the applied axial force, A is the cross-sectional area of the rod, and E is the modulus of elasticity. Assume that the change in length of the rod must not exceed 0.15 in. Using 2000
For the rod in Problem 14.15, study the effect of increasing the number of simulation cycles on the estimated failure probability. Use the following numbers of simulation cycles: 20, 100, 500, 1000, and 2000. Provide your results in the form of plots of estimated failure probability and its COV as
The ultimate moment capacity, M, of an under-reinforced concrete rectangular section is given bywherein which the following are random variables: As is the cross-sectional area of the reinforcing steel, fy is the yield stress (strength) of the steel, d is the distance from the reinforcing steel to
Use the performance function and probabilistic characteristics of the random variables of Problem 14.5, except for the mean value of X1, to compute the partial safety factors for a target reliability level of 3.0.
Use the performance function and probabilistic characteristics of the random variables of Problem 14.6, except for the mean value of X1, to compute the partial safety factors for a target reliability level of 3.0.
Use the performance function and probabilistic characteristics of the random variables of Problem 14.1, except for the mean value of X1, to compute the partial safety factors for a target reliability level of 3.0.
Use the performance function and probabilistic characteristics of the random variables of Problem 14.2, except for the mean value of X1, to compute the partial safety factors for a target reliability level of 3.0.
The following system consists of components with the indicated nonfailure probabilities:Compute the reliability of the system assuming independent failure events for the components. 0.9 0.80 0.85 0.99 0.95 0.95
For the system described in Problem 15-1, develop a fault tree model and evaluate the minimal cut set. Determine the reliability of the system.
The following system consists of components with the indicated nonfailure probabilities:Compute the reliability of the system assuming independent failure events for the components. A B E C D F
For the system described in Problem 15-3, develop a fault tree model and evaluate the minimal cut set. Determine the reliability of the system.
For the system described in Problem 15-3, develop a fault tree model and evaluate the minimal cut set using the algorithm at the end of Section 15.2.4. Determine the reliability of the system.
Develop a FTA program that evaluates the minimal cut set using the algorithm at the end of Section 15.2.4. Determine the reliability of several systems to demonstrate your program.
Select an engineering system that interests you, define the system, define a system failure criterion, and develop (a) a fault tree model for the system, and (b) an event tree model.
The following system consists of six components with the following reliability values and associated costs:(a) Develop an event tree model for the system, and (b) using risk analysis, select one of the following two improvement options: A B E C D F
A system consists of n identical components in series with each component having a reliability of p.The failure events of the components can be assumed to be independent. Compute the reliability of the system as functions of p and n. Plot system reliability as functions of p and n for the ranges of
A system consists of n identical components in parallel with each component having a reliability of p. The failure events of the components can be assumed to be independent. Compute the reliability of the system as functions of p and n. Plot system reliability as functions of p and n for the ranges
A system consists of n1 sets of n2 identical components in series with each component having a reliability of p. The n1 sets are in parallel. The failure events of the components can be assumed to be independent. Compute the reliability of the system as functions of p, n1, and n2. Plot system
A system consists of n1 sets of n2 identical components in parallel, with each component having a reliability of p. The n1 sets are in series. The failure events of the components can be assumed to be independent. Compute the reliability of the system as functions of p, n1, and n2. Plot system
A system consists of N identical components, with each component having a reliability of p. The failure of the system is defined as the failure of any n out of N components. The failure events of the components can be assumed to be independent. Compute the reliability of the system as functions of
For the system described in Problem 15.13, develop a fault tree model, and evaluate the minimal cut set for N = 20, n = 3, and p = .9. Determine the reliability of the system.
For the system described in Problem 15.13, develop a fault tree model and evaluate the minimal cut set for N = 20, n = 5, and p = .9. Determine the reliability of the system.
For the system described in Problem 15.13, develop a fault tree model and evaluate the minimal cut set for N = 20, n = 10, and p = .9. Determine the reliability of the system.

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