- 15.1: In 13, determine if the data to be collected is univariate or bivar...
- 15.2: In 13, determine if the data to be collected is univariate or bivar...
- 15.3: In 13, determine if the data to be collected is univariate or bivar...
- 15.4: Name and describe four common ways of obtaining data for a statisti...
- 15.5: In order to determine the average grade for all students who took a...
- 15.6: Sues grades are 88, 87, 85, 82, 80, 80, 78, and 60. a. What is the ...
- 15.7: The hours, xi , that Peg worked for each of the last 15 weeks are s...
- 15.8: Each time Kurt swims, he times a few random laps. His times, in sec...
- 15.9: In 912, for each of the given scatter plots: a. Describe the correl...
- 15.10: In 912, for each of the given scatter plots: a. Describe the correl...
- 15.11: In 912, for each of the given scatter plots: a. Describe the correl...
- 15.12: In 912, for each of the given scatter plots: a. Describe the correl...
- 15.13: A school investment club chose nine stocks at the beginning of the ...
- 15.14: The table lists the six states that had the largest percent of incr...
- 15.15: The salaries that Aaron earned for four years are shown in the tabl...
- 15.16: Mrs. Brudek bakes and sells cookies. The table shows the number of ...
- 15.17: In a local park, an attempt is being made to control the deer popul...
- 15.18: In 1821, use your answers to Exercises 1417 to estimate the require...
- 15.19: In 1821, use your answers to Exercises 1417 to estimate the require...
- 15.20: In 1821, use your answers to Exercises 1417 to estimate the require...
- 15.21: In 1821, use your answers to Exercises 1417 to estimate the require...
Solutions for Chapter 15: STATISTICS
Full solutions for Amsco's Algebra 2 and Trigonometry | 1st Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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 •
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.