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mathematics
advanced engineering mathematics

Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions

How did we obtain numeric methods from the Taylor series?
Do 10 steps. Solve exactly. Compute the error. Show details.y' + 0.2y = 0, y(0) = 5, h = 0.2
Solve by the Euler’s method. Graph the solution in the y1y2-plane. Calculate the errors.y'1 = 2y1 - 4y2, y'2 = y1 - 3y2, y1(0) = 3, y2(0) = 0, h = 0.1, 10 steps
Show that the heat equation u∼t∼ = c2u∼xx, 0 ≤ x∼ ≤ L, can be transformed to the “nondimensional” standard form ut = uxx, 0 ≤ x ≤ 1, by setting x = x∼/L, t = c2t∼/L2, u = u∼/u0, where u0 is any constant temperature.
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are giveny' = y, y(0) = 1, h = 0.1, (1.105171, 1.221403, 1.349858)
Check the values for the Poisson equation at the end of Example 1 by solving (3) by Gauss elimination.
Explain the Euler and improved Euler methods in geometrical terms. Why did we consider these methods?
Using the present method, solve (1)–(4) with h = k = 0.2 for the given initial deflection and initial velocity 0 on the given t-interval.f(x) = x if 0 = x < 1/5, f(x) = 1/4(1 - x) if 1/5 ≤ x ≤ 1, 0 ≤ t ≤ 1
Find and graph three circular disks that must contain all the eigenvalues of the matrix:Of the coefficients in Prob. 14Data from Prob. 14Solve 3x2 - 6x3 = 0 4x1 - x2 + 2x3 = 16-5x1 + 2x2 - 4x3 = -20
Find and graph three circular disks that must contain all the eigenvalues of the matrix: In Prob. 19 Data from Prob. 19 Compute the inverse of: 15 20 10 20 35 15 10 15 90
Fit and graph:A quadratic parabola to the data in Prob. 34.Data from Prob. 34Fit and graph:A straight line to (1, 0), (0, 2), (1, 2), (2, 3), (3, 3)
Compute the condition number (corresponding to the l∞-vector norm) of the coefficient matrix:In Prob. 21Data from Prob. 21Do 3 steps without scaling, starting from [1 1 1]T.4x1 - x2 = 22.0 4x2 - x3 =
Compute the condition number (corresponding to the l?-vector norm) of the coefficient matrix: In Prob. 18 Data from Prob. 18 Compute the inverse of: 2.0 1.6 0.3 0.1 4.4 -4.3 3.3 0.5 2.8
Compute the condition number (corresponding to the l?-vector norm) of the coefficient matrix: In Prob. 19 Data from Prob. 19 Compute the inverse of: 15 20 10 20 35 15 10 15 90
Compute the matrix norm corresponding to the l∞-vector norm for the coefficient matrix:In Prob. 22Data from Prob. 22Do 3 steps without scaling, starting from [1 1 1]T.0.2x1 + 4.0x2 - 0.4x3 = 32.00.5x1 - 0.2x2 + 2.5x3 = -5.17.5x1 + 0.1x2 - 1.5x3 = -12.7
Compute the matrix norm corresponding to the l∞-vector norm for the coefficient matrix:In Prob. 21Data from Prob. 21Do 3 steps without scaling, starting from [1 1 1]T.4x1 - x2 = 22.0 4x2 - x3 =
Compute the matrix norm corresponding to the l∞-vector norm for the coefficient matrix:In Prob. 17Data from Prob. 17Solve 42x1 + 74x2 + 36x3 = 96-46x1 - 12x2 - 2x3 = 82 3x1 + 25x2 + 5x3 = 19
Compute the matrix norm corresponding to the l∞-vector norm for the coefficient matrix:In Prob. 15Data from Prob. 15Solve 8x2 - 6x3 = 23.610x1 + 6x2 + 2x3 = 68.412x1 - 14x2 + 4x3 =
What is the consumer’s risk in Prob. 12 if we want the RQL to be 12%? Use c = 9 from the answer of Prob. 12.Data from Prob. 12If in a sampling plan for large lots of spark plugs, the sample size is 100 and we want the AQL to be 5% and the producer’s risk 2% what acceptance number c should we
Lots of kitchen knives are inspected by a sampling plan that uses a sample of size 20 and the acceptance number c = 1. What is the probability of accepting a lot with 1%, 2%, 10% defectives (knives with dull blades)? Use Table A6 of the Poisson distribution. Graph the OC curve. Probability function
Why are interval estimates generally more useful than point estimates?
If we have several samples from the same population, do they have the same sample distribution function? The same mean and variance?
Make a list of methods in this section, each with the distribution needed in testing.
Show that in Prob. 1, the requirement of the significance level α = 0.3% leads to LCL = μ - 3σ/√n and UCL = μ + 3σ/√n, and find the corresponding numeric values.Data from Prob. 1Suppose a machine for filling cans with lubricating oil is set so that it will generate fillings which form a
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work. x = Revolutions per minute, y = Power of a Diesel engine [hp] y 400 500 600 700 5800 10,300 14,200 18,800 750 21,000
Can we develop statistical methods without using probability theory? Apply the methods without using a sample?
Test μ = 0 assuming μ > 0, normality and using the sample 0, 1, -1, 3, -8, 6, 1 (deviations of the azimuth [multiples of 0.01 radian] in some revolution of a satellite). Choose α = 5%.
How will the probabilities in Prob. 1 with n = 20 change (up or down) if we decrease c to zero? First guess.Data from Prob. 1Lots of kitchen knives are inspected by a sampling plan that uses a sample of size 20 and the acceptance number c = 1. What is the probability of accepting a lot with 1%, 2%,
Derive the maximum likelihood estimator for μ. Apply it to the sample (10, 25, 26, 17, 10, 4), giving numbers of minutes with 0–10, 11–20, 21–30, 31–40, 41–50, more than 50 fliers per minute, respectively, checking in at some airport check-in.
If 100 flips of a coin result in 40 heads and 60 tails, can we assert on the 5% level that the coin is fair?
Are oil filters of type A better than type B filters if in 11 trials, A gave cleaner oil than B in 7 cases, B gave cleaner oil than A in 1 case, whereas in 3 of the trials the results for A and B were practically the same?
By what factor does the length of the interval in Prob. 2 change if we double the sample size?Data from Prob. 2Find a 95% confidence interval for the mean of a normal population with standard deviation 4.00 from the sample 39, 51, 49, 43, 57, 59. Does that interval get longer or shorter if we take
What is the idea of the maximum likelihood method? Why do we say “likelihood” rather than “probability”?
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work. x = Brinell hardness, y = Tensile strength [in 1000 psi (pounds per square inch)] of steel with 0.45% C tempered for 1 hour X y 200 110 300 150 400 190 500 280
Couldn’t we make the error of interval estimation zero simply by choosing the confidence level 1?
Do the same test as in Prob. 4, using a result by K. Pearson, who obtained 6019 heads in 12,000 trials.Data from Prob. 4In one of his classical experiments Buffon obtained 2048 heads in tossing a coin 4040 times. Was the coin fair?
How should we change the sample size in controlling the mean of a normal population if we want UCL - LCL to decrease to half its original value?
Lots of copper pipes are inspected according to a sample plan that uses sample size 25 and acceptance number 1. Graph the OC curve of the plan, using the Poisson approximation. Find the producer’s risk if the AQL is 1.5%.
Derive a maximum likelihood estimate for p.
Can you claim, on a 5% level, that a die is fair if 60 trials give 1,· · ·, 6 with absolute frequencies 10, 13, 9, 11, 9, 8?
Do the computations in Prob. 4 without the use of the DeMoivre–Laplace limit theorem.Data from Prob. 4Does a process of producing stainless steel pipes of length 20 ft for nuclear reactors need adjustment if, in a sample, 4 pipes have the exact length and 15 are shorter and 3 longer than 20 ft?
What sample size would be needed for obtaining 95% a confidence interval (3) of length 2σ? Of length σ?
What is testing? Why do we test? What are the errors involved?
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work. x = Voltage [V], y = Current [A]. Also find the resistance R [?]. X 40 5.1 y 40 4.8 80 80 0.0 10.3 110 13.0 110 12.7
When did we use the t-distribution? The F-distribution?
How does the result in Prob. 6 change if we use a smaller sample, say, of size 5, the other data (x̅ = 58.05, α = 5%, etc.) remaining as before?Data from Prob. 6Assuming normality and known variance σ2 = 9, test the hypothesis μ = 60.0 against the alternative μ = 57.0 using a sample of size 20
Graph the ranges of the samples in Prob. 6 on a control chart for ranges. Data from Prob. 6 Graph the means of the following 10 samples (thickness of gaskets, coded values) on a control chart for means, assuming that the population is normal with mean 5 and standard deviation 1.16. Time 10:00 5
Suppose that in Prob. 6 we made 3 times 4 trials and A happened 2, 3, 2 times, respectively. Estimate p.Data from Prob. 6Extend Prob. 5 as follows. Suppose that m times n trials were made and in the first n trials A happened k1 times, in the second n trials A happened k2 times, · · ·, in
If a service station had served 60, 49, 56, 46, 68, 39 cars from Monday through Friday between 1 P.M. and 2 P.M., can one claim on 5% a level that the differences are due to randomness? First guess. Then calculate.
Assuming normality, solve Prob. 6 by a suitable test. Data from Prob. 6 Thirty new employees were grouped into 15 pairs of similar intelligence and experience and were then instructed in data processing by an old method (A) applied to one (randomly selected) person of each pair, and by a new
Find a 95% confidence interval for the percentage of cars on a certain highway that have poorly adjusted brakes, using a random sample of 800 cars stopped at a roadblock on that highway, 126 of which had poorly adjusted brakes.
What is the chi-square (χ2) test? Give a sample example from memory.
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work. x = Temperature [?F], y = Conductivity [Btu/(hr ? ft ? ?F)]. Also find y at room temperature 66?F. X y 32 50 0.337 0.345 100 0.365 150 0.380 212 0.395
What are one-sided and two-sided tests? Give typical examples.
What is the rejection region in Prob. 6 in the case of a two-sided test with α = 5%?Data from Prob 6Assuming normality and known variance σ2 = 9, test the hypothesis μ = 60.0 against the alternative μ = 57.0 using a sample of size 20 with mean x̅ = 58.50 and choosing α = 5%.
Eight samples of size 2 were taken from a lot of screws. The values (length in inches) are Assuming that the population is normal with mean 3.500 and variance 0.0004 and using (1), set up a control chart for the mean and graph the sample means on the chart. Sample No. 12 3 4 5 6 7 8 3.50 3.51 3.49
In Prob. 8, show that f(1) + f(2) + · · · = 1 (as it should be!). Calculate independently of Prob. 8 the maximum likelihood of p in Prob. 8 corresponding to a single observed value of X.Data from Prob. 8Let X = Number of independent trials until an event A occurs. Show that X has a geometric
In a table of properly rounded function values, even and odd last decimals should appear about equally often. Test this for the 90 values of J1(x) in Table A1 in App. 5.Table A1 For more extensive tables see Ref. [GenRefl] in App. 1. x Jo(x) Jo(x) (4) 0.0 1.0000 -0.2601 0.3391 0.1 0.9975 -0.2921
Assuming that the populations corresponding to the samples in Prob. 8 are normal, apply a suitable test for the normal distribution. Data from Prob. 8 In a clinical experiment, each of 10 patients were given two different sedatives A and B. The following table shows the effect (increase of sleeping
Find a 99% confidence interval for the mean of a normal population from the sample:Copper content (%) of brass 66, 66, 65, 64, 66, 67, 64, 65, 63, 64
How do we test in quality control? In acceptance sampling?
(a) Obtain 100 samples of size 10 each from the normal distribution with mean 100 and variance 25. For each sample, test the hypothesis μ0 = 100 against the alternative μ1 > 100 at the level of α = 10%. Record the number of rejections of the hypothesis. Do the whole experiment once more and
Take a sample, for instance, that in Prob. 4, and investigate and graph the effect of changing y-values (a) For small x (b) For large x (c) In the middle of the sample. Data from Prob. 4 Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the
What is the power of a test? What could you perhaps do when it is low?
A firm sells oil in cans containing 5000 g oil per can and is interested to know whether the mean weight differs significantly from 5000 g at the 5% level, in which case the filling machine has to be adjusted. Set up a hypothesis and an alternative and perform the test, assuming normality and using
Find formulas for the UCL, CL, and LCL (corresponding to 3σ-limits) in the case of a control chart for the number of defectives, assuming that, in a state of statistical control, the fraction of defectives is p.
Samples of 3 fuses are drawn from lots and a lot is accepted if in the corresponding sample we find no more than 1 defective fuse. Criticize this sampling plan. In particular, find the probability of accepting a lot that is 50% defective. (Use the binomial distribution (7).)
Find the maximum likelihood estimate of θ in the density f(x) = θe-θx if x ≥ 0 and f(x) = 0 if x < 0.
Check your generator experimentally by imitating results of n trials of rolling a fair die, with a convenient n (e.g., 60 or 300 or the like). Do this many times and see whether you can notice any “nonrandomness” features, for example, too few Sixes, too many even numbers, etc., or whether your
How would you proceed in the sign test if the hypothesis is μ∼= μ∼0 (any number) instead of μ∼ = 0?
Find a 99% confidence interval for the mean of a normal population from the sample:Knoop hardness of diamond 9500, 9800, 9750, 9200, 9400, 9550
What is Gauss’s least squares principle (which he found at age 18)?
(a) Obtain 100 samples of 4 values each from the normal distribution with mean 8.0 and variance 0.16 and their means, variances, and ranges.(b) Use these samples for making up a control chart for the mean.(c) Use them on a control chart for the standard deviation.(d) Make up a control chart for the
Obtain 100 samples of size 10 of the standardized normal distribution. Calculate from them and graph the corresponding 95% confidence intervals for the mean and count how many of them do not contain 0. Does the result support the theory? Repeat the whole experiment, compare and comment.
Find a 95% confidence interval for the regression coefficient ?1, assuming (A2) and (A3) hold and using the sample. In Prob. 2 Data from Prob. 3 Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work. x = Revolutions per
What is the difference between regression and correlation?
If simultaneous measurements of electric voltage by two different types of voltmeter yield the differences (in volts) 0.4, -0.6, 0.2, 0.0, 1.0, 1.4, 0.4, 1.6, can we assert at the 5% level that there is no significant difference in the calibration of the two types of instruments? Assume normality.
Since the presence of a point outside control limits for the mean indicates trouble, how often would we be making the mistake of looking for nonexistent trouble if we used(a) 1-sigma limits.(b) 2-sigma limits? Assume normality.
Compute in Prob. 11 from the sample 1.9, 0.4, 0.7, 0.6, 1.4. Graph the sample distribution function F̂(x) and the distribution function F(x) of the random variable, with θ = θ̂, on the same axes. Do they agree reasonably well?Data from Prob. 11Find the maximum likelihood estimate of θ in the
In a famous plant-crossing experiment, the Austrian Augustinian father Gregor Mendel (1822–1884) obtained 355 yellow and 123 green peas. Test whether this agrees with Mendel’s theory according to which the ratio should be 3:1.
Does the amount of fertilizer increase the yield of wheat X [kg/plot]? Use a sample of values ordered according to increasing amounts of fertilizer:33.4 35.3 31.6 35.0 36.1 37.6 36.5 38.7.
Find a 95% confidence interval for the variance of a normal population from the sample:Length of 20 bolts with sample mean 20.2 cm and sample variance 0.04 cm2
Find a 95% confidence interval for the regression coefficient.? x = Humidity of air [%], y = Expansion of gelatin [%], X y 10 0.8 20 1.6 30 2.3 40 2.8
Assuming normality, find the maximum likelihood estimates of mean and variance from the sample in Prob. 14.Data from Prob. 14Find the mean, variance, and standard derivation of the sample 21.0 21.6 19.9 19.6 15.6 20.6 22.1 22.2.
Suppose that in the past the standard deviation of weights of certain 100.0-oz packages filled by a machine was 0.8 oz. Test the hypothesis H0: σ = 0.8 against the alternative H1: σ > 0.8 (an undesirable increase), using a sample of 20 packages with standard deviation 1.0 oz and assuming
A so-called c-chart or defects-per-unit chart is used for the control of the number X of defects per unit (for instance, the number of defects per 100 meters of paper, the number of missing rivets in an airplane wing, etc.).(a) Set up formulas for CL and LCL, UCL corresponding to μ ± 3σ,
Using the given sample, test that the corresponding population has a Poisson distribution. x is the number of alpha particles per 7.5-s intervals observed by E. Rutherford and H. Geiger in one of their classical experiments in 1910, and ?(x) is the absolute frequency (= number of time periods
Find experimentally how much MLEs can differ depending on the sample size. Generate many samples of the same size n, e.g., of the standardized normal distribution, and record x̅ and s2. Then increase n.
Does an increase in temperature cause an increase of the yield of a chemical reaction from which the following sample was taken? Temperature [°C] 10 20 30 40 60 80 Yield [kg/min] 0.6 1.1 0.9 1.6 1.2 2.0
Find a 95% confidence interval for the variance of a normal population from the sample:Mean energy (keV) of delayed neutron group (Group 3, half-life 6.2 s) for uranium U235 fission: a sample of 100 values with mean 442.5 and variance 9.3
Determine a 99% confidence interval for the mean of a normal population, using the sample 32, 33, 32, 34, 35, 29, 29, 27.
Brand A gasoline was used in 16 similar automobiles under identical conditions. The corresponding sample of 16 values (miles per gallon) had mean 19.6 and standard deviation 0.4. Under the same conditions, high-power brand B gasoline gave a sample of 16 values with mean 20.2 and standard deviation
Find a 95% confidence interval for the variance of a normal population from the sample:The sample in Prob. 9Data from Prob. 9Find a 99% confidence interval for the mean of a normal population from the sample:Copper content (%) of brass 66, 66, 65, 64, 66, 67, 64, 65, 63, 64
A machine fills boxes weighing Y lb with X lb of salt, where X and Y are normal with mean 100 lb and 5 lb and standard deviation 1 lb and 0.5 lb, respectively. What percent of filled boxes weighing between 104 lb and 106 lb are to be expected?
Using a sample of 10 values with mean 14.5 from a normal population with variance σ2 = 0.25, test the hypothesis μ0 = 15.0 against the alternative μ1 = 14.5 on the 5% level. Find the power.
Show that for a normal distribution the two types of errors in a test of a hypothesis H0: μ = μ0 against an alternative H1: μ = μ1 can be made as small as one pleases (not zero!) by taking the sample sufficiently large.
Assume the thickness X of washers to be normal with mean 2.75 mm and variance 0.00024 mm2. Set up a control chart for μ and graph the means of the five samples (2.74, 2.76), (2.74, 2.74), (2.79, 2.81), (2.78, 2.76), (2.71, 2.75) on the chart.
Find the risks in the sampling plan with n = 6 and c = 0, assuming that the AQL is θ0 = 1% and the RQL is θ1 = 15%. How do the risks change if we increase n?
Find the regression line of y on x for the data (x, y) = (0, 4), (2, 0), (4, -5), (6, -9), (8, -10).
In numerics we use partial sums of power series. To get a feel for the accuracy for various x, experiment with sin x. Graph partial sums of the Maclaurin series of an increasing number of terms, describing qualitatively the ?breakaway points? of these graphs from the graph of sin x. Consider other

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