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statistics for experimentert

Questions and Answers of Statistics For Experimentert

3. Record the head sizes for the individuals on your team as measured by the team leader.
2. The team leader should measure and record the head size (measured as the circumference at the widest part of the forehead) of each of the other members of his or her team.
1. Designate a team leader for your team by choosing the person on your team who celebrated his or her last birthday most recently.
1.31 “Ozzie and Harriet Don’t Live Here Anymore” (San Luis Obispo Tribune, February 26, 2002) is the title of an article that looked at the changing makeup of America’s suburbs. The article
1.29 The article “Where College Students Buy Textbooks”(USA Today, October 14, 2010) gave data on where students purchased books. The accompanying frequency table summarizes data from a sample of
1.28 The report “Trends in Education 2010: Community Colleges” (www.collegeboard.com/trends) included the accompanying information on student debt for students graduating with an AA degree from a
1.27 The article “Fliers Trapped on Tarmac Push for Rules on Release” (USA Today, July 28, 2009) gave the following data for 17 airlines on number of flights that were delayed on the tarmac for
1.26 Example 1.5 gave the accompanying data on violent crime on college campuses in Florida during continued 2012 (from the FBI web site):University/College Student Enrollment Number of Violent
1.25 The article “Going Wireless” (AARP Bulletin, June 2009)reported the estimated percentage of households with only wireless phone service (no landline) for the 50 states and the District of
1.24 Heal the Bay is an environmental organization that releases an annual beach report card based on water quality (Heal the Bay Beach Report Card, May 2009). The 2009 ratings for 14 beaches in San
1.22 Figure EX-1.22 is a graph similar to one that appeared in USA Today (June 29, 2009). This graph is meant to be a bar graph of responses to the question shown in the graph.a. Is response to the
1.21 About 38,000 students attend Grant MacEwan College in Edmonton, Canada. In 2004, the college surveyed non-returning students to find out why they did not complete their degree (Grant MacEwan
1.20 Box Office Mojo (www.boxofficemojo.com) tracks movie ticket sales. Ticket sales (in millions of dollars)for each of the top 20 movies in 2007 and 2008 are shown in the accompanying table.Movie
1.19 The article “Feasting on Protein” (AARP Bulletin, September 2009) gave the cost (in cents per gram) of protein for 19 common food sources of protein.Food Cost Food Cost Chicken 1.8 Yogurt
1.18 The report “Findings from the 2008 Administration of the College Senior Survey” (Higher Education Research Institute, UCLA, June 2009) gave the following relative frequency distribution
1.12 Classify each of the following variables as either categorical or numerical. For those that are numerical, determine whether they are discrete or continuous.a. Number of students in a class of
1.11 In a study of whether taking a garlic supplement reduces the risk of getting a cold, participants were assigned to either a garlic supplement group or to a group that did not take a garlic
1.9 A building contractor has a chance to buy an odd lot of 5000 used bricks at an auction. She is interested in determining the proportion of bricks in the lot that are cracked and therefore
1.8 A consumer group conducts crash tests of new model cars. To determine the severity of damage to 2014 Toyota Camrys resulting from a 10-mph crash into a concrete wall, the research group tests six
1.6 The increasing popularity of online shopping has many consumers using Internet access at work to browse and shop online. In fact, the Monday after Thanksgiving has been nicknamed “Cyber
1.5 The student senate at a university with 15,000 students is interested in the proportion of students who favor a change in the grading system to allow for plus and minus grades (e.g., B1, B, B2,
1.2 Give a brief deﬁnition of the terms population and sample.
1.1 Give a brief deﬁnition of the terms descriptive statistics and inferential statistics.
●● construct a dotplot and describe the distribution of a numerical variable.
●● construct a frequency distribution and a bar chart and describe the distribution of a categorical variable.
●● distinguish between categorical, discrete numerical, and continuous numerical data.
●● the steps in the data analysis process.Students will be able to:●● distinguish between a population and a sample.
Describe Statistics Use And Interpretation
117. Exhibit a 3 x 3 matrix with real characteristic roots such that tr(A²) = tr(A³)= tr(A) = k and A is not idempotent.
116. In Prob. 115, show that the roots of the polynomial |A - λB| are greater than or equal to 1.
115. If A and B are positive definite n x n matrices and A - B is a non-negative n x n matrix, show that det(A) ≥ det(B).
114. Prove Theorem 12.6.5.
113. Prove Theorem 12.6.4.
112. Prove Theorem 12.6.2.
111. Prove Theorem 12.3.21.
110. Let A = BC, where A is n x n of rank p and C is p x n of rank p. Show that A is idempotent if and only if CB = I.
109. Prove Corollary 12.6.12.2.
108. Prove Corollary 12.6.12.1.
107. Let A be an n x n matrix with characteristic roots 0 and +1. Show that A is not necessarily idempotent.
106. Let C be an n x n matrix with characteristic roots -1, 0, and +1. Show that C is not necessarily tripotent.
105. Show that if A = A', then A is a tripotent matrix if and only if A = A'.
104. If A and B are n x n symmetric matrices and if A + B (or A - B) is positive definite, show that there exists a nonsingular matrix R such that R'AR and R'BR are each diagonal.
103. Consider the following Toeplitz matrix, which occurs in time series.$$R =\begin{bmatrix}1 & ho_1 & ho_2 & \dots & ho_{n-1} \\ho_1 & 1 & ho_1 & \dots & ho_{n-2} \\ho_2 & ho_1 & 1 & \dots &
102. Let Y be an n X n positive definite matrix. For each positive integer q, show that there exists an n x n unique positive definite matrix B such that Bq= V.
101. Let X be an $n \times p$ matrix such that X = [X₁, X₂], where X₁ is $n \times p₁$ and X₂ is $n \times p₂$ and $p₁ + p₂ = p$. Show that a c-inverse of X'X is given by$$(XX)^{c} =
100. If V is an $n \times n$ positive definite matrix and A is a symmetric $n \times n$ matrix show that there exists a nonsingular matrix R such that RAVR-1 = D, where D is a diagonal matrix.
99. Work Prob. 98 if the words positive definite are replaced by non-negative.
98. If A is an $n \times n$ positive definite matrix, show that there exists a positive definite matrix B such that A = B2.
97. Let A be an $n \times n$ nonsingular matrix, B be an $n \times n$ symmetric matrix, and AB be an idempotent matrix. Show that A'B is also an idempotent matrix.
96. If A is an $n \times n$ non-negative matrix and$$A = \begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix},$$where A₁₁ is an n₁ x n₁ matrix, show that |A| ≤ |A₁₁||A₂₂|.
95. If A is an $m \times n$ matrix and B is an $n \times m$ matrix such that I + BA is nonsingular, show that I + AB is also nonsingular and(I + AB)-1 = I - A(I - BA)-1B.
94. If A is an $n \times n$ nonsingular matrix, B is an $m \times m$ matrix, and C is an $n \times m$ matrix, show that[I - ACBC'] = [I - C'ACB].
93. If A is a positive definite $n \times n$ matrix, show for any positive integer $m$ that B is positive definite where $b_{ij} = a_{ij}^{m}$.
92. In Prob. 91, show that if the equal sign holds, then AB = BA = 0.
91. If A and B are *n* x *n* non-negative matrices, show that$$∑_{i=1}^n ∑_{j=1}^n a_{ij}b_{ij} ≥ 0.$$
90. If an *n* x *n* matrix T is an upper triangular, idempotent matrix with the first *k*diagonal elements equal to unity and the remaining diagonal elements equal to zero and T is partitioned so
90. If an n x n matrix T is an upper triangular, idempotent matrix with the first k diagonal elements equal to unity and the remaining diagonal elements equal to zero and T is partitioned so that$$T
89. If C₁ and C₂ are n x n symmetric matrices such that C₁C₂ = 0 and C₁ + C₂is tripotent, show that C₁ and C₂ are each symmetric tripotent matrices.
88. In Prob. 87, show that tr (AB) = tr (A) if and only if AB = A.
87. If A and B are each n x n symmetric idempotent matrices, show that tr (AB)≤ tr (A).
86. If A is a symmetric idempotent matrix, show that A'A - A and AA' - A are also symmetric idempotent matrices.
85. Show that A A' - A'BA is an idempotent matrix if A'BA is an idempotent matrix.
84. Show that an n x n matrix is idempotent if and only if its transpose is idempotent.
83. Let A be any n x n non-negative matrix such that A = C'C where C has size n x n and let B be any c-inverse of A. Show that (CB)'(CB) is also a c-inverse of A.
82. Let A be a symmetric matrix. Show that if a c-inverse of A exists that is non-negative, then A must be non-negative.
81. Let A₁₁ be any n x n symmetric idempotent matrix and let B be an (m + n)x (m + n) matrix defined by$$B =\begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}.$$Show that B is a
80. If A is an n x n matrix and A² has real, non-negative characteristic roots, show that A also has real roots.
79. Let C be any tripotent matrix. Show that rank (C) = rank (C²).
78. Show that C is an idempotent matrix if and only if there exist two matrices A and B, each symmetric idempotent, such that C =(AB.)-. Show that C = B
77. IfP is an orthogonal symmetric matrix and P oF ±l, show that J + P and I - P are both singular. .
76. If P is a symmetric orthogonal matrix, show that the characteristic roots (are either +1 or -1. If P ≠ ±I, show that P has at least one root of each.
75. Let a be a *k* *×* 1 vector such that a'a = 1. Show that I - 2aa' is an orthog matrix.
74. Let A be a *k* *×* *n* matrix. Show that I - 2AA- is an orthogonal matrix. Show 1 I - 2AA- is orthogonal for any L-inverse of A.
73. Prove Theorem 12.3.3 by using A-, the *g*-inverse of A.
72. Let A be a non-negative *k* *×* *k* matrix and define δ*i* as in Prob. 71. Show thiδ*i* = 0 for *i* = *t*, then δ*i* = 0 for all *i* > *t*.
71. Let A be a *k* *×* *k* symmetric matrix and define δ*i* by$$δ_1 = a_{11}, δ_2 =\begin{vmatrix}a_{11} & a_{12} \\a_{21} & a_{22}\end{vmatrix}, ..., δ_k = |A|.$$Show that δ*i* ≥ 0 for *i* =
70. Let C₁ and C₂ be *k* *×* *k* symmetric disjoint matrices such that C₁ + C₂ is potent. Show that C₁ and C₂ are tripotent.
69. For any symmetric matrix C, show that there exist two non-negative matri A and B such that for each and every positive integer *m*$$C^m = A^m + (-B)^m.$$
68. Let A be a non-negative matrix. Show that B is positive definite for each *a*every *a* such that 0 < *a* ≤ 1 where$$B = aI + (1 - a)A.$$
67. Show that the rank of an *n* *×* *n* upper triangular idempotent matrix is equal the number of nonzero diagonal elements.
66. If A and B are positive definite *k* *×* *k* matrices, show that$$[A + B]^{1/k} ≥ [A]^{1/k} + [B]^{1/k},$$where *k* is any positive integer.
65. Let C1 and C2 be (symmetric) disjoint tripotent matrices. Show that C1 + C2 and C1 - C2 are (symmetric) tripotenl matrices.
64. Let P be a *k* × *k* orthogonal matrix and let P₁ and P₂ be respectively *k*₁ and *k*₂distinct rows from P such that P₁P₂ = 0. Show that A is a symmetric tripotent matrix where?
63. Use the result of Theorem 12.4.4 for the matrix C in Prob. 60 to show that rank (C) = 2.
62. In Prob. 60, find two matrices A and B that are symmetric, idempotent, and disjoint such that C = A - B. See Theorem 12.4.3.
61. In Prob. 60, show that C² is a symmetric idempotent matrix.
60. Show that the matrix C below is tripotent and find its characteristic roots.$$C = \frac{1}{12} \begin{bmatrix} 1 & 1 & -5 & 3 \\ 1 & 1 & -5 & 3 \\ -5 & -5 & 7 & 3 \\ 3 & 3 & 3 & -9 \end{bmatrix}$$
59. Let B be a *k* × *k* matrix and A be a *k* × *n* matrix. Show that A'BA is an idem-potent matrix if BAA' is an idempotent matrix.
58. Let A be any *n* × *n* idempotent matrix and let *t* be any real number. Show that(I - A)(I - *t*A) is an idempotent matrix.
57. If A and B are symmetric *n* × *n* matrices and AB is idempotent (not necessarily symmetric), show that BA is also idempotent.
56. Let V be an *n* × *n* symmetric positive definite matrix such that V = P'P, where P has size *n* × *n*, and let A be a symmetric *n* × *n* matrix. Show that PAP is a symmetric idempotent
55. If A is any *k* × *k* matrix such that A = A*+1 for some positive integer *n*, show that tr (A) = tr (A²) = tr (A³) = … = tr (A*+1).
54. Let A be a symmetric idempotent matrix. Prove that the sum of squares of the off-diagonal element of any row (or column) is less than or equal to 1/4.
53. Let A be a symmetric idempotent matrix. Prove that$$-\frac{1}{2} \le a_{ij} \le \frac{1}{2}$$ for all i ≠ j.
51. Let A and V be n x n matrices and let V be nonsingular. If AV is idempotent--- OCR End ---
50. Exhibit a 3 x 3 matrix A of rank 2 that has two roots equal to +1 and one root equal to zero such that A is not idempotent. This illustrates the fact that (1) of Theorem 12.3.2 is not a
49. Prove Theorem 12.3.8.
48. Prove Theorem 12.3.7.
47. Show that Theorem 12.3.4 is not necessarily true if B is an idempotent matrix, but not symmetric.
46. In Prob. 45 let B = C−1 and partition B as$$B = \begin{bmatrix} B_{11} & b_{12} \\ b_{21} & b \end{bmatrix}$$where B₁₁ is an n x n submatrix. Show that(1) b = 0,(2) b12 = (1/n)1,(3) 1'B11 =

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