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statistics for experimentert

Matrices With Applications In Statistics 2nd Edition Franklin A Graybill - Solutions

1.33 The U.S. Department of Education reported that 14% of adults were classiﬁed as being below a basic literacy level, 29% were classiﬁed as being at a basic literacy level, 44% were classiﬁed as being at an intermediate literacy level, and 13% were classiﬁed as being at a proﬁcient
1.32 The report “Testing the Waters 2009” (www.nrdc.org)included information on the water quality at the 82 most popular swimming beaches in California.Thirty-eight of these beaches are in Los Angeles County. For each beach, water quality was tested weekly and the data below are the percent of
9. Does your answer from Step 8 surprise you? Explain why or why not.
8. For this step, work with three or four other students from your class. For each of the 10 shapes, form a new size estimate by computing the average of the size estimates for that shape made by the individuals in your group. Is this new set of estimates more accurate than your own individual
7. Use the last column of the activity sheet to record the squared differences (for example, if the difference for shape 1 was 23, the squared difference would be(23)2 5 9. Explain why the sum of the squared differences can also be used to assess how accurate your shape estimates were.
6. Compare your estimates with those of another person in the class by comparing the sum of the absolute values of the differences between estimates and corresponding actual values. Who was better at estimating shape sizes? How can you tell?
5. Would the sum of the differences tell you if the estimates and actual values were in close agreement?Does a sum of 0 for the differences indicate that all the estimates were equal to the actual value? Explain.
4. What would cause a difference to be negative? What would cause a difference to be positive?
3. Your instructor will provide the actual sizes for the 10 shapes, which should be entered into the “Actual Size” column of the activity sheet. Now complete the“Difference” column by subtracting the actual value from your estimate for each of the 10 shapes.
2. Next you will be visually estimating the sizes of the shapes in Figure 1.9. Size will be described as the number of squares of this size that would ﬁt in the shape. For example, the shape would be size 3, as illustrated by You should now quickly visually estimate the sizes of the shapes in
1. Construct an activity sheet that consists of a table that has 6 columns and 10 rows. Label the columns of the table with the following six headings:(1) Shape, (2) Estimated Size, (3) Actual Size,(4) Difference (Estimated 2 Actual), (5) Absolute Difference, and (6) Squared Difference.Enter the
10. Then person 1 measured person 2’s head size, person 2 measured person 3’s head size, and so on, with person 10 ﬁnally measuring person 1’s head size. Do you think that the resulting head size measurements would be more variable, less variable, or show about the same amount of
9. Consider the following scheme (you don’t actually have to carry this out): Suppose that a group of 10 people measured head sizes by ﬁrst assigning each person in the group a number between 1 and
8. Which data set was more variable—head size measurements of the different individuals on your team or the different measurements of the team leader’s head size? Explain the basis for your choice.
7. Do you think the team leader’s head size changed in between measurements? If not, explain why the measurements of the team leader’s head size are not all the same.
6. Using the data from Step 3, construct a dotplot of the team leader’s measurements of team head sizes. Then, using the same scale, construct a separate dotplot of the different measurements of the team leader’s head size (from Step 5).Now use the available information to answer the following
5. After all team members have measured the team leader’s head, record the different team leader head size measurements obtained by the individuals on your team.
4. Next, each individual on the team should measure the head size of the team leader. Do not share your measurement with the other team members until all team members have measured the team leader’s head size.
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 states that nonfamily households (for example, homes headed by a single professional or an elderly
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 1152 full-time college students.Where Books Purchased Frequency Campus bookstore 576 Campus
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 public community college in 2008.Debt Relative Frequency None 0.62 Less than $10,000 0.23 Between
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 at least 3 hours for the period from October 2008 to May 2009:Airline Number of Delays Rate per
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 Crimes Reported in 2012 Edison State College 17,107 4 Florida A&M University 13,204 14 Florida Atlantic
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 Columbia. In the accompanying data table, each state was also classified into one of three
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 Francisco County during wet weather were:A1 C B A A1 A1 A A1 B D C D F Fa. Would it be appropriate
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 question a categorical or numerical variable?b. Explain why a bar chart rather than a dotplot was
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 College Early Leaver Survey Report, 2004). Sixty-three students gave a personal (rather than an
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 (2007) 2007 Sales (millions of dollars)Spider-Man 3 336.5 Shrek the Third 322.7 Transformers 319.2
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 5.0 Salmon 5.8 Milk 2.5 Turkey 1.5 Peas 5.2 Soybeans 3.1 Tofu 6.9 Roast beef 2.7 Cheddar cheese 3.6
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 summarizing student responses to the question “If you could make your college choice over, would you
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 35 who turn in a term paper before the due dateb. Gender of the next baby born at a particular
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 supplement(“Garlic for the Common Cold,” Cochrane Database of Systematic Reviews, 2009). Based on the
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 unusable for her current project, but she does not have enough time to inspect all 5000 bricks. Instead,
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 cars of this type and assesses the amount of damage.Describe the population and sample for this
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 Monday”because of the large increase in online purchases that occurs on that day. Data from a large-scale
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, rather than just B). Two hundred students are interviewed to determine their attitude toward this
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.$$R =\begin{bmatrix}1 & ho_1 & ho_2 & \dots & ho_{n-1} \\ho_1 & 1 & ho_1 & \dots & ho_{n-2} \\ho_2 & ho_1 & 1 & \dots & ho_{n-3} \\\vdots & \vdots & \vdots & \ddots & \vdots \\ho_{n-1} & ho_{n-2} & ho_{n-3} & \dots &
102. Let Y be an n X n positive definite matrix. For each positive integer q, show that there exists an n x n unique positive definite matrix B such that Bq= V.
101. Let X be an $n \times p$ matrix such that X = [X₁, X₂], where X₁ is $n \times p₁$ and X₂ is $n \times p₂$ and $p₁ + p₂ = p$. Show that a c-inverse of X'X is given by$$(XX)^{c} = \begin{bmatrix}X_{1}^{'}X_{1} & X_{1}^{'}X_{2} \\X_{2}^{'}X_{1} & X_{2}^{'}X_{2}\end{bmatrix}^{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.$$
90. If an *n* x *n* matrix T is an upper triangular, idempotent matrix with the first *k*diagonal elements equal to unity and the remaining diagonal elements equal to zero and T is partitioned so that T =[T₁₁ T₁₂0 T₂₂]where T₁₁ is a *k* x *k* matrix, show that T₁₁ = I, T₂₂ =
90. If an n x n matrix T is an upper triangular, idempotent matrix with the first k diagonal elements equal to unity and the remaining diagonal elements equal to zero and T is partitioned so that$$T = \begin{bmatrix}T_{11} & T_{12} \\0 & T_{22}\end{bmatrix}$$
89. If C₁ and C₂ are n x n symmetric matrices such that C₁C₂ = 0 and C₁ + C₂is tripotent, show that C₁ and C₂ are each symmetric tripotent matrices.
88. In Prob. 87, show that tr (AB) = tr (A) if and only if AB = A.
87. If A and B are each n x n symmetric idempotent matrices, show that tr (AB)≤ tr (A).
86. If A is a symmetric idempotent matrix, show that A'A - A and AA' - A are also symmetric idempotent matrices.
85. Show that A A' - A'BA is an idempotent matrix if A'BA is an idempotent matrix.
84. Show that an n x n matrix is idempotent if and only if its transpose is idempotent.
83. Let A be any n x n non-negative matrix such that A = C'C where C has size n x n and let B be any c-inverse of A. Show that (CB)'(CB) is also a c-inverse of A.
82. Let A be a symmetric matrix. Show that if a c-inverse of A exists that is non-negative, then A must be non-negative.
81. Let A₁₁ be any n x n symmetric idempotent matrix and let B be an (m + n)x (m + n) matrix defined by$$B =\begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}.$$Show that B is a symmetric idempotent matrix if and only if A₂₂ is a symmetric idempotent matrix and A₁₂ = 0.
80. If A is an n x n matrix and A² has real, non-negative characteristic roots, show that A also has real roots.
79. Let C be any tripotent matrix. Show that rank (C) = rank (C²).
78. Show that C is an idempotent matrix if and only if there exist two matrices A and B, each symmetric idempotent, such that C =(AB.)-. Show that C = B
77. IfP is an orthogonal symmetric matrix and P oF ±l, show that J + P and I - P are both singular. .
76. If P is a symmetric orthogonal matrix, show that the characteristic roots (are either +1 or -1. If P ≠ ±I, show that P has at least one root of each.
75. Let a be a *k* *×* 1 vector such that a'a = 1. Show that I - 2aa' is an orthog matrix.
74. Let A be a *k* *×* *n* matrix. Show that I - 2AA- is an orthogonal matrix. Show 1 I - 2AA- is orthogonal for any L-inverse of A.
73. Prove Theorem 12.3.3 by using A-, the *g*-inverse of A.
72. Let A be a non-negative *k* *×* *k* matrix and define δ*i* as in Prob. 71. Show thiδ*i* = 0 for *i* = *t*, then δ*i* = 0 for all *i* > *t*.
71. Let A be a *k* *×* *k* symmetric matrix and define δ*i* by$$δ_1 = a_{11}, δ_2 =\begin{vmatrix}a_{11} & a_{12} \\a_{21} & a_{22}\end{vmatrix}, ..., δ_k = |A|.$$Show that δ*i* ≥ 0 for *i* = 1, 2, ..., *k* is not a sufficient condition for A to non-negative.
70. Let C₁ and C₂ be *k* *×* *k* symmetric disjoint matrices such that C₁ + C₂ is potent. Show that C₁ and C₂ are tripotent.
69. For any symmetric matrix C, show that there exist two non-negative matri A and B such that for each and every positive integer *m*$$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.

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