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Probability And Statistics For Engineers 5th Edition Richard L Scheaffer, Madhuri Mulekar, James T McClave, Cecie Starr - Solutions

(a) Show that for Example 1.0.1 the perpendicular projection operator onto C(X) is 25 15 5 -5 -15 -25 15 9 3 -3 -9 -15 16 1 5 3 1 -1 -3 -5 M= (/6 + 70 -5 -3 -1 1 3 5-15 -9 -3 3 9 15-25 -15 -5 5 15 25(b) Show that for Example 1.0.2 the perpendicular projection operator onto C(X) is 1/3 1/3 1/3 0 0 0
Consider a regression model Y = X(J +e, e '"N(O, a2 J) and suppose that we want to predict the value of a future observation, say Yo, that will be independent of Y and be distributed N(x~(J, ( 2).(a) Find the distribution of Yo - x~{3(b) Find a 95% symmetric two-sided prediction interval for
Consider the model Y=X(J+b+e, E(e) = 0, Cov(e) = a 2 J, where b is a known vector. Show that Proposition 2.1.3 is not valid for this model by producing a linear unbiased estimate of p' X(J, say ao + a'Y, for which ao ¥- O.Hint: Modify p' MY.
Consider the model Yi = (JIXil + (J2Xi2 + ei, eiS i.i.d. N(O, ( 2 ). Use the data given below to answer (a) through (d). Show your work, i.e., do not use a regression or general linear models computer program.(a) Estimate (31, (32, and 172.(b) Give 95% confidence intervals for (31 and (31 +
Consider the model Yi = (31xi1 + (32xi2 + ei, eiS i.i.d. N(O, (72 ). There are 15 observations and the sum of the squared observations is y'y = 3.03. Use the normal equations given below to answer parts (a) through (d).(a) Estimate (31, (32, and 172.(b) Give 98% confidence intervals for (32 and (32
Consider the model where the predictor variables take on the following values.i 1 2 3 4 5 6 7 Xil 1 1 -1 -1 0 0 0 Xi2 1 -1 1 -1 0 0 0 Show that (30, (31, (32, (311 + (322, (312 are estimable and find (nonmatrix)algebraic forms for the estimates of these parameters. Find the MSE and the standard
Show that the F test developed in the first part of this section is equivalent to the (generalized) likelihood ratio test for the same hypothesis.
Redo the tests in Exercise 2.2 using the theory of Section 3.2. Write down the models and explain the procedure.
Redo the tests in Exercise 2.3 using the procedures of Section 3.2. Write down the models and explain the procedure.Hints: (a) Let A be a matrix of zeros, the generalized inverse of A, A-, can be anything at all because AA - A = A for any choice of A -. (b) There is no reason why Xo cannot be a
In testing a reduced model Y = Xo'Y + e against a full model Y = X {3 +e, what linear parametric function of the parameters is being tested?
Show that p'MY = p'[Mp(p'Mp)-p'M]Y so that to estimate p'X{3, one only needs the perpendicular projection of Y onto C(Mp).
Show that Y'(A - AO)'V-l(A - Ao)Y equals the difference in the SSEs for models (3) and (1).
If X is defined as in equation (3.6.1) and Xo is defined in (3.6.2), find a matrix U such that Xo = XU. Note that C(X) =C(Xo), so (3.6.2) is a reparameterization of (3.6.1). Clearly, Xo"! = XU"! =X;3, so ;3 = u"! + v for some v with Xv = O. Find a matrix U* so that Hints: U is easy to find, and the
Verify that the estimate of /-L + lli is fk and that the algebraic formulas for the sums of squares in the ANOVA table are correct.Hint: To find, for example, Y (M - [l/n]J;;') Y = Y'MaY, use Exercise 4.2 to get MaY and recall that Y'MaY = [MaY]' [MaY].
Verify that the formulas for expected mean squares in the ANOVA table are correct.Hint: Use Theorem 1.3.2 and Exercise 4.3.
Using the theories of Sections 3.2 and 2.6, respectively, find the F test and the t test for the hypothesis Ho : E!=l Aiai =d, in terms of the MSE, the fks, and the AiS.
Suppose N1 = N2 = ... = Nt == N. Rewrite the ANOVA table incorporating any simplifications due to this assumption.
If N1 = N2 = ... = Nt == N, show that two contrasts A~fl and A~fl are orthogonal if and only if A~ A2 = O.
Find the least squares estimates of /1-, a1, and at using the side condition a1 = O.
Using p as defined by (4.2.1) and X as defined in Section 1 of this chapter [especially (4.1.1)], show that(a) pi X = N, where X = (0, A1, ... , At).(b) p E C(X).
An experiment was conducted to see which of four brands of blue jeans were most resistant to wearing out while students knelt before their linear models instructor begging for additional test points. In a class of 32 students, 8 students were randomly assigned to each brand of jeans. Before being
After the final exam of spring quarter, 30 of the subjects of the previous experiment decided to test the sturdiness of 3 brands of sport coats and 2 brands of shirts. In this study, sturdiness was measured as the length of time before tearing when the instructor was hung by his collar out of his
In the model of Exercise 5.1, let t = 4. Suppose we want to use the LSD method to test contrasts defined by Name Al A2 A3 A4 All -1 -1 BOO 1 -1 C 1/3 1/3 1/3 -1 Describe the procedure. Give test statistics for each test that is to be performed.
Show that for testing all hypotheses in a six dimensional space with 30 degrees of freedom for error, if the subspace F test is omitted and the nominal LSD level is a = .005, then the true error rate must be less than .25.Hint: Try to find a Scheffe rejection region that is comparable to the LSD
Compare all pairs of means for the blue jeans exercise of Chapter 4. Use the following methods:(a) Scheffe's method, a = 0.01,(b) the LSD method, a = 0.01, (c) the Bonferroni method, a = 0.012, (d) Thkey's HSD method, a = 0.01, (e) the Newman-Keuls method, a = 0.01.
Test whether the four orthogonal contrasts you chose for the blue jeans exercise of Chapter 4 equal zero. Use all of the appropriate multiple comparison methods discussed in this chapter to control the experimentwise error rate at a = .05 (or thereabouts).
Compare all pairs of means in the coat-shirt exercise of Chapter 4. Use all of the appropriate multiple comparison methods discussed in this chapter to control the experimentwise error rate at a = .05(or thereabouts).
Suppose that in a balanced one-way ANOVA the treatment means 1h., ... ,flt- are not independent but have some nondiag-onal covariance matrix V. How can Tukey's HSD method be modified to accommodate this situation?
For an unbalanced one-way ANOVA, give the contrast coefficients for the contrast whose sum of squares equals the sum of squares for treatments. Show the equality of the sums of squares.
Find the MSE, Var(/Jo), Var(/Jd, CoV(/JO,/Jl)'
Use Scheffe's method of multiple comparisons to derive the Working-Hotelling simultaneous confidence band for a simple linear regression line.
For predicting Y = (Yl,"" Yq)' from the value of x = (Xl, ... , Xp-d' we say that a predictor f(x) is best if the scalar E{[y - f(x)]'[y - f(x)]} is minimized. Show that with simple modifications, Theorems 6.3.1 and 6.3.2 hold for the extended problem, as does Proposition 6.3.3.
Assume that Vxx is nonsingular. Show that Py.x = 0 if and only if the best linear predictor of Yl based on X and Y2 equals the best linear predictor of Yl based on x alone.
If (Yil, Yi2, Xil, Xi2, ..• , Xi,p-l)', i = 1, ... , n are independent N(/-l, V), find the distribution of v'n - p - 1 Ty ·x / VI - T~.x when Py.x = O.
Show that if M is the perpendicular projection operator onto C(X) with then Wi = Wj if and only if Ti = Tj.
Discuss the application of the traditional lack of fit test to the problem where Y = X f3 + e is a simple linear regression model.As we have seen, in the traditional method of testing for lack of fit, the row structure of the design matrix X completely determines the choice of Z. Now, suppose that
Let Mi be the perpendicular projection operator onto C(Xi ), i = 1,2. Show that the perpendicular projection operator onto C(Z)is Mz = [~l ~2]'Show that SSE(Z) = SSE(XI) + SSE(X2 ), where SSE(Xi ) is the sum of squares for error from fitting Yi = Xi/3i + ei, i = 1,2.
Test the model Yij = /30 + /3lxi + /32X~ + eij for lack of fit using the data:Xi 1.00 2.00 0.00 -3.00 2.50 Yij 3.41 22.26 -1.74 79.47 37.96 2.12 14.91 1.32 80.04 44.23 6.26 23.41 -2.55 81.63 18.39
Using the following data, test the model Yij = /30 +/3IXil + /32Xi2 + eij for lack of fit. Explain and justify your method.Xl X2 Y Xl X2 Y 31 9.0 122.41 61 2.2 70.08 43 8.0 115.12 36 4.7 66.42 50 2.8 64.90 52 9.4 150.15 38 5.0 64.91 38 1.5 38.15 38 5.1 74.52 41 1.0 45.67 51 4.6 75.02 41 5.0 68.66
(a) Find the design matrix for the orthogonal polynomial model Y = T,",{+e corresponding to the model Yij = f30 + f31 x i + f32x~ + f33x~ + eij, i = 1,2,3,4, j = 1, ... , N, where Xi = a + (i - l)t.Hint: First consider the case N = l.(b) For the model Yij = /.L + Qi + eij, i = 1,2,3,4, j = 1, ...
Maximizing a Quadratic Response.Consider the model, Yi = flo + fl1Xi + fl2XT + ei, eiS Li.d. N(O, a2), i =1,2,3, ... , n. Let Xo be the value at which the function E(y) = flo + fl1X +fl2X2 is maximized (or minimized).(a) Find the maximum likelihood estimate of Xo.(b) Find a (1 - 0:)100% confidence
Find E[Y - E(Ylx)f in terms of the variances and covariances of X and y. Give a "natural" estimate of E [Y _ E(ylx)] 2.
Test whether the data of Example 6.2.1 indicate that the multiple correlation coefficient is different from zero.
Test whether the data of Example 6.2.1 indicate that the partial correlation coefficient Pyl.2 is different from zero.
Show that in Section 2, 'Y* = f3* and f30 = 'Yo -(1jn)Ji' Z'Y*.Hint: Examine the corresponding argument given in Section 1 for simple linear regression.
Does the statement "the interactions add nothing to the model" mean that ')'11 = ')'12 = ... = ')'11? If it does, justify the statement. If it does not, what does the statement mean?
Find the ANOVA table for the two-way ANOVA without interaction model when there are proportional numbers. Find the least squares estimate of a contrast in the aiS. Find the variance of the contrast and give a definition of orthogonal contrasts that depends only on the contrast coefficients and the
Find the ANOVA table for the two-way ANOVA with interaction model when there are proportional numbers.
Analyze the following data as a two-factor ANOVA where the subscripts i and j indicate the two factors.Yijk S 1 2 3 j 1 0.620 1.228 0.615 1.342 3.762 2.245 0.669 2.219 2.077 0.687 4.207 3.357 0.155 2.000 2 1.182 3.080 2.240 1.068 2.741 0.330 2.545 2.522 3.453 2.233 1.647 1.527 2.664 1.999 0.809
Analyze the following data as a two-factor ANOVA where the subscripts i and j indicate the two factors.Yijk S 1 2 3 j 1 1.620 2.228 2.999 1.669 3.219 1.615 1.155 4.080 2.182 3.545 2 1.342 3.762 2.939 0.687 4.207 2.245 2.000 2.741 1.527 1.068 0.809 2.233 1.942 2.664 1.002 The dependent variable is
In the mid-1970s, a study on the prices of various motor oils was conducted in (what passes for) a large town in Montana.The study consisted of pricing four brands of oil at each of nine stores. The data are given below.Brand Store P H V Q 1 87 95 95 82 2 96 104 106 97 3 75 87 81 70 4 81 94 91 77 5
An experiment was conducted to examine thrust forces when drilling under different conditions. Data were collected for four drilling speeds and three feeds. The data are given below.Speed Feed 100 250 400 550 121 98 83 58 124 108 81 59.005 104 87 88 60 124 94 90 66 110 91 86 56 329 291 281 265 331
Consider the model Yijk = J.L + Qi + 'Tlj + "tij + eijk, i = 1,2,3,4, j = 1,2,3, k = 1, ... , Nij , where for i # 1 # j, Nij = Nand Nu = 2N. This model could arise from an experimental design having Q treatments of No Treatment (NT), al, a2, a3 and 'Tl treatments of NT, bll b2 . This gives a total
Consider the linear model Yij = P, + Qi + 'f/j + eij, i =1, ... ,a, j = 1, ... ,b. As in Section 1, we can write the design matrix as X =[Xo, Xl"'" Xa, Xa+l, ... , Xa+b]' If we write the observations in the usual order, we can use Kronecker products to write the design matrix. Write X = [J, X*,
Consider the balanced two-way ANOVA with interaction model Yijk = P, + Qi + 'f/j + "Iij + eijk, i = 1, ... ,a, j = 1, ... ,b, k = 1, ... , N with eijkS independent N(O, 0'2). Find E[Y'(~J~ + MoJY] in terms of p" the QiS, the 'f/jS, and the "IijS.
9.87 A factory operates with two machines of type A and one machine of type B. The weekly repair costs Y for type A machines are normally distributed with mean and variance . The weekly repair costs X for machines of type B are also normally distributed, but with mean and variance . The expected
9.86 Suppose the sample variances given in Exercise 9.35 are good estimates of the population variances.Using the allocation scheme of Exercise 9.85, find the number of measurements to be taken on each coupling agent to estimate the true difference in means to within 1 unit with confidence
9.85 Suppose that, in a large-sample estimate of for two populations with respective variances and , a total of n observations is to be selected. How should these n observations be allocated to the two populations so that the length of the resulting confidence interval will be minimized?
9.84 It is desired to estimate the proportion of defective items produced by a certain assembly line to within 0.1 with confidence coefficient 0.95. What is the smallest sample size that will guarantee this accuracy no matter where the true proportion of defectives might lie? [Hint: Find the value
9.83 Two types of aluminum powders are blended before a sintering process is begun to form solid aluminum. The adequacy of the blending is gauged by taking numerous small samples from the blend and measuring the weight proportion of one type of powder. For adequate blending, the weight proportions
9.82 The weights of aluminum grains follow a lognormal distribution, which means that the natural logarithms of the grain weights follow a normal distribution. A sample of 177 aluminum grains has logweights averaging 3.04 with a standard deviation of 0.25. (The original weight measurements were in
9.81 Window sizes in integrated circuit chips must be of fairly uniform size in order for the circuits to function properly. Thus, the variance of the window width measurements is an important quantity. Refer to the data of Exercise 9.79.a Estimate the true variance of the post-etch window widths
9.80 In the setting of Exercise 9.79, a second important problem is to compare the average window widths before and after etching. Using the data given there, estimate the true difference in average window widths in a 95% confidence interval.
9.79 In the fabrication of integrated circuit chips, it is of great importance to form contact windows of precise width. (These contact windows facilitate the interconnections that make up the circuits.)Complementary metal-oxide semiconductor(CMOS) circuits are fabricated by using a
9.78 Twenty specimens of a slightly different steel from that used in Exercise 9.77 were observed and yielded Rockwell hardness measurements with a mean of 72 and a variance of 94. Estimate the difference between the mean hardness for the two varieties of steel in a 95% confidence interval.
9.77 The Rockwell hardness measure of steel ingots is produced by pressing a diamond point into the steel and measuring the depth of penetration. A sample of 15 Rockwell hardness measurements on specimens of steel gave a sample mean of 65 and a sample variance of 90. Estimate the true mean hardness
9.76 Suppose the sample mean and standard deviation of Exercise 9.75 had come from a sample of 100 measurements. Construct a 90% confidence interval for the average diameter of the cable.What assumption must necessarily be made?
9.75 The diameter measurements of an armored electric cable, taken at 10 points along the cable, yield a sample mean of 2.1 centimeters and a sample standard deviation of 0.3 centimeter.Estimate the average diameter of the cable in a confidence interval with a confidence coefficient of 0.90. What
9.74 The number of improperly soldered connections per microchip in an electronics manufacturing operation follows a binomial distribution with and p unknown. The cost of correcting these malfunctions, per microchip, is Find the maximum likelihood estimate of E(C) if pN is available as an estimate
9.73 The absolute errors in the measuring of the diameters of steel rods are uniformly distributed between zero and . There such measurements on a standard rod produced errors of 0.02, 0.06, and 0.05 centimeter. What is the maximum likelihood estimate of ? [Hint: This problem cannot be solved by
9.72 If X denotes the number of the trial on which the first defective is found in a series of independent quality-control tests, find the maximum likelihood estimator of p, the true probability of observing a defective.
9.71 The stress resistances for specimens of a certain type of plastic tend to have a gamma distribution with , but may change with certain changes in the manufacturing process.For eight specimens independently selected from a certain process, the resistances (in psi)were Find a 95% confidence
9.70 Suppose ,…, denotes a random sample from the gamma distribution with a known but unknown . Find the maximum likelihood estimator of .
9.69 Suppose ,…, denotes a random sample from the normal distribution with mean and variance . Find the maximum lilelihood estimators of and .
9.68 Because in the Poisson case, it follows from the Central Limit Theorem that will be approximately normally distributed with mean and variance l>n, for large n.l XV(Xi) = l ll X1 Xn a Use the above facts to construct a largesample confidence interval for .b Suppose that 100 reinforced concrete
9.67 If ,…, denotes a random sample from a Poisson distribution with mean , find the maximum likelihood estimator of .
9.64 In the setting of Exercise 9.62, construct the tolerance interval if the sample size is as follows:a bDoes the sample size seem to have a great effect on the length of the tolerance interval, as long as it is “large”?
9.63 In the setting of Exercise 9.62, construct the tolerance interval if the sample standard deviation is a 0.015 inch b 0.06 inch Compare these answers to that of Exercise 9.62.Does the size of s seem to have a large effect on the length of the tolerance interval?
9.62 A cutoff operation is supposed to cut dowel pins to 1.2 inches in length. A random sample of 60 such pins gave an average length of 1.1 inches and a standard deviation of 0.03 inch. Assuming normality of the pin lengths, find a tolerance interval for 90% of the pin lengths in the population
9.61 The average thickness of the plastic coating on electrical wire is an important variable in determining the wearing characteristics of the wire.Ten thickness measurements from randomly selected points along a wire gave the following(in thousandths of an inch):Construct a 99% tolerance interval
9.60 A bottle-filling machine is set to dispense 10.0 cc of liquid into each bottle. A random sample of 100 filled bottles gives a mean fill of 10.1 cc and a standard deviation of 0.02 cc. Assuming the amounts dispensed to be normally distributed, a Construct a 95% tolerance interval for 90% of the
9.59 A particular model of subcompact automobile has been tested for gas mileage 50 times. These mileage figures have a mean of 39.4 mpg and a standard deviation of 2.6 mpg. Predict the gas mileage to be obtained on the next test, with 1 - a = 0.90.
9.58 In designing concrete structures, the most important property of the concrete is its compressive strength. Six tested concrete beams showed compressive strengths (in thousand psi) of 3.9, 3.8, 4.4, 4.2, 3.8, and 5.4. Another beam of this type is to be used in a construction project.Predict its
9.57 It is extremely important for a business firm to be able to predict the amount of downtime for its computer system over the next month. A study of the past five months has shown the downtime to have a mean of 42 hours and a standard deviation of 3 hours. Predict the downtime for next month in
9.55 In studying the properties of a certain type of resistor, the actual resistances produced were measured on a sample of 15 such resistors.These resistances had a mean of 9.8 ohms and a standard deviation of 0.5 ohm. One resistor of this type is to be used in a circuit. Find a 95%prediction
9.54 An electric circuit contains three resistors, each of a different type. Tests on 10 type I resistors showed a sample mean resistance of 9.1 ohms with a sample standard deviation of 0.2 ohm;tests on 8 type II resistors yielded a sample mean of 14.3 ohms and a sample standard deviation of 0.4
9.53 Time-Yankelovich surveys, regularly seen in the news magazine Time, report on telephone surveys of approximately 1,000 respondents. In December 1983, 60% of the respondents said that they worry about nuclear war. In a similar survey in June 1983, only 50% said that they worry about nuclear
9.52 Silicon wafers are scored and then broken into the many small microchips that will he mounted into circuits. Two breaking methods are being compared. Out of 400 microchips broken by method A, 32 are unusable because of faulty breaks. Out of 400 microchips broken by method B, only 28 are
9.50 The number of cycles to failure for reinforced concrete beams was measured in seawater and in air. The data (in thousands) are as follows:Seawater: 774, 633, 477, 268, 407, 576, 659, 963, 193 Air: 734, 571, 520, 792, 773, 276, 411, 500, 672 Estimate the difference between mean cycles to
9.49 Using the data presented in Exercise 9.48, estimate the ratio of variances for the two methods of removing acid gases in a 90% confidence interval.
9.48 Acid gases must be removed from other refinery gases in chemical production facilities in order to minimize corrosion of the plants. Two methods for removing acid gases produced the corrosion rates(in mm/yr) are listed below in experimental tests:Method A: 0.3, 0.7, 0.5, 0.8, 0.9, 0.7, 0.8
9.46 One-hour carbon monoxide concentrations in 45 air samples from a section of a city showed an average of 11.6 ppm and a variance of 82.4. After a traffic control strategy was put into place, 19 air samples showed an average carbon monoxide concentration of 6.2 ppm and a variance of
9.45 Unaltered, altered, and partly altered bitumens are found in carbonate-hosted lead-zinc deposits and may aid in the production of sulfide necessary to precipitate ore bodies in carbonate rocks. (See Powell, Science, April 6, 1984, p. 63.) The atomic hydrogen/carbon (H/C) ratios for 15 samples
9.44 Seasonal ranges (in hectares) for alligators were monitored on a lake outside Gainesville, Florida, by biologists from the Florida Game and Fish Commission. Six alligators monitored in the spring showed ranges of 8.0, 12.1, 8.1, 18.1, 18.2, and 31.7. Four different alligators monitored in the
9.43 Research Quarterly, May 1979, reports on a study of impulses applied to a ball by tennis rackets of various construction. Three measurements on ball impulses were taken on each type of racket. For a Classic (wood) racket, the mean was 2.41 and the standard deviation was 0.02.For a Yamaha
9.42 For a certain species of fish, the LC50 measurements(in parts per million) for DDT in 12 experiments were as follows, according to the EPA:16, 5, 21, 19, 10, 5, 8, 2, 7, 2, 4, 9 Another common insecticide, Diazinon, gave LC50 measurements of 7.8, 1.6, and 1.3 in three independent experiments.a
9.40 A large firm made up of several companies has instituted a new quality-control inspection policy.Among 30 artisans sampled in Company A, only 5 objected to the new policy. Among 35 artisans sampled in Company B, 10 objected to the policy.Estimate the true difference between the proportions
9.39 In studying the proportion of water samples containing harmful bacteria, how many samples should be selected before and after a chemical is added if we want to estimate the true difference between proportions to within 0.1 with a 95% confidence coefficient? (Assume the sample sizes are to be
9.38 Bacteria in water samples are sometimes difficult to count, but their presence can easily be detected by culturing. In 50 independently selected water samples from a certain lake, 43 contained certain harmful bacteria. After adding a chemical to the lake water, another 50 water samples showed
9.37 Two different types of coating for pipes are to be compared with respect to their ability to aid in resistance to corrosion. The amount of corrosion on a pipe specimen is quantified by measuring the maximum pit depth. For coating A, 35 specimens showed an average maximum pit depth of 0.18 cm.

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