Applied Statistics and Probability for Engineers 6th Edition solutions

Applied Statistics and Probability for Engineers 6
An article entitled "A Method for Improving the Accuracy of Polynomial Regression Analysis" in the Journal of Quality Technology (1971, pp. 149-155) reported the following data on \(y=\) ultimate shear strength of a rubber compound (psi) and \(x=\) cure temperature \(\left({ }^{\circ} F\right)\)

Applied Statistics and Probability for Engineers 6
Consider the following data, which result from an experiment to determine the effect of \(x=\) test time in hours at a particular temperature on \(y=\) change in oil viscosity: (a) Fit a second-order polynomial to the data. (b) Test for significance of regression usin

Applied Statistics and Probability for Engineers 6
The following data were collected during an experiment to determine the change in thrust efficiency (y, in percent) as the divergence angle of a rocket nozzle (x) changes: (a) Fit a second-order model to the data. (b) Test for significance of regression and lack of fit using \(\alpha

Applied Statistics and Probability for Engineers 6
An article in the Journal of Pharmaceuticals Sciences (1991, Vol. 80, pp. 971-977) presents data on the observed mole fraction solubility of a solute at a constant temperature and the dispersion, dipolar, and hydrogen-bonding Hansen partial solubility parameters. The data are as shown in the Table E1

Applied Statistics and Probability for Engineers 6
Consider the arsenic concentration data in Exercise 12-16. (a) Discuss how you would model the information about the persons sex. (b) Fit a regression model to the arsenic in nails using age, drink use, cook use, and the persons sex as the regressors. (c) Is there evidence that the persons sex affect

Applied Statistics and Probability for Engineers 6
Consider the gasoline mileage data in Exercise 12-11. (a) Discuss how you would model the information about the type of transmission in the car. (b) Fit a regression model to the gasoline mileage using \(\text { cid, etw }\) and the type of transmission in the car as the regressors. (c) Is there evid

Applied Statistics and Probability for Engineers 6
Consider the surface finish data in Example 12-13. Test the hypothesis that two different regression models (with different slopes and intercepts) are required to adequately model the data. Use indicator variables in answering this question.

Applied Statistics and Probability for Engineers 6
Consider the X-ray inspection data in Exercise 12-15. Use rads as the response. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the models obtained. Which model

Applied Statistics and Probability for Engineers 6
Consider the electric power data in Exercise 12-10. Build regression models for the data using the following techniques: (a) All possible regressions. Find the minimum \(\mathrm{C}_{\mathrm{p}}\) and minimum \(\mathrm{MS}_{\mathrm{E}}\) equations. (b) Stepwise regression. (c) Forward selection. (d) B

Applied Statistics and Probability for Engineers 6
Consider the regression model fit to the coal and limestone mixture data in Exercise 12-17. Use density as the response. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Com

Applied Statistics and Probability for Engineers 6
Consider the wire bond pull strength data in Exercise 12-12. Build regression models for the data using the following methods: (a) All possible regressions. Find the minimum \(\mathrm{C}_{\mathrm{p}}\) and minimum \(\mathrm{MS}_{\mathrm{E}}\) equations. (b) Stepwise regression. (c) Forward selection.

Applied Statistics and Probability for Engineers 6
Consider the gray range modulation data in Exercise 12-19. Use the useful range as the response. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the models obtai

Applied Statistics and Probability for Engineers 6
Consider the nisin extraction data in Exercise 12-18. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the models obtained. Which model would you prefer? Why?

Applied Statistics and Probability for Engineers 6
Consider the stack loss data in Exercise 12-20. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the models obtained. Which model would you prefer? Why? (f) Remov

Applied Statistics and Probability for Engineers 6
Consider the NHL data in Exercise 12-22. Build regression models for these data with regressors GF through FG using the following methods: (a) All possible regressions. Find the minimum \(\mathrm{C}_{\mathrm{p}}\) and minimum \(\mathrm{MS}_{\mathrm{E}}\) equations. (b) Stepwise regression. (c) Forwar

Applied Statistics and Probability for Engineers 6
Use the football data in Exercise 12-21 to build regression models using the following techniques: (a) All possible regressions. Find the equations that minimize \(\mathrm{MS}_{\mathrm{E}}\) and that minimize \(\mathrm{C}_{\mathrm{p}}\). (b) Stepwise regression. (c) Forward selection. (d) Backward e

Applied Statistics and Probability for Engineers 6
Consider the arsenic data in Exercise 12-16. Use arsenic in nails as the response and age, drink use, and cook use as the regressors. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimina

Applied Statistics and Probability for Engineers 6
Consider the gas mileage data in Exercise 12-11. Build regression models for the data from the numerical regressors using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the models obtained. Whi

Applied Statistics and Probability for Engineers 6
When fitting polynomial regression models, we often subtract \(\bar{x}\) from each \(\bar{x}\) value to produce a "centered" regressor \(x^{\prime}=x-\bar{x}\). This reduces the effects of dependencies among the model terms and often leads to more accurate estimates of the regression coefficients. Us

Applied Statistics and Probability for Engineers 6
Consider the data in Exercise . Use all the terms in the full quadratic model as the candidate regressors. (a) Use forward selection to identify a model. (b) Use backward elimination to identify a model. (c) Compare the two models obtained in parts (a) and (b). Which model would you prefer and why?

Applied Statistics and Probability for Engineers 6
We have used a sample of 30 observations to fit a regression model. The full model has nine regressors, the variance estimate is \(\hat{\sigma}^{2}=M S_{E}=100\), and \(R^{2}=0.92\). (a) Calculate the \(F\) -statistic for testing significance of regression. Using \(\alpha=0.05\), what would you concl

Applied Statistics and Probability for Engineers 6
Problem 105E A sample of 25 observations is used to fit a regression model in seven variables. The estimate of ?2 for this full model is MSE = 10. (a) A forward selection algorithm has put three of the original seven regressors in the model. The error sum of squares for the three-variable model is SS

Applied Statistics and Probability for Engineers 6
Consider the following computer output. (a) Fill in the missing values. Use bounds for the P-values. (b) Is the overall model significant at \(\alpha = 0.05\)? Is it significant at \alpha = 0.01? (c) Discuss the contribution of the individual regressors to the model.

Applied Statistics and Probability for Engineers 6
Consider the following inverse of the model matrix: \(\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}=\left[\begin{array}{rcc} 0.893758 & -0.028245 & -0.0175641 \\ -0.028245 & 0.0013329 & 0.0001547 \\ -0.017564 & 0.0001547 & 0.0009108 \end{array}\right]\) (a) How many variables are in the regre

Applied Statistics and Probability for Engineers 6
The data shown in Table E12-14 represent the thrust of a jet-turbine engine \(\text { (y) }\) and six candidate regressors: \(\mathrm{X}_{1}\) = primary speed of rotation, \(\mathrm{X}_{2}\) = secondary speed of rotation, \(\mathrm{X}_{3}\) = fuel flow rate, \(\mathrm{X}_{4}\) = pressure, \(\mathrm{X

Applied Statistics and Probability for Engineers 6
Consider the engine thrust data in Exercise 12-108. Refit the model using \(y^{*}=\ln y\) as the response variable and \(x_{3}^{*}=\ln _{3}\) as the regressor (along with \(x_{4} and x_{5}\)). (a) Test for significance of regression using \(\alpha=0.01\). Find the \(P \text {-value }\) for this test

Applied Statistics and Probability for Engineers 6
Transient points of an electronic inverter are influenced by many factors. Table E12-15 gives data on the transient point , in volts) of PMOS-NMOS inverters and five candidate regressors: \(x_{1}=\) width of the NMOS device, \(x_{2}=\) length of the NMOS device, \(x_{3}=\) width of the PMOS device,

Applied Statistics and Probability for Engineers 6
Consider the inverter data in Exercise 12-110. Delete observation 2 from the original data. Define new variables as follows: \(y^{*}=\ln y, x_{1}^{*}=1 / \sqrt{x_{1}} x_{2}^{*}=\sqrt{x_{2}} x_{3}^{*}=1 / \sqrt{x_{3}}, \text { and } x_{4}^{*}=\sqrt{x_{4}}\) (a)

Applied Statistics and Probability for Engineers 6
Following are data on \(y=\) green liquor \((g / I)\) and \(x=\) paper machine speed (feet per minute) from a Kraft paper machine. (The data were read from a graph in an article in the Tappi Journal, March 1986.) (a) Fit the model \(Y=\beta_{0}+\beta_{1} x+\beta_{2} x^{2}+\epsilon

Applied Statistics and Probability for Engineers 6
Consider the jet engine thrust data in Exercises 12-108 and 12-109. Define the response and regressors as in Exercise 12-109. (a) Use all possible regressions to select the best regression equation, where the model with the minimum value of \(M S_{E}\) is to be selected as best. (b) Repeat part (a)

Applied Statistics and Probability for Engineers 6
Consider the electronic inverter data in Exercises and Define the response and regressors variables as in Exercise , and delete the second observation in the sample. (a) Use all possible regressions to find the equation that minimizes \(C_{p}\) (b) Use all possible regressions to find the equation

Applied Statistics and Probability for Engineers 6
A multiple regression model was used to relate y = viscosity of a chemical product to \(x_{1}\) = temperature and \(x_{2}\) = reaction time. The data set consisted of n = 15 observations. (a) The estimated regression coefficients were \(\hat{\beta}_{0}=300.00\), \(\hat{\beta}_{1}=0.85\), and \(\ha

Applied Statistics and Probability for Engineers 6
Tables E12-16 and E12-17 present statistics for the Major League Baseball 2005 season (The Sports Network). (a) Consider the batting data. Use model-building methods to predict wins from the other variables. Check that the assumptions for your model are valid. (b) Repeat part (a) for the pitching dat

Applied Statistics and Probability for Engineers 6
An article in the Journal of the American Ceramics Society (1992, Vol. 75, pp. 112116) described a process for immobilizing chemical or nuclear wastes in soil by dissolving the contaminated soil into a glass block. The authors mix \(\mathrm{CaO}\) and \(\mathrm{Na}_2 \mathrm{O}\) with soil and model

Applied Statistics and Probability for Engineers 6
Exercise 12-9 introduced the hospital patient satisfaction survey data. One of the variables in that data set is a categorical variable indicating whether the patient is a medical patient or a surgical patient. Fit a model including this indicator variable to the data using all three of the other reg

Applied Statistics and Probability for Engineers 6
Consider the following inverse model matrix. \(\left(X^{\prime} X\right)^{-1}=\left[\begin{array}{cccc} 0.125 & 0 & 0 & 0 \\ & 0.125 & 0 & 0 \\ 0 & 0 & 0.125 & 0 \\ 0 & 0 & 0 & 0.125 \end{array}\right]\) (a) How many regressors are in this model? (b) What was the sample size? (c) Notice

Applied Statistics and Probability for Engineers 6
Consider a multiple regression model with \(k\) regressors. Show that the test statistic for significance of regression can be written as \(F_{0}=\frac{R^{2} / k}{\left(1-R^{2}\right) /(n-k-1)}\) Suppose that \(n=20, k=4\), and \(R^{2}=0.90\). If \(\alpha=0.05\), what conclusion would you draw about

Applied Statistics and Probability for Engineers 6
Problem 121MEE A regression model is used to relate a response to k = 4 regressors with n = 20. What is the smallest value of R2 that will result in a significant regression if ? = 0.05? Use the results of the previous exercise. Are you surprised by how small the value of R2 is?

Applied Statistics and Probability for Engineers 6
Problem 122MEE Show that can express the residuals from a multiple regression model as e = (I ? H)y where H = X(X X) ?1 X .

Applied Statistics and Probability for Engineers 6
Problem 123MEE Show that the variance of the ith residual ei in a multiple regression model is ?2 (1? hii) and that the covariance between ei and ej is ??2 hij where the hs are the elements of H = X(X X)?1X

Applied Statistics and Probability for Engineers 6
Consider the multiple linear regression model \(\mathbf{y}=\mathbf{X} \boldsymbol{\beta}+\boldsymbol{\epsilon}\). If \(\boldsymbol{\beta}\) denotes the least squares estimator of \(\boldsymbol{\beta}\), show that \(\hat{\beta}=\beta+\mathbf{R} \epsilon\), where \(\mathbf{R}=\left(\mathbf{X}^{\prime}

Applied Statistics and Probability for Engineers 6
Piecewise Linear Regression. Suppose that \(y\) is piecewise linearly related to \(x\). That is, different linear relationships are appropriate over the intervals \(-\infty<x \leq x^{*}\) and \(x^{*}<x<\infty\). (a) Show how indicator variables can be used to fit such a piecewise linear regression mo