 Chapter 25.25.5: Suppose a machine for filling cans with lubricating oil is set so t...
 Chapter 25.25.6: Lots of knives are inspected by a sampling plan that uses a sample ...
 Chapter 25.25.7: If 100 flips of a coin result in 30 heads and 70 tails. can we asse...
 Chapter 25.25.8: What would change in Example 1. had we observed only 5 positive val...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.1: What is a sample? Why do we take samples?
 Chapter 25.25.2: Find the maximum likelihood estimate for the parameter f.L of a nOl...
 Chapter 25.25.3: Find a 95% confidence interval for the mean JL of a normal populati...
 Chapter 25.25.4: Test JL = a against JL > 0, assuming normality and using the sample...
 Chapter 25.25.5: (Threesigma control chart) Show that in Prob. I, the requirement o...
 Chapter 25.25.6: What happens in Prob. I if the sample size is increased to 50? Firs...
 Chapter 25.25.7: If in 10 flips of a coin we get the same ratio as in Prob. I (3 hea...
 Chapter 25.25.8: Does a process of producing plastic pipes of length f.L = 2 meters ...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.2: What is the role of probability theory in statistics?
 Chapter 25.25.2: Apply the maximum likelihood method to the normal distribution with...
 Chapter 25.25.3: Does the interval in Prob. I gel longer or shorter if we take 'Y = ...
 Chapter 25.25.4: In one of his classical experiments Buffon obtained 2048 heads in t...
 Chapter 25.25.5: What sample size should we choose in Prob. 1 if we want LCL and VCL...
 Chapter 25.25.6: How will the probabilities in Prob. I with 11 = 20 change (up or do...
 Chapter 25.25.7: What would be the smallest number of heads in Prob. I under which t...
 Chapter 25.25.8: Do the computations in Prob. 2 without the use of the DeMoivreLapl...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.3: Will you get better results by taking larger samples? Explain.
 Chapter 25.25.2: (Binomial distribution) Derive a maximum likelihood estimate for p.
 Chapter 25.25.3: By what factor does the length of the interval in Prob. 1 change if...
 Chapter 25.25.4: Do the same test as in Prob. 2. using a result by K. Pearson. who o...
 Chapter 25.25.5: How does the meaning of the control limits ( I ) change if we apply...
 Chapter 25.25.6: What are the producer's and consumer's risks in Prob. I if the AQL ...
 Chapter 25.25.7: If in rolling a die 180 times we get 39. 22. 41. 26. 20, 32. can we...
 Chapter 25.25.8: Test whether a thermostatic switch is properly set to 200 e against...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.4: Do several samples from a certain popUlation have the same mean? Th...
 Chapter 25.25.2: Extend Prob. 3 as follows. Suppose that 111 times 11 trials were ma...
 Chapter 25.25.3: Find a 90% confidence interval for the mean JL of a nonnal populati...
 Chapter 25.25.4: Assuming normality and known variance u 2 = 4. test the hypothesis ...
 Chapter 25.25.5: How should we change the sample size in controlling the mean of a n...
 Chapter 25.25.6: Large lots of batterie~ are inspected according to the following pl...
 Chapter 25.25.7: Solve Prob. 4 if the sample is 25, 31. 33, 27, 29. 35.
 Chapter 25.25.8: Are air filters of type A better than type B filters if in IO trial...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.5: What is a parameter? How can we estimate it? Give an example.
 Chapter 25.25.2: Suppose that in Prob. 4 we made 4 times 5 trials and A happened 2, ...
 Chapter 25.25.3: What sample size would be needed for obtaining a 95% confidence int...
 Chapter 25.25.4: How does the result in Prob. 4(a) change if we use a smaller sample...
 Chapter 25.25.5: What LCL and UCL should we use instead of (1) if instead of x we us...
 Chapter 25.25.6: Graph the AOQ curve in Prob. 5. Determine the AOQL, assuming that r...
 Chapter 25.25.7: A manufacturer claims that in a process of producing kitchen knives...
 Chapter 25.25.8: In a clinical experiment. each of 10 patients were given two differ...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.6: What is a statisticaL test? What errors occur in testing?
 Chapter 25.25.2: Consider X = Number of independent trials IImil all el'ent A occurs...
 Chapter 25.25.3: (Use of Fig. 525) Find a 95% confidence interval for a sample of 20...
 Chapter 25.25.4: Detemine the power of the test in Prob. 4(a).
 Chapter 25.25.5: Ten samples of size 2 were taken from a production lot of bolts. Th...
 Chapter 25.25.6: Do the work required in Prob. 5 if 11 = 50 and c = O.
 Chapter 25.25.7: Between 1 P.M. and 2 P.M. on five consecutive days (Monday through ...
 Chapter 25.25.8: Assuming that the populations corresponding to the samples in Prob....
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.7: How do we test in quality control?
 Chapter 25.25.2: In Prob. 6 find the maximum likelihood estimate of p corresponding ...
 Chapter 25.25.3: What sample size i~ needed to obtain a 99% confidence interval of l...
 Chapter 25.25.4: What is the rejection region in Prob. 4 in the case of a twosided ...
 Chapter 25.25.5: Graph the means of the following 10 samples (thickness of washers. ...
 Chapter 25.25.6: Find the binomial approximation of the hypergeometric distribution ...
 Chapter 25.25.7: Test for normality at the I % level using a sample of /I = 79 (roun...
 Chapter 25.25.8: Thirty new employees were grouped into 15 pairs of similar intellig...
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.8: What is the x2test? Give a simple example hom memory.
 Chapter 25.25.2: In rolling a die. suppose that we get the first Six in the 7th tria...
 Chapter 25.25.3: Fwd a 99% confidence interval for the mean of a nonnal popUlation f...
 Chapter 25.25.4: Using the sample 0.80. 0.81, 0.81. 0.82, 0.81. 0.82, 0.80,0.82. 0.8...
 Chapter 25.25.5: Graph the ranges of the samples in Prob. 8 on a control chart for r...
 Chapter 25.25.6: In Example 1, what are the producer's and consumer's risks if the A...
 Chapter 25.25.7: In a sample of 100 patients having a certain disease 45 are men and...
 Chapter 25.25.8: Assuming normality. solve Prob. 8 by a suitable test from Sec. 25.4.
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.9: What are nonparametric tests? When would you apply them?
 Chapter 25.25.2: (Poisson distribution) Apply the maximum likelihood method to the P...
 Chapter 25.25.3: Fwd a 99% confidence interval for the mean of a nonnal popUlation f...
 Chapter 25.25.4: A firm sells oil in cans containing 1000 g oil per can and is inter...
 Chapter 25.25.5: What effect on UCL  LCL does it have if we double the sample size?...
 Chapter 25.25.6: Calculate peA; 0) in Example 1 if the sample size is increased from...
 Chapter 25.25.7: In Prob. 9 find the smallest number (>50) of women that leads to th...
 Chapter 25.25.8: Set up a sign test for the lower q
 Chapter 25.25.9: Find and sketch or graph the sample regression line of Y and x and ...
 Chapter 25.10: In what tests did we use the Idistribution? The X2 distribution?
 Chapter 25.25.2: (Uniform distribution) Show that in the case of the parameters a an...
 Chapter 25.25.3: Fwd a 99% confidence interval for the mean of a nonnal popUlation f...
 Chapter 25.25.4: If a sample of 50 tires of a certain kind has a mean life of 32 000...
 Chapter 25.25.5: Since the presence of a point outside control limits for the mean i...
 Chapter 25.25.6: Samples of 5 screws are drawn from a lot with fraction defective O....
 Chapter 25.25.7: Verify the calculations in Example 1 of the text.
 Chapter 25.25.8: How would you proceed in the sign test if the hypothesis is fi = fi...
 Chapter 25.25.9: Find a 95% confidence interval for the regression coefficient Kl, a...
 Chapter 25.11: What are onesided and twosided tests? Give typical examples.
 Chapter 25.25.2: Find the maximum likelihood estimate of e in the density f(x) = ee...
 Chapter 25.25.3: Fwd a 99% confidence interval for the mean of a nonnal popUlation f...
 Chapter 25.25.4: If simultaneous measurements of electric voltage by two different t...
 Chapter 25.25.5: Graph An = uIE(R*) as a function of 11. Why is An a monotone decrea...
 Chapter 25.25.6: Find the risks in the single sampling plan with 11 = 5 and c = 0, a...
 Chapter 25.25.7: Does the random variable X = Number of accide/lTs per week in 1I ce...
 Chapter 25.25.8: Check the table in Example 2 of the text.
 Chapter 25.25.9: Find a 95% confidence interval for the regression coefficient Kl, a...
 Chapter 25.12: List some areas of application of statistical tests.
 Chapter 25.25.2: In Prob. I I. find the mean f.L. substitute it in fex). find the ma...
 Chapter 25.25.3: Fwd a 99% confidence interval for the mean of a nonnal popUlation f...
 Chapter 25.25.4: If a standard medication cures about 70% of patients with a certain...
 Chapter 25.25.5: (Number of defectives) Find formulas for the UCL, CL, and LCL (corr...
 Chapter 25.25.6: Why is it impossible for an OC curve to have a vertical portion sep...
 Chapter 25.25.7: Using the given sample, test that the corresponding population has ...
 Chapter 25.25.8: Apply the test in Example 2 to the following data (x = disulfide co...
 Chapter 25.25.9: Find a 95% confidence interval for the regression coefficient Kl, a...
 Chapter 25.13: What do we mean by "goodness of fiC?
 Chapter 25.25.2: Compute e in Prob. 11 from the sanlple 1.8, 0.4. 0.8. 0.6. 1.4. Gra...
 Chapter 25.25.3: Find a 95% confidence interval for the percentage of cars on a cert...
 Chapter 25.25.4: Suppose that in the past the standard deviation of weights of certa...
 Chapter 25.25.5: How would progressive tool wear in an automatic lathe operation be ...
 Chapter 25.25.6: If in a single sampling plan for large lots of spark plugs, the sam...
 Chapter 25.25.7: Can we assert that the traffic on the three lanes of an expressway ...
 Chapter 25.25.8: Test the hypothesis that for a certain type of voltmeter. readings ...
 Chapter 25.25.9: Derive the second expression for Sx 2 in (9a) from the first one.
 Chapter 25.14: Acceptance sampling uses principles of testing. Explain.
 Chapter 25.25.2: Do the same task as in Frob. 13 if the given sample is 0.5.0.7.0.1....
 Chapter 25.25.3: Find a 99% confidence interval for p in the binomial distribution f...
 Chapter 25.25.4: Suppose that in operating batterypowered electrical equipment, it ...
 Chapter 25.25.5: (Number of defects per unit) A socalled cchart or defectsperuni...
 Chapter 25.25.6: What is the consumer's risk in Prob. 14 if we want the RQL to be 12%?
 Chapter 25.25.7: If it i5 known that 25% of certain steel rod~ produced by a standar...
 Chapter 25.25.8: In a swinefeeding experiment. the following gains in weight [kg] o...
 Chapter 25.25.9: CAS EXPERIMENT. Moving Data. Take a sample, for instance, that in P...
 Chapter 25.15: What is the power of a test? What can you do if the power is low?
 Chapter 25.25.2: CAS EXPERIMENT. Maximum Likelihood Estimates. (MLEs). Find experime...
 Chapter 25.25.3: Find a Y5% confidence interval for the variance of a normal populat...
 Chapter 25.25.4: Brand A gasoline was used in 9 automobiles of the same model under ...
 Chapter 25.25.5: (Attribute control charts). Twenty samples of size 100 were taken f...
 Chapter 25.25.6: Graph and compare sampling plans with c = I and increasing values o...
 Chapter 25.25.7: Three samples of 200 livets each were taken from a large production...
 Chapter 25.25.8: Apply the test explained in Example 2 to the following data (x = di...
 Chapter 25.16: Explain the idea of a maximum likelihood estimate from memory
 Chapter 25.25.3: Find a Y5% confidence interval for the variance of a normal populat...
 Chapter 25.25.4: The two samples 70. 80. 30. 70. 60. 80 and 140. 120, 130. 120. 120....
 Chapter 25.25.4: Using samples of sizes 10 and 16 with variances Sx 2 = 50 and Sy 2 ...
 Chapter 25.25.5: CAS PROJECT. Control Charts. (a) Obtain 100 samples of 4 values eac...
 Chapter 25.25.6: Samples of 3 fuses are drawn from lots and a lot is accepted if in ...
 Chapter 25.25.7: In a table of properly rounded function values, even and odd last d...
 Chapter 25.25.8: Does an increase in temperature cause an increase of the yield of a...
 Chapter 25.17: How does the length of a confidence interval depend on the sample s...
 Chapter 25.25.3: Find a Y5% confidence interval for the variance of a normal populat...
 Chapter 25.25.4: Assuming normality and equal varIance and usmg independent samples ...
 Chapter 25.25.6: Graph the OC curve and [he AOQ curve for the single sampling plan f...
 Chapter 25.25.7: Are the 5 tellers in a ceI1ain bank equally timeefficient if durin...
 Chapter 25.25.8: Does the amount of feltilizer increase the yield of wheat X [kg/plo...
 Chapter 25.18: Couldn't we make the error in interval estimation zero simply by ch...
 Chapter 25.25.3: Find a Y5% confidence interval for the variance of a normal populat...
 Chapter 25.25.4: Show that for a nonnal distribution the two types of en'ors in a te...
 Chapter 25.25.7: CAS EXPERIMENT. Random Number Generator. Check your generator expel...
 Chapter 25.19: What is the Least squares principle? Give applications.
 Chapter 25.25.3: Find a Y5% confidence interval for the variance of a normal populat...
 Chapter 25.25.4: CAS EXPERIMENT. Tests of Means and Variances. (a) Obtain 100 sample...
 Chapter 25.25.7: TEAM PROJECT. Difficulty with Random Selection. 77 students were as...
 Chapter 25.20: What is the difference between regression and cOIl'elation analysis?
 Chapter 25.25.3: Find a Y5% confidence interval for the variance of a normal populat...
 Chapter 25.21: Find the maximum likelihood estimates of mean and variance of a nor...
 Chapter 25.25.3: If X is normal with mean 27 and variance 16, what distributions do ...
 Chapter 25.22: Determine a 95% confidence interval for the mean fL of a normal pop...
 Chapter 25.25.3: If Xl and X2 are independent normal random variables with mean 23 a...
 Chapter 25.23: What will happen to the length of the interval in Prob. 22 if we re...
 Chapter 25.25.3: A machine fills boxes weighing Y lb with X lb of salt, where X and ...
 Chapter 25.24: Determine a 99% confidence interval for the mean of a normal popula...
 Chapter 25.25.3: If the weight X of bags of cement is normally distributed with a me...
 Chapter 25.25: What confidence interval do we obtain in Prob. 24 if we assume the ...
 Chapter 25.25.3: CAS EXPERIMENT. Confidence lntervals. Obtain 100 samples of size 10...
 Chapter 25.26: Assuming normality, find a 95% confidence interval for the variance...
 Chapter 25.27: Find a 95% confidence interval for the mean fL, assuming normality ...
 Chapter 25.28: Find a 95% confidence interval for the mean fL, assuming normality ...
 Chapter 25.29: Find a 95% confidence interval for the mean fL, assuming normality ...
 Chapter 25.30: What sample size should we use in Prob. 28 if we want to obtain a c...
 Chapter 25.31: Find a 99'1t confidence interval for the variance 02 assuning norm...
 Chapter 25.32: Find a 99'1t confidence interval for the variance 02 assuning norm...
 Chapter 25.33: Using a sample of IO values with mean 14.5 from a normal population...
 Chapter 25.34: In Prob. 33. change the alternative to fL =1= 15.0 and test as before.
 Chapter 25.35: Find the power in Prob. 33.
 Chapter 25.36: Using a sample of 15 values with mean 36.2 and variance 0.9. te~t t...
 Chapter 25.37: Using a sample of 20 values with variance 8.25 from a normal popula...
 Chapter 25.38: A firm sells paint in cans containing I kg of paint per can and is ...
 Chapter 25.39: Using samples of sizes to and 5 with variances s/ = 50 and Sy 2 = 2...
 Chapter 25.40: Assume the thickness X of washers to be normal with mean 2.75 mm an...
 Chapter 25.41: What effect on UCL  LCL in a control chart for the mean does it ha...
 Chapter 25.42: The following sample~ of screws (length in inches) were taken from ...
 Chapter 25.43: A purchaser checks gaskets by a single sampling plan that uses a sa...
 Chapter 25.44: Does an automatic cutter have the tendency of cutting longer and lo...
 Chapter 25.45: ind the least squares regression line to the data (2. I). (0. 1). ...
Solutions for Chapter Chapter 25: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 25
Get Full SolutionsSince 193 problems in chapter Chapter 25 have been answered, more than 46265 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter Chapter 25 includes 193 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.