Introduction To Probability And Statistics For Science Engineering And Finance 1st Edition Walter A. Rosenkrantz - Solutions

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Suppose that you are inspecting a lot of 1000 light bulbs, among which 40 are defectives. You choose two light bulbs randomly from the lot without replacement. Let Find the probability that at least one light bulb chosen is defective. [Hint: Compute P(X + X ≥ 1).]
Assume that two random variables (X, Y) are uniformly distributed on a circle with radiusa. Then the joint probability density function is Find μ , the expected value of X.
Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function
The density function of the continuous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year, is given in Exercise 3.7 on page 112 as Find the average number of hours per year that families run their vacuum cleaners.
If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function X Y find the average profit per automobile.
The density function of coded measurements of the pitch diameter of threads of a fitting is Find the expected value of X.
Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table gives the joint distribution for X and Y.Find μ and μ .
A private pilot wishes to insure his airplane for $200,000.The insurance company estimates that a total loss will occur with probability 0.001, a 50% loss with probability 0.01, and a 25% loss with probability 0.25. Ignoring all other partial losses, what premium should the insurance company charge
Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.2, 0.3, 0.4, and 0.1, respectively, that she will be able to sell it for a profit of $250, sell it for a profit of $150, break even, or sell it for a loss of $100. What is her
In a gambling game, a woman is paid $2 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should she pay to play if the game is fair?
A coin is biased such that a tail is two times as likely to occur as a head. Find the expected number of tails when this coin is tossed twice.
Find the mean of the random variable T representing the total of the three coins in Exercise 3.25 on page 113.
The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given in Exercise 3.13 on page 112 as Find the average number of imperfections per 10 meters of this fabric.
Problem 13.27 Consider a 24 agricultural experiment with four factors A, B, C, D. The experiment is to be carried out on two plots of land each of which is large enough to run only eight factor-level combinations. The scientists choose to confound the ABCD interac[1]tion with the block effect.
Problem 13.26 The data in the following table come from a complete 23 factorial exper[1]iment to study the effect of cutting speed (A), tool geometry (B), and cutting angle (C) on the life of a machine tool. Two levels of each factor were chosen and there were three replicates for each factor-level
Problem 13.25 Analyze the data of the preceding problem as a half replicate with defining relation I = ABC. That is, compute the fractional replicates lA, lB, lC and the correspond[1]ing sum of squares; then compute SST and SSE for this half replicate. Determine the proportion of variability that
Problem 13.24 The data in the following table come from an experiment to study the ef[1]fects of temperature (factor A), processing time (factor B), and the rate of temperature rise (factor C) on the amount dye (the response variable) left in the residue bath of a dyeing process. The experiment was
Problem 13.23 Analyze the data of the preceding problem as a half replicate with defining relation I = ABC. That is, compute the fractional replicates lA, lB, lC and the correspond[1]ing sum of squares; then compute SST and SSE for this half replicate. Determine the proportion of variability that
Problem 13.21 Refer to Example 13.16. An exploratory analysis of the data indicated that the effect of the factor X4 (deposition time) on Y1 was significant, but its effect on Y2 was not. This suggests that we study both Model (Y1 | X4, X8, X4*X8) and Model (Y2 | X4, X8, X4*X8). (a) Rearrange the
Problem 13.20 Refer to the Model (Y1 | X1, X8, X1*X8) of Example 13.16. (a) Estimate the main effects and interactions by first computing the contrasts LX1, LX8, LX1X8. (b) Compute SSX1, SSX8, and SSX1X8. Your answers should agree with those in Table 13.18. Hint: Use Equations 13.33, 13.34, and
Problem 13.19 Refer to the Model (Y2 | X1, X8, X1*X8) of Example 13.16. (a) Estimate the main effects and interactions by first computing the contrasts LX1, LX8, LX1X8. (b) Compute SSX1, SSX8, and SSX1X8. Your answers should agree with those in Table 13.16. Hint: Use Equations 13.33, 13.34, and
Problem 13.18 Refer to Problem 13.17. (a) Compute Tukey’s HSDA, HSDB. Use α = 0.05. (b) Using Tukey’s HSD determine all pairs of means that are significantly different.
Problem 13.17 The data in the table below come from an experiment to determine the amount of warping (the response variable) in a metal plate as a function of the following two factors: (A) temperature (measured in degrees Celsius), and (B) copper content (mea[1]sured as a percentage). (a) Compute
Problem 13.16 The data in the table below record the service time (in minutes) required to service disk drives from three different manufacturers by three different technicians. Each technician was randomly assigned to repair five disk drives for each brand of disk drive. (a) Compute the estimated
Problem 13.15 The data in the next table are the results of an experiment to investigate the sources of variability in testing the strength of Portland cement. A sample of cement was divided into small samples for testing. The cement was “gauged” (mixed with water) by three different men called
Problem 13.14 Verify Equation 13.23.
Problem 13.13 Verify that interaction effects (αβ)ij satisfy Equation 13.18.
Problem 13.12(a) Show that Y ij. is an unbiased estimator of the (i, j)th cell mean µij (i = 1,...,a); (j = 1,...,b). (b) Compute V ar(Y ij.).
Problem 13.11 Show that Y i.. and Y .j. are unbiased estimators of the factor A means µi. (i = 1,...,a) and the factor B means µ.j (j = 1,...,b), respectively.
Problem 13.10 Refer to Problem 13.9. (a) Compute Tukey’s HSDA, HSDB. Use α = 0.05. (b) Using Tukey’s HSD determine all pairs of means that are significantly different.
Problem 13.9 Refer to the data of Table 13.5. The residuals plot 13.7 indicates that the hypothesis of equal variances is invalid. To stabilize the variance we use the transformation u = 1 + y. The transformed data set is given in the next table. (a) Compute the two-way ANOVA table for the
Problem 13.8 The data below come from an experiment to compare four methods for manufacturing penicillin. Note that the blocks are displayed as rows and the treatments are displayed as columns. The raw material consisted of b = 5 blends consisting of enough material so that all a = 4 methods could
Problem 13.7 The data below come from a larger experiment to determine the effective[1]ness of blast furnace slags (A) as agricultural liming materials on three types of soil (B): sandy loam (1), sandy clay loam (2), and loamy sand (3). The treatments were all applied at 4000 lbs per acre and the
Problem 13.6 Two types of fertilizer are being compared against a control (no fertilizer). The response variable is the yield (tons) in sugar beets. The results are displayed in the next table.(a) Calculate the ANOVA table and test whether there are differences among the treatments. Use α = 0.05.
Problem 13.5 The data below come from a comparison study of the execution times (mea[1]sured in milliseconds) of five different work loads on three different configurations of a pro[1]cessor. The goal was to study the effect of processor configuration on computer performance. (a) Identify the
Problem 13.4 Show that the residuals satisfy both conditions displayed in Equation 13.5.
Problem 13.3 The table below lists the final weight (in grams) of 18 hamsters fed three different diets varying in fat content. The hamsters were blocked by weight in groups of three; that is, the hamsters in each block had the same weights at the start of the experiment. (a) Estimate the overall
Problem 13.2 In the table below the counts of the numbers of insects (Leptinotarsa dec[1]imlineata) were recorded after four treatments were applied to six areas of a field. Area Treatment 1 23456 1 492 410 475 895 401 330 2 111 67 233 218 28 18 3 58 267 283 278 392 141 4 4 1 53 14 138 11(a)
Problem 13.1 With reference to the fruit fly mortality data (Table 13.1) perform a one[1]way ANOVA for the Model (Y = mortality rate |A = dosage level). That is, drop the blocking variable B (genetic strain) from the model and analyze the data using a one– way ANOVA model to determine whether or
Problem 12.25(a) Show that E(Y i.) = µ + αi. (b) Show that E(Y i. −Y ..) = αi, i.e., Y i. −Y .. is an unbiased estimator of the ith treatment mean.
Problem 12.24 Derive Equation 12.26.
Problem 12.23 A manufacturing engineer wants to study the amount of time required to assemble a piece of equipment; he is particularly interested in how the assembly time varies among workers. To study this question he selects four workers at random and records the assembly times for each of the
Problem 12.22 The following data give the amount, in parts per million (ppm), of a toxic substance as measured by four randomly selected laboratories. Each laboratory made six measurements of the toxic substance. The question to be studied was how much of the variability is due to the different
Problem 12.21 The following data give the polysilicon resistivities measured at three dif[1]ferent positions for each of six randomly selected silicon wafers. Wafer 1 Wafer 2 Wafer 3 Wafer 4 Wafer 5 Wafer 6 Resistivity 47.337 56.053 44.594 44.430 47.093 46.825 Resistivity 46.815 55.640 42.759
Problem 12.20 The following data give the tensile strengths (psi) of cotton yarn produced by four different looms selected at random from a large group of such looms at a textile mill. Loom Tensile strength (psi) 1 18 18 20 16 14 17 2 16 14 15 20 16 14 3 11 8 8 5 13 8 4 24 23 20 19 16 19(a) Is this
Problem 12.19 Verify that, under the assumptions of the single factor ANOVA model, the random variable ˆθ = i ciY i is normally distributed with mean and variance given by Equations 12.15 and 12.16.
Problem 12.18 A high voltage electric cable consists of 12 wires. The following data are the tensile strengths of the 12 wires in each of 9 cables. (a) Use a computer to draw side-by-side box plots for the nine cables. Do the box plots indicate that the strength of the wires varies among the
Problem 12.17 Continuation of Problem 12.9. (a) Construct 95% confidence intervals for each of the treatment means. (b) Use Tukey’s method to determine those differences in the treatment means that are significant. Use α = 0.05.
Problem 12.16 Continuation of Problem 12.8. (a) Construct 95% confidence intervals for the treatment means. (b) Use Tukey’s method to determine those differences in the treatment means that are significant. Use α = 0.05.
Problem 12.15 Continuation of Problem 12.7. (a) Construct 95% confidence intervals for each of the measurement methods. (b) Use Tukey’s method to identify those differences in the measurement methods that are significant. Use α = 0.05.
Problem 12.14 Continuation of Problem 12.6. (a) Construct 95% confidence intervals for each of the treatment means. (b) Use Tukey’s method to identify those differences in the treatment means that are signif[1]icant. Use α = 0.05.
Problem 12.13 Continuation of Problem 12.4. (a) Construct 95% confidence intervals for the treatment means. (b) Use an appropriate t statistic to construct 95% confidence intervals for all differences µi − µk. (c) Use Tukey’s method to identify those differences in the treatment means that
Problem 12.12 The analytical methods committee of the Royal Society of Chemistry re[1]ported the following results on the levels of tin recovered from samples of blended fish–meat product after boiling with hydrochloric acid for 30, 45, 60, and 70 minutes, respectively. Refluxing time (min) Tin
Problem 12.11(a) Show that E(yi.) = µ + αi. (b) Show that E(yi. − y..) = αi, i.e., yi. − y.. is an unbiased estimator of the ith treatment mean.
Problem 12.10 The analytical methods committee of the Royal Society of Chemistry re[1]ported the following results on the levels of tin recovered after samples (taken from the same product) were boiled with hydrochloric acid for 30, 45, 60, and 75 minutes, respectively. Refluxing time (min) Tin
Problem 12.9 In a weight loss experiment men were given one of three different diets: 10 men followed diet A, 7 men followed diet B, and 8 men followed diet C. The weight losses after one month are displayed in the following table. Notice that the observations at each factor level now appear as
Problem 12.8 The following data give the results of an experiment to study the effects of three different word processing programs on the length of time (measured in minutes) to type a text file. A group of 24 secretaries were randomly divided into 3 groups of 8 secretaries each, and each group was
Problem 12.7 The following data give four measurements of the ionization potential, mea[1]sured in electron volts (eV), of a gaseous ion for three different methods: photoionization (PI), extrapolated voltage difference (EVD), and semi log plot (SL). Method Ionization potential (eV) PI 10.68 9.22
Problem 12.6 The following data give the burn times, in seconds, of four fabrics tested for flammability. Fabric Burn times (sec) 1 18 17 18 17 14 18 2 12 11 11 11 12 11 3 15 9 13 7 12 9 4 14 12 8 13 15 9 Construct the ANOVA table and test at the level α = 0.05, the null hypothesis that there is
Problem 12.5 The data in the following table are the results of a test in which the burst strengths of cardboard boxes made with four different types of wood pulp were studied; three measurements were taken for each wood pulp mixture. Wood pulp Burst strength (psi) 1 237 219 209 2 205 202 201 3 144
Problem 12.4 The effect of four different mixing techniques on the compressive strengths of concrete is investigated and the following data are collected. Mixing technique Compressive strength 1 2868 2913 2844 2887 2 3250 3262 3110 2996 3 2838 3049 3030 2832 4 2847 2721 2530 2598 Does the mixing
Problem 12.3 (a) Table 12.1 indicates that the first two students taught by method 3 obtained different scores (28 and 30) on the arithmetic test. What feature of the statistical model (Equation 12.1) accounts for this? (b) As shown in Table 12.1, three students taught by method 1 scored a 24 on
Problem 12.2 The following table lists the tensile strengths of a rubber compound produced by three different methods, denoted A, B, and C. Treatment Tensile strength A 4038 4364 4410 4242 B 4347 3859 3432 3720 C 3652 3829 4071 4232 (a) Compute the three treatment means and the grand mean y1., y2.,
Problem 12.1 The side-by-side box plot displayed in Figure 12.1 reveals an outlier in the group of students taught by method 3 (praised). (Outliers are discussed in Section 1.5.) Verify that 23 is the outlier by showing that it lies outside the interval (Q1 − 1.5 × IQR, Q3 + 1.5 × IQR), where
Problem 11.14 Show that the variance–covariance matrix of the residual vector is Cov(e) = σ2(I − H) where H is the hat matrix. Hint: Use the representation e = (I − H)Y , Equation 11.26, and Equation 11.16.
Problem 11.13 The data listed in the next table come from an experiment to study the effects of three environmental variables on exhaust emissions of light-duty diesel trucks. Fit the Model (Y = NOx| X1 = Humidity, X2 = Temperature, X3 = Barometric pressure). Test the null hypothesis H0 : βi = 0,
Problem 11.12 The data in the next table were obtained in a forestry study of certain characteristics of a stand of pine trees including A, age of a particular pine stand; HD, the average height of dominant trees in feet; N, the number of pine trees per acre at age A; and MDBH, the average diameter
Problem 11.11 The following data come from an experiment to determine the relation between Y, the shear strength (psi) of a rubber compound, and X, the cure temperature (oF). (a) Draw the scatter plot to see if the simple linear regression model is appropriate. (b) Fit the quadratic polynomial
Problem 11.10 Refer to Example 11.6 and the associated data set Table 11.1. (a) Fit the quadratic polynomial Model (Y = MPG | X1 = speed, X2 = speed2) and state the estimated regression function. (b) Draw the scatter plot and the graph of the fitted polynomial on the same coordinate axes. (c)
Problem 11.9 Refer to the ground water quality data set of Problem 10.12. (a) Fit the multiple linear regression Model (Y = TDS | X1 = SEC, X2 = SiO2) and state the estimated regression function. (b) Estimate the change in the total dissolved solids (TDS) when the specific electrical
Problem 11.8 Refer to Problem 10.23 of Chapter 10. (a) State the matrix formulation of the the Model (Y = hardness |X = time). (b) State the normal equations for this model in the matrix format 11.4. (c) Verify that βˆ0 = 153.916667 and βˆ1 = 2.416667 satisfy the normal equations given in part
Problem 11.7 Derive the properties of the hat matrix H listed in Equations 11.14-11.17.
Problem 11.6 Refer to Example 11.4. Compute the estimated standard errors using the equations (derived in Chapter 10) s(βˆ0) = s ‑ 1 n + x2 Sxx ; s(βˆ1) = s √Sxx . Your results should agree with those obtained in the solution to Example 11.4.
Problem 11.5 Show that in the case of simple linear regression det(X X) = nSxx.
Problem 11.4 Refer to Problem 10.12 of Chapter 10. (a) State the matrix formulation of the the Model (Y = TDS |X = SEC). (b) State the normal equations for this model in the matrix format 11.4. (c) Verify that βˆ0 = 29.268569 and βˆ1 = 0.597884 satisfy the normal equations given in part (b).
Problem 11.3 Refer to Problem 10.4 of Chapter 10. (a) State the matrix formulation of the the Model (Y = CPU time |X = Disk I/Os). (b) State the normal equations for this model in the matrix format 11.4. (c) Verify that βˆ0 = −0.008282 and βˆ1 = 0.243756 satisfy the normal equations given in
Problem 11.2 Refer to Example 10.3 of Chapter 10. (a) State the matrix formulation of the the Model (Y = shear strength |X = weld diameter). (b) Write down the normal equations for this model in the matrix format 11.4. (c) Verify that βˆ0 = −216.325, βˆ1 = 3.9904 satisfy the normal equations.
Problem 11.1 Refer to Example 11.2. Show by direct substitution that βˆ0 = 38.78474, βˆ1 = −3.434047 is a solution to the normal Equation 11.19.
Problem 10.47 Perform the necessary algebraic manipulations in order to derive Equation 10.58.
Problem 10.46 Verify the representation for βˆ0 displayed in Equation 10.56.
Problem 10.45 Show that E(Y ) = β0 + β1x.
Problem 10.44 Refer to the 1999 weekly returns for the bond fund PTTRX listed in Table 1.19. Compute PTTRX’s β for the year 1999.
Problem 10.43 Cirrhosis, a liver disease, is a side effect of excessive alcohol consumption. Table 10.14 in Section 10.9, based on statistics compiled by the Commodity Board of the Spirits Industry, records the rate of consumption of pure alcohol per capita for various countries and the cirrhosis
Problem 10.42 Table 10.13 in Section 10.9 lists the number of vehicles per 10,000 popu[1]lation (VPP), the number of deaths per 10,000 vehicles (DPV), and the number of deaths per 10,000 population for twelve countries for the year 1987. In this problem we apply the techniques of correlation
Problem 10.41 Refer back to of the CO2 emissions data in Problem 1.17. In this problem we explore the relationships between some of the variables X1,...,X4. (a) Draw the scatter plot of (X3, X2). In this context would you expect a positive, zero, or negative correlation? Justify your answer. (b)
Problem 10.40 Refer to the fathers’ heights X and weights Y shown in the StatLab data set (Table 1.16). (a) Draw the scatter plot and guess the value of r. (b) Calculate r.
Problem 10.39 Refer to the girls’ heights X and weights Y shown in the StatLab data set (Table 1.16). (a) Draw the scatter plot and guess the value of r. (b) Calculate r.
Problem 10.38 Refer to the StatLab data set in Table 1.16. Psychologists are interested in exploring the relationship between the girls’ scores on the Raven test (Y ) and the Peabody test (X). The Peabody test measures verbal facility; the Raven test measures a different sort of mental ability,
Problem 10.37 Calculate the sample correlation coefficient r for the following artificial data: x 123456 y 223546
Problem 10.36 Refer back to the UNIX data of Problem 10.21. (a) Draw the studentized residual and normal probability plots for Model (Y = time |X = data bytes) (UNIX DATA). (b) Do these plots and the scatter plot indicate any departures from the simple linear re[1]gression model? Summarize your
Problem 10.35 Refer back to Problem 10.12. Do you agree with the authors’ conclusion that TDS can be estimated from the regression models with acceptable error? In particular, are the data consistent with the hypotheses of the simple linear regression model? Justify your conclusions by model
Problem 10.34 Consider the following two bivariate data sets each consisting of eleven (x, y) pairs. The two data sets have the same x values but different y values. Dataset I Dataset II xy y 4 4.26 3.10 5 5.68 4.74 6 7.24 6.13 7 4.82 7.26 8 6.95 8.14 9 8.81 8.77 10 8.04 9.14 11 8.33 9.26 12 10.84
Problem 10.33 Refer back to Problem 10.23. (a) Plot the studentized residuals against predicted hardness for the regression Model(Y = hardness |X = time). (b) Draw the normal probability plots of the studentized residuals. (c) Are these plots consistent with the hypotheses of the simple linear
Problem 10.32(a) A biologist believes that the curve defined by the function v = u a+bu provides a good fit to a data set. Show that the transformed variables x = 1/u, y = 1/v lie on a straight line y = β0 + β1x and express β0 and β1 in terms of a andb. (b) Sketch the curve v = u/(1 + u) for 0
Problem 10.31(a) The data in the following table are pressures, in atmospheres, of oxygen gas kept at 25 degrees C and made to occupy various volumes, measured in liters. Draw the scatter plot of (vi, pi). Is the simple linear regression model appropriate? (b) Chemists believe that the relation
Problem 10.30 The data in the following table come from measuring the average speeds (s) (in meters/minute) for racing shells seating different numbers of rowers (n). From calculations based on fluid mechanics, it is known that the speed is proportional to the 1/9th power of the number of rowers;
Problem 10.29 Kepler’s third law of planetary motion states that the square of the period (P) of revolution of a planet is proportional to the cube of its mean distance (D) from the sun; that is, P2 = kD3. Taking the square root of both sides yields the formula P = cD3/2; c = √ 2π γM , where
Problem 10.28 Refer to Example 10.11 and Table 10.8. Fit the curve of Equation 10.47 by carrying out a linear regression analysis on the transformed data y = ln v, x = ln t. Graph the scatter plot and estimated regression line for the transformed data.
Problem 10.27(This is a continuation of Problem 10.11.) (a) Display the results of your regression analysis in the format of Table 10.5. (b) Compute 95% confidence intervals for the regression coefficients. (c) Compute a 95% confidence interval for the mean gpm (gallons per hundred miles) of an
Problem 10.26(This is a continuation of Problem 10.10.) (a) Display the results of your regression analysis in the format of Table 10.5. (b) Compute 95% confidence intervals for the regression coefficients. (c) Compute a 95% confidence interval for the mean mpg of an automobile that weighs 3655

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