Statistics Without Maths For Psychology 5th Edition Christine Dancey - Solutions

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41. Does there ever exist a c-inverse of Aº that is equal to A if A is singular?
40. Show that there does not always exist a c-inverse of a matrix A that equals A.
39. For the matrix in Frob. 15, find a nonsingular matrix B such that BA ... H where H is in Hermite form?
38. For the matrix$$A = \begin{bmatrix}1 & 2 \\1 & 1 \\-1 & 0\end{bmatrix}.$$
37. Show rank (A') ≥ rank (A) for any c-inverse of A. (See Theorem 6.6.8.)
36. Find a c-inverse of the matrix A in Prob. 6.
35. Let A1' and A2' be any two c-inverses of the matrix A, and let g be any vector such that AA1'g = g. Show that AA2'g = g.
34. Let A be an matrix and let P and Q be orthogonal matrices such that PAQ = D where D is a diagonal matrix. Show that QD'P = A'.
33. If A is a positive semidefinite matrix, show that A' is also a positive semidefinite matrix.
32. Let A be an matrix of rank such that A = BC where B and C each has rank . Show that (BC)' = C'B'.
31. Let A be an matrix and let B be an matrix. Define F and G by G = A'AB; F = AGG', and show that AB = FG and (FG)' = G'F'.
30. In Prob. 28, if PP' = I, show that P'A' is a c-inverse of AP.
29. In Prob. 28, if and P is nonsingular, show that P-1A' is a c-inverse of AP.
28. Let A be an matrix and let P be any matrix of rank . Show that PA' is a c-inverse of AP, where A' and P' are any c-inverses of A and P, respectively.
27. If A is an matrix, B is an matrix and AB' = 0, and B'A = 0, show that(1) AB' = 0,(2) B'A = 0,(3) AB = 0.(4) BA' = 0,(5) B'A' = 0.(6) A'B = 0.
26. If A is an I'll x I'll symmetric matrix such that I'A = 0, sbow tha?
25. If A is an mxm symmetric matrix such that a'A=0, show that a'A=0(a is an m x 1 vector).
24. Let λi (i = 1,2,..., r) be the nonzero characteristic roots of an mx m symmetric matrix A. Show that λi (i = 1, 2, ..., r) are the nonzero characteristic roots of A'.
23. Let A be an m x m symmetric matrix and P be an orthogonal matrix such that P'AP = D, where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that P'A P is also a diagonal matrix.
22. Prove Theorem 6.3.3.
21. If A is a given mxn matrix, find conditions on a matrix X so that the system AX= I is consistent.
20. Find a solution to the system of equations 2x1-x2+x3=8, x1+2x2-x3=-5.
19. Let A be an mxn given matrix, and let X be any nx m matrix such that A'AX = A'is satisfied, and let Y be any n x m matrix such that YAA' = A'is satisfied. Show that the g-inverse of A is given by A = YAX.
18. Find the g-inverse of the matrix A where$$A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & -2 & 0 \\ 3 & 3 & -3 \end{bmatrix}$$
17. Find the g-inverse of the matrix A where--- OCR End ---
16. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & 1 & 1 & 0 & 0\\1 & 1 & 1 & 0 & 0\\1 & 1 & 1 & 0 & 0\\0 & 0 & 0 & 2 & 2\\0 & 0 & 0 & 2 & 2\end{bmatrix}$$
15. Find the g-inverse of the symmetric matrix A where$$A = \begin{bmatrix}3 & 1 & 0 & 1\\1 & 4 & -1 & 2\\0 & -1 & 6 & 2\\1 & 2 & 2 & 4\end{bmatrix}$$
14. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}6 & 1 & 2 & 4 & 9\\-3 & 1 & 5 & 2 & 7\\1 & 0 & 3 & -4 & 1\\1 & 3 & 17 & 4 & 24\\1 & -1 & -13 & 6 & -9\end{bmatrix}$$
13. Prove the following: Let A be an m x n matrix, X be an n x r matrix, C be an m x r matrix, B be an r x g matrix, and D be an n x g matrix. A necessary and sufficient condition that the two equations AX = C and XB = D have a common solution X is (1) each equation has a solution and (2) AD = CB.
12. Let the m x 2 (m≥2) matrix A of rank 1 be defined by A = [a, ca]where c is a scalar and a is an m x 1 nonzero vector. Find the g-inverse of A in terms of a and c.
11. In Prob, 10,show that the first three equations are consistent.
10. Show that the system of equations given below is not consistent.6x1 + x2 - 3x3 + x4 = 0, 4x1 - x2 + x3 - 2x4 = 5,--- OCR End ---
9. Find the general solution to the system of equations in Prob. 7.
8. Find a solution to the system of equations in Prob. 7.
7. Use Theorem 6.3.1 to show that the system of equations given below is consistent.3x1 - 2x2 + x3 = 3, 3x1 + x2 + 2x3 = 5, 3x1 + 10x2 + 5x3 = 11.
6. Find the g-inverse of the 3 x 3 matrix A where$$A = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & -1 \\ 3 & 1 & -1 \end{bmatrix}$$using the methods presented in each of the Theorems 6.5.1, 6.5.2, 6.5.5, and 6.5.8.
5. Show that the g-inverse of a general 2 x 2 symmetric matrix A of rank 1 defined by$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$is given by$$A = \begin{bmatrix} a_{11} & a_{12} \\ T & T \\ a_{21} & a_{22} \\ T & T \end{bmatrix}$$where T = a11 + a12 + a21 + a22 = tr
4. Find the *g*-inverse of the 5 x 2 matrix *A* where
3. Find the *g*-inverse of the 6 x 2 matrix *A* where 1 1 3 3 5 2*A* =2 1 0 6 1 5 Use Theorem 6.2.16.
2. Find the *g*-inverse of the 2 x 2 matrix *A* where-1 -1*A* =-1 -1 Use Theorems 6.4.1 and 6.4.9.
1. Find the *g*-inverse of the vector *a* where 13*a* = 1 52 Use Theorem 6.4.8.
23. By the method in Prob. 22, find the inverse of I - A where$$A = \begin{bmatrix}0 & 1 & -1 \\0 & 0 & 2 \\0 & 0 & 0\end{bmatrix}$$
22. In Prob. 21, show (I - A)^-1 = Σ^(k-1)_i=0 A^i.
21. Suppose A is an n x n matrix and A^k = 0 for a positive integer k. Show that I - A is nonsingular.
20. Prove Theorem 5.6.4.
19. Prove Theorem 5.6.2.
18. Prove Corollary 5.3.3.
17. In Prob. 16, find a matrix C such that AC = B.
16. Let A and B be defined by$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3\end{bmatrix}$$$$B = \begin{bmatrix}0 & 2 & 1 \\0 & 4 & 2 \\0 & -2 & -1\end{bmatrix}$$Show that the column space of B is a subspace of the column space of A.
15. Prove Theorem 5.4.5.
14. Let A and B be m x n matrices of rank n where n < m. Show that A and B may not have the same column space.
13. If A and B are nonsingular n x n matrices, prove that they have the same column space.
12. Find the dimension of the column space of A where$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3 \\1 & 2 & 0\end{bmatrix}$$
11. In Prob. 10, show that z₁ is the projection of y into S by using the results of Theorem 4.4.1.
10. Let S be spanned by x, and x2 where x\ = [1,1,0, -1]; x2 = (0, I, 1, IJ; the vector y is defined by y' = (I, 2, - I, I]. Find vectors ZI and Zl such that y =ZI + Z2 where Zt is in Sand Z2 is in SJ..
9. Find the vector z and the scalar λ such that y = λx + z--- OCR End ---
8. Find a basis for the orthogonal complement of the subspace in E₃ spanned by x where x' = [1, 1, -1].
7. Prove Theorem 5.2.5.
6. Use the results of Probs. 4 and 5 to demonstrate Theorem 5.2.5.
5. In Prob. 1, find the dimension of S₁ ⊕ S₂.
4. In Prob. 1, find the dimension of S₁ ∩ S₂.
3. If {a₁, ..., aₚ} spans S₁ and {b₁, ..., bₛ} spans S₂, prove that {a₁, ..., aₚ, b₁, ..., bₛ} spans S₁ ⊕ S₂.
2. In Prob. 1, find a basis for S₁ ⊕ S₂.
1. If a₁, a₂, a₃ span S₁ and b₁, b₂, b₃ span S₂, find a basis for S₁ ∩ S₂.$$a_1 = \begin{bmatrix}1 \\ 3 \\ -1 \\ 2\end{bmatrix}; a_2 = \begin{bmatrix}0 \\ -1 \\ 2 \\ 1 \end{bmatrix}; a_3 = \begin{bmatrix}2 \\ 7 \\ 2 \\ -3 \end{bmatrix};$$$$b_1 = \begin{bmatrix}1 \\ 8 \\ 0 \\
20. Consider the plane &'1 through the points ai' 32, 0 wherea, = [I, 1,0,1], 32 = [I, -I, 1,0).Find the projection ?
19. Find the projection of the directed line segment froma, to a2 onto the plane &'~where ei'i goes through 0, xI = [I, 1,I] and X2= [1, -I, I] and where a1 =[1,0, -I], a~ = [2,-I, 1]-
18. In Prob. 16, find the angle between t and fil.
17. In Prob. 16, sketch the three lines.
16. Find the projection of the directed line segment ( froma, to a1 onto the line fil that goes through 0 and b where al = [1,2, -1], a2 ... [I, -1,2), b' = [I, I,ll
15. Let fil be a line through 0 anda, where a' = [I, -1, 1]. Find the equation of the plane r!J that is perpendicular to ~ and passes through the point [2, 1, -1].
14. In Prob. 13, show that this value of d is the minimum distance between 0 and the plane fi'.
13. In Prob. 12, verify that the distance between 0 and tbe point of intersection of the:Iine ~ and the plane f!; is d = (l'(B'B)-ll]-lfl.
12. In Prob. II, find the point on 11 where 2 intersects {?
11. rn Prob. 10,find the equation of a line !l' tbat goes through the origin and is perpendicular to every line in the plane {?
10.Find the equation of the plane {}'that goes through the three points bj = [I, I, 1], bi = [I, -1,0), bJ = [0, 1, -1]. Write the equation in the form a'y = c.
9. Find the equation of the line ~ that is parallel to fI!I and 21, of Prob. 3, equidistant from the two lines, and in the same plane as !i'l and 21.
8. Find the distance from the point x' = [2, 1, -IJ to the line 5£ that goes through the two points a'". [I, 1,0] and b' = [I, -1,2].
7. Find the direction angles of the line through the points a' = [2,0,-I]and b' = [I, -1,3],
6. Find the direction angles of II in Prob, 5.
5. Find the angle between the two directed.line segments II and 12where II is the directed line segment from 0to a' = [1,-I, I] and 12 is the directed line segment from 0 to b' = [2, I, -1].
4. Do the two lines 2, and 21 intersect where fL'1 goes through the two points a' = [I, I, -1]; b'= [2,0,1]and fL' 2 goes through the two points c' = [I, I, I];d' = (2, 0, 2]7
3. Use Def. 4.2.2 to show that fill and fI!1 aTeparallel where 5£, goes through the two points a' = [1,0,I]; b' = [0, I, - I] and 5£2 goes 'through the two points c' = [1,2, -I]; d'= [2, I, I).
2. Find the equation of a line fI!I that goes through the point x' = [I, - 1, 1], intersects another line fI!2, and is perpendicular to fI!2' The line fI! 2 goes through the points a' = 0 and b' = [2, I, 1]-
1. Find the equation of the line in E3 through the point x' = [I, 1,0) and parallel to the line through the two points a' = [0, I, -IJ, b' = [1,0,1]-
Look at the following table. Professor Green predicted that strength would relate positively to motivation.Unfortunately the professor meant to obtain one-tailed p-values. Professor Green wants you to interpret the results below, for a one-tailed hypothesis.The relationship between Strength and
Look at the output below.Which is the most appropriate statement? The relationship between the two ratings is (a) Strong (rho = 0.7, p (b) Strong (rho = 0.6, p (c) Moderate (r = 0.7, p (d) Moderate (r = 0.6, p Correlations Rating1 Rating2 Spearman's rho Rating1 Correlation 1.000 Coefficient Sig.
Look at the following text, taken from Daley, Sonuga-Barke and Thompson (2003). They were making comparisons between mothers of children with and without behavioural problems.Mann–Whitney U tests were used . . . signifi cant differences indicated that mothers of children with behavioural problems
The strongest relationship is between pain and:(a) Tension(b) Autonomic(c) Fear(d) Punishment
Which is the most appropriate statement? In general terms, the affective measures and pain show a:(a) Weak relationship(b) Moderate relationship(c) Strong relationship(d) Perfect relationship
The participants were measured:(a) At two timepoints(b) At three timepoints(c) At four timepoints(d) Cannot tell
How many participants were in the study?(a) 7(b) 14(c) 21(d) Cannot tell
Which is the most sensible conclusion?(a) There are differences between the groups, but these stand a 21% chance of being due to sampling error(b) There are differences between the groups, and these are unlikely to be due to sampling error(c) There are no differences between the three groups at
How many participants were in the study?(a) 5(b) 10(c) 15(d) 20
Which group had the highest scores?(a) Group 1(b) Group 2(c) Group 3(d) Cannot tell
Which is the most sensible conclusion?(a) There are no signifi cant differences between the three groups, p > 0.05(b) There are signifi cant differences between the groups, p = 0.003(c) There are no signifi cant differences between the groups, p = 0.003(d) Impossible to tell
A t-value of 3 has been converted into a z-score of -3.2. This means:(a) The calculations are incorrect(b) There is not likely to be a statistically signifi cant difference between conditions(c) There is likely to be a statistically signifi cant difference between conditions(d) The results are
If a Wilcoxon test shows that t = 3 with an associated probability of 0.02, this means:(a) Assuming the null hypothesis to be true, a t-value of 3 would occur 2% of the time through sampling variation(b) We are 98% certain that our results are statistically signifi cant(c) Given our data, we expect
If, in a repeated-measures design with two conditions, you have a small number of participants, with skewed, ordinal data, the most appropriate inferential test is:(a) Unrelated t-test(b) Related t-test(c) Mann–Whitney U test(d) Wilcoxon
A Mann–Whitney test gives the following result:U = 9, p = 0.1726 (2-tailed probability)The researcher, however, made a prediction of the direction of the difference, and therefore needs to know the one-tailed probability. This is:(a) 0.0863(b) 0.863(c) 0.1726(d) Indeterminate
The Mann–Whitney U involves:(a) The difference in the means for each condition(b) The sum of the ranks for each condition(c) Finding the difference in scores across conditions, then ranking these differences(d) The difference in ranks across conditions

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Statistics Without Maths For Psychology 5th Edition Christine Dancey - Solutions

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