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statistics principles and methods

Questions and Answers of Statistics Principles And Methods

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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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,
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
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
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
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
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
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
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
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
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
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
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
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
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
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 =
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
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)
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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
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
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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