 25.7.25.1.96: If 100 flips of a coin result in 30 heads and 70 tails. can we asse...
 25.7.25.1.97: If in 10 flips of a coin we get the same ratio as in Prob. I (3 hea...
 25.7.25.1.98: What would be the smallest number of heads in Prob. I under which t...
 25.7.25.1.99: If in rolling a die 180 times we get 39. 22. 41. 26. 20, 32. can we...
 25.7.25.1.100: Solve Prob. 4 if the sample is 25, 31. 33, 27, 29. 35.
 25.7.25.1.101: A manufacturer claims that in a process of producing kitchen knives...
 25.7.25.1.102: Between 1 P.M. and 2 P.M. on five consecutive days (Monday through ...
 25.7.25.1.103: Test for normality at the I % level using a sample of /I = 79 (roun...
 25.7.25.1.104: In a sample of 100 patients having a certain disease 45 are men and...
 25.7.25.1.105: In Prob. 9 find the smallest number (>50) of women that leads to th...
 25.7.25.1.106: Verify the calculations in Example 1 of the text.
 25.7.25.1.107: Does the random variable X = Number of accide/lTs per week in 1I ce...
 25.7.25.1.108: Using the given sample, test that the corresponding population has ...
 25.7.25.1.109: Can we assert that the traffic on the three lanes of an expressway ...
 25.7.25.1.110: If it i5 known that 25% of certain steel rod~ produced by a standar...
 25.7.25.1.111: Three samples of 200 livets each were taken from a large production...
 25.7.25.1.112: In a table of properly rounded function values, even and odd last d...
 25.7.25.1.113: Are the 5 tellers in a ceI1ain bank equally timeefficient if durin...
 25.7.25.1.114: CAS EXPERIMENT. Random Number Generator. Check your generator expel...
 25.7.25.1.115: TEAM PROJECT. Difficulty with Random Selection. 77 students were as...
Solutions for Chapter 25.7: Goodness of Fit. x2Test
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 25.7: Goodness of Fit. x2Test
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Chapter 25.7: Goodness of Fit. x2Test includes 20 full stepbystep solutions. Since 20 problems in chapter 25.7: Goodness of Fit. x2Test have been answered, more than 46428 students have viewed full stepbystep solutions from this chapter.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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 •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.