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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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,
Solve the given ODE. Show the details of your work.y"' + 4y" + 13y' = 0
Are the given functions linearly independent or dependent on the half-axis x > 0? Give reason.cosh 2x, sinh 2x, e2x
Find out, without calculation, whether doubling the flow rate in Example 1 has the same effect as halfing the tank sizes. (Give a reason.)
Find a general solution by conversion to a single ODE.The system in Example 5 of the text.
Determine the radius of convergence. Show the details of your work. m-0 (-1) km 2m x
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.y" + (x - 1)y = 0
List some properties of the Legendre polynomials.
Graph Q0(x), Q1(x), and some further Legendre functions.
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + y' - xy = 0
Why did we introduce two kinds of Bessel functions?
Substitute αsxs + αs+1xs+1 + αs+2xs+2 into Legendre’s equation and obtain the coefficient recursion (4).
Apply the power series method. Do this by hand, not by a CAS, to get a feel for the method, e.g., why a series may terminate, or has even powers only, etc. Show the details.y" + y = 0
Find a general solution in terms of Jv and Yv. Indicate whether you could also use J-v instead of Yv. Use the indicated substitution. Show the details of your work.xy" - 5y' + xy = 0 (y = x3u)
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.2x(x - 1)y" - (x + 1)y' + y = 0
This is just a sample of such ODEs; some more follow in the next problem set. Find a general solution in terms of Jv and J-v or indicate when this is not possible. Use the indicated
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.y" + 4y = 0
Find a power series solution in powers of x. Show the details.y" - y' + x2y = 0
Find a solution of (α2 - x2)y" - 2xy' + n(n+1)y = 0, α ≠ 0, by reduction to the Legendre equation.
Show that the Hankel functions (10) form a basis of solutions of Bessel’s equation for any v.
Find and graph (on common axes) the solutions of for k = 0, 1, 2, . . . ., 10 (or as far as you get useful graphs). For what k do you get elementary functions? Why? Try for noninteger k,
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + (2 - 2x)y' + (x - 2)y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.(x - 1)2y" - (x - 1) y' - 35y = 0
Show that Iv(x) has the representation 1,(x) = Σ m-0 2m + v X 22m+ +vm! F(m + v + 1)
Find a basis of solutions by the Frobenius method. Try to identify the series as expansions of known functions. Show the details of your work.xy" + (1 - 2x)y' + (x - 1)y = 0
Find a power series solution in powers of x. Show the details.y" - 4xy' + (4x2 - 2)y = 0
Find a general solution in terms of hyper geometric functions.2x(1 - x)y" - (1 - 6x)y' - 2y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.x2y" + xy' + (x2 - 5)y = 0
Shifting summation indices is often convenient or necessary in the power series method. Shift the index so that the power under the summation sign is Xm. Check by writing the first few terms
Associated Legendre functions Pkn(x)?are needed, e.g., in quantum physics. They are defined by and are solutions of the ODE where q(x) = n(n + 1) - k2/(1 - x2). Find P11(x), P12(x), P22(x), and
Modified Bessel functions of the third kind (Sometimes called of the second kind) are defined by the formula (14) below. Show that they satisfy the ODE (12). K₂(x) = TT 2 sin VTT - [I_₂(x) —
ZEROS of Bessel functions play a key role in modelingUsing (21) and Rolle’s theorem, show that between any two consecutive zeros of J0(x) there is precisely one zero of J1(x).
Find a power series solution in powers of x. Show the details.y" + (1 + x2)y = 0
Find a general solution in terms of hyper geometric functions.x(1 - x)y" + (1/2 + 2x)y' - 2y = 0
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.y' + 4y = 1, y(0)= 1.25, x1 = 0.2
Show that y = uv with v(x) = exp (-1/2 ∫ p(x) dx) gives from the ODE y" + p(x)y' + q(x)y = 0 the ODEu" + [q(x) - 1/4p(x)2 - 1/2p' (x)] u = 0,not containing the first derivative of u.
Find a general solution in terms of hyper geometric functions.4x(1 - x)y" + y' + 8y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.xy" - (x + 1)y' + y = 0
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.y" + 3xy' + 2y = 0, y(0)= 1, y' (0) = 1, x
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.y" + 3xy' + 2y = 0, y(0)= 1, y' (0) = 1, x
Find a general solution in terms of hyper geometric functions.2(t2 - 5t + 6)ÿ - (2t - 3)ý - 8y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.y" + (1/4x)y = 0
Solve the initial value problem by a power series. Graph the partial sums of the powers up to and including x5. Find the value of the sum s (5 digits) at x1.(x - 2)y' = xy, y(0) = 4, x1 = 2
Find a general solution in terms of hyper geometric functions.3t(1 + t)ÿ + tý - y = 0
Find a basis of solutions. Try to identify the series as expansions of known functions. Show the details of your work.xy" + y' - xy = 0
Linear Independence is of basic importance, in this chapter, in connection with general solutions, as explained in the text. Are the following functions linearly independent on the given interval?
Linear Independence is of basic importance, in this chapter, in connection with general solutions, as explained in the text. Are the following functions linearly independent on the given interval?
If the mathematics scores of the SAT college entrance exams are normal with mean 480 and standard deviation 100 (these are about the actual values over the past years) and if some college sets 500 as
State the main theorems on probability. Illustrate them by simple examples.
Determine the number of different bridge hands. (A bridge hand consists of 13 cards selected from a full deck of 52 cards.)
Let X be the number of years before a certain kind of pump needs replacement. Let X have the probability function f(x) = kx3, x = 0, 1, 2, ,3, 4. Find k. Sketch f and F.
Graph a sample space for the experiments:Tossing a coin until the first Head appears
Graph a sample space for the experiments:Rolling 2 dice
Graph f and F when the density of X is f(x) = k = const if -2 ≤ x ≤ 2 and 0 elsewhere. Find P(0 ≤ X ≤ 2).
Let f(x, y) = k if x > 0, y > 0, x + y < 3 and 0 otherwise. Find k. Sketch f(x, y). Find P(X + Y ≤ 1), P(Y > X).

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