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statistics for experimentert

Statistics 4th Edition David Freedman, Robert Pisani, Roger Purves - Solutions

58. If AB = 0 show that(A x F)(B x G) = 0 for any matrices F and G whose sizes are such that multiplication is defined.
57. Find det (A) in Prob. 31.
56. Find the inverse of the matrix B defined by$$B =\begin{bmatrix}4 & 1 & 3 & 2 & 1 \\1 & 2 & 6 & 4 & 2 \\3 & 6 & 12 & 8 & 4 \\2 & 4 & 8 & 24 & 12 \\1 & 2 & 4 & 12 & 12\end{bmatrix}$$Use Theorem 8.3.7.
55. Let A be partitioned as in Prob. 50, where A12 = 0 and det (A22) ≠ 0. Find A-1in terms of A11, A21, A22.
54. In Prob. 53, find A-1 and show that B-1A-1 = C-1 where C-1 is defined in Eq. (8.3.2).
53. Find matrices A and B such that AB = C, where C is defined in Eq. (8.3.1), A is lower triangular and does not involve the bi, and B is a diagonal matrix and does not involve the ai.
52. Let the n x n matrix A be defined by A =$$\begin{bmatrix}I_1 & 0 \\B & I_2\end{bmatrix}$$where B is an n1 x n2 matrix and the size of the other submatrices are thus determined. Show that A-1 exists and find it.
51. Let A be partitioned as in Prob. 50. If rank (A) = rank (A11), show that A22 =A21A11-1A12.
50. Let A be an n x n matrix that is partitioned as follows:
49. Find the characteristic vectors of the matrix in Prob. 45.
48. Find the characteristic roots of the matrix in Prob. 31.
47. Use Theorem 8.8.10 to find the determinant of the matrix in Example 8.2.1.
46. Use Theorem 8.8.7 to find the inverse of the matrix B in Example 8.2.1. Assume that the inverse exists.
45. Find the characteristic roots of the matrix A where A =$$\begin{bmatrix}2 & 1 & -1 & 0 \\0 & 2 & 1 & -1 \\-1 & 0 & 2 & 1 \\1 & -1 & 0 & 2\end{bmatrix}$$.
44. For the matrices in Prob. 38, demonstrate Theorem 8.8.13.
43. In Prob. 41, find the characteristic roots of A.
42. In Prob. 41, find det (A).
40. In Prob. 39 find det (A x B) in terms of the elements of A and B.41. For the matrix A below, the identity matrices are each 3 x 3. Find the inverse of A.A =$$\begin{bmatrix}31 & 21 \\-1 & 41\end{bmatrix}$$.
39. Let A be an m x m matrix, and B an n x n upper triangular matrix; show that A x B is an upper triangular block matrix.
38. For the matrices defined below demonstrate (A x B)¹ = A¹ x B¹.A =$$\begin{bmatrix}3 & 2 \\1 & 1\end{bmatrix}$$; B =$$\begin{bmatrix}-6 & 0 \\-1 & 8\end{bmatrix}$$.
37. For the matrices defined below demonstrate that (A x B) (F x G) = (AF) × (BG).A =$$\begin{bmatrix}2 & 1 \\1 & 3\end{bmatrix}$$; B =$$\begin{bmatrix}1 & 3 & -2 \\2 & 0 & -1\end{bmatrix}$$; F =$$\begin{bmatrix}3 \\1\end{bmatrix}$$; G =$$\begin{bmatrix}4 \\-1 \\0\end{bmatrix}$$.
36. For the matrices A and B in Prob. 33 and C defined below, demonstrate that(A x B) x C = A x (B x C).
35. Find det (A x B) and det (B x A) if$$A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \\ \end{bmatrix} B = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 2 & 4 & 1 \\ \end{bmatrix}$$
34. In Prob. 33 demonstrate that A x B = (A x I)(I x B).
33. Compute A x B and B x A for the matrices below.$$A = [1, -1, 0]; B = \begin{bmatrix} 3 & 1 \\ 4 & 2 \\ 2 & 0 \\ \end{bmatrix}$$
32. If A and B are symmetric matrices, show that A x B is symmetric.
31. Use the details of the proof of Theorem 8.6.1 to find a lower triangular matrix R and an upper triangular matrix T such that A = RT where$$A = \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ \end{bmatrix}$$
30. Generalize Prob. 29; that is, find the inverse of the matrix T, where T is a *k* x *k* lower triangular matrix, where each element on and below the main diagonal is equal to unity.
29. Find the inverse of the 5 x 5 lower triangular matrix T where$$T = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ \end{bmatrix}.$$
28. Work Probs. 26 and 27 when A is a k x k matrix.
27. Find the inverse of A in Prob. 26 assuming conditions on the 0i andb, such that the inverse exists.
26. Find the determinant of the matrix$$A =\begin{bmatrix}0 & a_1 & 0 & 0\\b_1 & 0 & a_2 & 0\\0 & b_2 & 0 & a_3\\0 & 0 & b_3 & 0\end{bmatrix}$$
25. Let the 2*k* x 2*k* matrix A be partitioned as follows$$A=\begin{bmatrix}A_{11} & A_{12}\\A_{21} & A_{22}\end{bmatrix}$$where *A11* is a *k* x *k* matrix; further suppose that *A21A22* = *A22A21* and let|*A22*| ≠ 0. Show that det (A) = det (A11A22 - A12A21).(Use Theorem 8.2.1.)
24. Evaluate the determinant of A where *xi* = *i* and *k* = 4.$$A =\begin{bmatrix}1 & 1 & 1 & \cdots & 1 \\x_1 & x_2 & x_3 & \cdots & x_k \\x_1^2 & x_2^2 & x_3^2 & \cdots & x_k^2 \\\vdots & \vdots & \vdots & & \vdots \\x_1^{k-1} & x_2^{k-1} & x_3^{k-1} & \cdots & x_k^{k-1}\end{bmatrix}$$
23. Use Theorem 8.2.1, to find the determinant of the matrix A where A =$$\begin{bmatrix}1 & 3 & 1 & 3 \\4 & 2 & 2 & 1 \\4 & 2 & 2 & 3 \\3 & 1 & 4 & 1\end{bmatrix}$$
22. Evaluate the determinant of the matrix B where B is defined by
21. Use Theorem 8.4.3 to evaluate the determinant of the matrix V in Example 8.3.3
20. Use Theorem 8.9.3 to find the characteristic roots of the matrix in Prob. 19.
19. Use Theorem 8.9.3 to find the determinant of B where$$B = \begin{bmatrix} 2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 6 & 10 \end{bmatrix}$$.Note that B = I + bb' where b' = [1, 2, 3].
18. Prove Theorem 8.9.5 by using Theorem 8.2.1.
17. Use Theorem 8.2.1 to find the inverse of A where$$A = \begin{bmatrix} 1 & 0 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 2 \\ 3 & 2 & 1 & 2 & 4 \end{bmatrix}$$
16. Find A-1 in Prob. 14.
15. Find the inverse of the matrix in Prob. 3.
14. Find the determinant and characteristic roots of the matrix A where$$A = \begin{bmatrix} 0 & 1 & 1 & ... & 1 \\ 1 & 0 & 1 & ... & 1 \\ 1 & 1 & 0 & ... & 1 \\ . & . & . & ... & . \\ 1 & 1 & 1 & ... & 0 \end{bmatrix}$$each identity has size k x k, and there are n² submatrices.
13. Extend Prob. 12to the case in which there are 712 block matrices and the order of each is k x k.
12. Find the inverse of the triangular matrix T where [I J J T=0 I 0 0
11. If 6 6 6 6 6 6 8 8 8 8 B=6 8 3 3 3 6 8 3 2 2 6 8 3 2 4 find B.
10. Generalize Probs. 8 and 9 to a k x k matrix.
9. If the conditions on thea, in the matrix A in Prob. 8 are such that A is nonsingular, find A.
8. Let a a a a az az az A= a az az az a az az a what are the conditions on thea, so that A is nonsingular?
7. Find the determinant of the matrix. Tal J J J al J J C = J Jal J Jal JJ where a 0 and each matrix has dimension n x n. What are the conditions on a and n to insure that the inverse exists?
6. Find the inverse of the matrix C where$$C =\begin{bmatrix}a₁ & a₂ & a₂ & a₂ \\a₂ & a₃ & a₄ & a₄ \\a₂ & a₄ & a₃ & a₄ \\a₂ & a₄ & a₄ & a₃\end{bmatrix}$$if the ai are such that the inverse exists.
5. In Theorem 8.2.1, suppose n₁ = n₂ and B₁₁ = B₂₂ = 0. State a result for the existence of B-¹.
4. Let a k x k matrix C be defined by Eq. (8.3.13).(a) Find the conditions on the constantsa, b, and k such that C is positive definite.(b) Find the conditions on the constantsa, b, and k such that C is positive semi-definite.(c) Find the conditions on the constantsa, b, and k such that C² = C.
3. If B = $$\begin{bmatrix}-I & A-I \\A-I & A\end{bmatrix}$$, where A is an m x m symmetric matrix such that A² = A, show that |B| = (-1)m.
2. In Prob. 1, if ad - b² ≠ 0, find B-¹.
1. If B = $$\begin{bmatrix}a I & bI \\bI & dI\end{bmatrix}$$, where each identity matrix is of size m x m, find the character-istic roots of B.
If a study draws conclusions about the effects of age, find out whether the data are cross-sectional or longitudinal. p-968
Roughly 68% of the entries on a list of numbers are within one SD of the average, and about 95% are within two SDs of the average. This is so for many lists, but not all. p-968
The SD measures distance from the average. Each number on a list is off the average by some amount. The SD is a sort of average size for these amounts off. More technically, the SD is the r.m.s. size of the deviations from the average. p-968
r.m.s. size of a list average of (entries). p-968
The r.m.s. size of a list measures how big the entries are, neglecting signs. p-968
Half the area under a histogram lies to the left of the median, and half to the right. The median is another way to locate the center of a histogram. p-968
The average locates the center of a histogram, in the sense that the his- togram balances when supported at the average. p-968
Average of a list = sum of entries number of entries? p-968
A typical list of numbers can be summarized by its average and standard deviation (SD). p-968
Many observers think there is a permanent underclass in American society—most of those in poverty typically remain poor from year to year. Over the pe-riod 1970-2000, the percentage of the American population in poverty each year has been remarkably stable, at 12% or so. Income figures for each
As in exercise 10, but a whole series of students are chosen. The r.m.s. size of your losses should be around _____ Fill in the blank. p-968
Incoming students at a certain law school have an average LSAT (Law School Aptitude Test) score of 163 and an SD of 8. Tomorrow, one of these students? p-968
An investigator has a computer file showing family incomes for 1,000 sub-jects in a certain study. These range from $5,800 a year to $98,600 a year. By accident, the highest income in the file gets changed to $986,000.(a) Does this affect the average? If so, by how much?(b) Does this affect the
In the HANES5 sample, the average height of the boys was 137 cm at age 9 and 151 cm at age 11. At age 11, the average height of all the children was 151 cm. 14(a) On the average, are boys taller than girls at age 11?(b) Guess the average height of the 10-year-old boys. p-968
A study on college students found that the men had an average weight of about 66 kg and an SD of about 9 kg. The women had an average weight of about 55 kg and an SD of 9 kg.(a) Find the averages and SDs, in pounds (1 kg = 2.2 lb).(b) Just roughly, what percentage of the men weighed between 57 kg
For the men age 18-24 in HANESS, the average systolic blood pressure was 116 mm and the SD was 11 mm. 13 Say whether each of the following blood pressures is unusually high, unusually low, or about average: p-968 80 mm 115 mm 120 mm 210 mm
For persons age 25 and over in the U.S., would the average or the median be higher for income? for years of schooling completed?--- OCR End --- p-968
Here is a list of numbers:0.7 1.6 9.8 3.2 5.4 0.8 7.7 6.3 2.2 4.1 8.1 6.5 3.7 0.6 6.9 9.9 8.8 3.1 5.7 9.1(a) Without doing any arithmetic, guess whether the average is around 1, 5, or 10.(b) Without doing any arithmetic, guess whether the SD is around 1, 3, or 6. p-968
(a) Both of the following lists have the same average of 50. Which one has the smaller SD, and why? No computations are necessary. p-968(i) 50, 40, 60, 30, 70, 25, 75(ii) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50(b) Repeat, for the following two lists.(i) 50, 40, 60, 30, 70, 25, 75(ii) 50, 40, 60, 30,
(a) Find the average and SD of the list 41, 48, 50, 50, 54, 57.(b) Which numbers on the list are within 0.5 SDs of average? within 1.5 SDs of average? p-968
For a list of positive numbers, can the SD ever be larger than the average?The answers to these exercises are on pp. A50-51. p-968
Can the SD ever be negative? p-968
For the list 107, 98, 93, 101, 104, which is smaller—the r.m.s. size or the SD? No arithmetic is needed. p-968
What is the r.m.s. size of the list 17, 17, 17, 17, 17? the SD? p-968
(a) The Governor of California proposes to give all state employees a flat raise of $250 a month. What would this do to the average monthly salary of state employees? to the SD?(b) What would a 5% increase in the salaries, across the board, do to the average monthly salary? to the SD? p-968
Repeat exercise 5 for the following two lists: p-968(i) 5, -4, 3, -1, 7(ii) -5, 4, -3, 1, -7
(a) For each list below, work out the average, the deviations from average, and the SD.(i) 1, 3, 4, 5, 7(ii) 6, 8, 9, 10, 12(b) How is list (ii) related to list (i)? How does this relationship carry over to the average? the deviations from the average? the SD? p-968
Three instructors are comparing scores on their finals; each had 99 students. In class A, one student got 1 point, another got 99 points, and the rest got 50 points.In class B, 49 students got a score of 1, one student got a score of 50, and 49 students got a score of 99. In class C, one student
Someone is telling you how to calculate the SD of the list 1, 2, 3, 4, 5:The average is 3, so the deviations from average are-2 -1 0 1 2 The 0 doesn't count, so the r.m.s. deviation is$$\sqrt{\frac{4+1+1+4}{4}} = 1.6$$And that's the SD.Is this right? Answer yes or no, and explain briefly. p-968
Someone is telling you how to calculate the SD of the list 1, 2, 3, 4, 5:The average is 3, so the deviations from average are-2 -1 0 1 2 Drop the signs. The average deviation is$$ \frac{2+1+0+1+2}{5}=1.2 $$And that's the SD.Is this right? Answer yes or no, and explain briefly. p-968
Guess which of the following two lists has the larger SD. Check your guess by computing the SD for both lists. p-968(i) 9, 9, 10, 10, 10, 12(ii) 7, 8, 10, 11, 11, 13
As in exercise 9, but tomorrow a whole series of men will be chosen at random.After each man appears, his actual height will be compared with your guess to see how far off you were. The r.m.s. size of the amounts off should be ___.Fill in the blank. (Hint: Look at the bottom of this page.)The
The men in the HANESS sample had an average height of 69 inches, and the SD was 3 inches. Tomorrow, one of these men will be chosen at random. You have to guess his height. What should you guess? You have about 1 chance in 3 to be off by more than ___. Fill in the blank. Options: 1/2 inch, 3
One investigator takes a sample of 100 men age 18-24 in a certain town. Another takes a sample of 1,000 such men.(a) Which investigator will get a bigger average for the heights of the men in his sample? or should the averages be about the same?(b) Which investigator will get a bigger SD for the
Below are sketches of histograms for three lists. Match the sketch with the de-scription. Some descriptions will be left over. Give your reasoning in each case. p-968(i) ave ≈ 3.5, SD ≈ 1(ii) ave ≈ 3.5, SD ≈ 0.5(iii) ave ≈ 3.5, SD ≈ 2(iv) ave ≈ 2.5, SD ≈ 1(v) ave ≈ 2.5, SD ≈
The SD for the ages of the people in the HANES5 sample is around $$ \underline{ } $$. Fill in the blank, using one of the options below. Explain briefly. (This survey was discussed in section 2; the age range was 0-85 years.) p-968 5 years 25 years 50 years
Each of the following lists has an average of 50. For each one, guess whether the SD is around 1, 2, or 10. (This does not require any arithmetic.) p-968(a) 49, 51, 49, 51, 49, 51, 49, 51, 49, 51(b) 48, 52, 48, 52, 48, 52, 48, 52, 48, 52(c) 48, 51, 49, 52, 47, 52, 46, 51, 53, 51(d) 54, 49, 46, 49,
Each of the following lists has an average of 50. For which one is the spread of the numbers around the average biggest? smallest? p-968(i) 0, 20, 40, 50, 60, 80, 100(ii) 0, 48, 49, 50, 51, 52, 100(iii) 0, 1, 2, 50, 98, 99, 100
This continues exercise 1.(a) Here are the heights of four boys: 150 cm, 130 cm, 165 cm, 140 cm. Match the heights with the descriptions. A description may be used twice.unusually short about average unusually tall(b) About what percentage of boys age 11 in the study had heights between 138 cm and
The Public Health Service found that for boys age 11 in HANES2, the average height was 146 cm and the SD was 8 cm. Fill in the blanks.(a) One boy was 170 cm tall. He was above average, by $$ \underline{ } $$ SDs.(b) Another boy was 148 cm tall. He was above average, by $$ \underline{ } $$ SDs.(c) A
A computer is programmed to predict test scores, compare them with actual scores, and find the r.m.s. size of the prediction errors. Glancing at the printout, you see the r.m.s. size of the prediction errors is 3.6, and the following results for the first ten students:predicted score: 90 90 87 80
The list 103, 96, 101, 104 has an average. Find it. Each number in the list is off the average by some amount. Find the r.m.s. size of the amounts off. p-968
Each of the numbers 103, 96, 101, 104 is almost 100 but is off by some amount.Find the r.m.s. size of the amounts off. p-968

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