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statistics for experimentert

Questions and Answers of Statistics For Experimentert

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
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
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,
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$$A = \begin{bmatrix} a_{11} &
4. Find the *g*-inverse of the 5 x 2 matrix *A* where
3. Find the *g*-inverse of the 6 x 2 matrix *A* where 1 1 3 3 5 2*A* =2 1 0 6 1 5 Use Theorem 6.2.16.
2. Find the *g*-inverse of the 2 x 2 matrix *A* where-1 -1*A* =-1 -1 Use Theorems 6.4.1 and 6.4.9.
1. Find the *g*-inverse of the vector *a* where 13*a* = 1 52 Use Theorem 6.4.8.
23. By the method in Prob. 22, find the inverse of I - A where$$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
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
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
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' =
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
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
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
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
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
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)
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
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
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
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
The Wilcoxon matched-pairs signed ranks test can be used when:(a) There are two conditions(b) The same participants take part in both conditions(c) There is at least ordinal-level data(d) All of the
Look at the following partial printout of a Mann–Whitney U analysis from SPSS:The above information suggests that:(a) There will be a statistically signifi cant difference between conditions (b)
To assess the difference in scores from two conditions of a between-participants design, with ranked data, you would use:(a) The independent t-test(b) The Wilcoxon(c) The Related t-test(d)
The Wilcoxon matched pairs signed-ranks test (the Wilcoxon) is appropriate for:(a) Within-participants designs(b) Between-participants designs(c) Matched-participants designs(d) Both (a) and (c) above
Are there any univariate differences present?(a) Yes, for anxiety only(b) Yes, for depression only(c) Yes, for anxiety and depression(d) There are no univariate differences present
Which of the following would you report in a write-up?(a) Pillai’s trace = 0.497(b) Wilks’ lambda = 0.503(c) Hotelling’s trace = 0.989(d) Roy’s largest root = 0.989
Is there a multivariate difference between the conditions of the IV?(a) Yes(b) No(c) Can’t tell from the above printout(d) Yes but none of the DVs individually contribute signifi cantly to the
How many conditions are there in the IV?(a) 1(b) 2(c) 3(d) 4
What are the DVs in this study?(a) Condition and intercept(b) Anxiety and depression(c) Greenhouse and Geisser(d) None of the above
If you have correlated DVs, which of the following are applicable?(a) You should use t-tests to examine the contribution of the individual DVs to the linear combination of the DVs(b) You should not
Which part of the MANOVA printout gives us information about differences between the conditions of the IVs in terms of the linear combination of the DVs?(a) The Box’s M tests(b) The univariate
Which of the following are assumptions underlying the use of multivariate statistics?(a) Homogeneity of variance–covariance matrices(b) That we have equal sample sizes(c) That we have nominal-level
Which of the following are linear combinations?(a) A + B + C + D(b) b1x1 + b2x2 + b3x3 . . . + a(c) The Lottery numbers(d) Both (a) and (b) above
The assumption of multivariate normality means that:(a) Only the DVs should be normally distributed(b) All DVs and all IVs should be normally distributed(c) All DVs and all possible linear
Which of the following are true of MANOVA?(a) It forms a linear combination of the IVs(b) It forms a linear combination of the DVs(c) It is an extension of c2(d) It correlates the IVs with all of the
If we had three DVs and found a multivariate difference, what level of a would we set for each t-test to keep the overall a at 5%?(a) 5%(b) 1%(c) 1.67%(d) 3.33%
For uncorrelated DVs, how do we examine the relative contributions of the individual DVs to the combined DVs when our IV has only two conditions?(a) Conduct separate t-tests and adjust a to keep down
If you have correlated DVs in a MANOVA with a two-group IV, you should:(a) Cry(b) Conduct t-test analyses of the single DVs(c) Conduct c2 analyses of the DVs followed by t-tests(d) None of the above
Box’s M test:(a) Is a test of the homogeneity of variance assumption underlying ANOVA(b) Should be ignored at all times(c) Is a test of the homogeneity of variance–covariance matrices(d) Is
Which of the following are multivariate methods of calculating F?(a) Wilks’ lambda(b) Pillai’s trace(c) Hotelling’s trace(d) All of the above
Which of the following is true of MANOVA?(a) It analyses multiple IVs only(b) It analyses multiple DVs with one or more IVs(c) It can be used only with categorical data(d) All of the above
What is the most appropriate conclusion to be drawn from the above printout?(a) There is a multivariate difference, not attributable to sampling error, between males and females(b) Writing skills but

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