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### lecture13

Course: MA 2612, Fall 2008
School: Uni. Worcester
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Word Count: 737

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13 Lecture Administrative 1. Lab 3 tomorrow. MA 2612 - Applied Statistics II D 2004 2. Exam 1 on Monday. Today 1. Example using SLR model. 2. Inference about regression parameters. Exam 1: Monday 11:00am (60 minutes, no extra time) Chapters 6 and 9 Open book/notes Some problems similar to quizzes Some true/false or multiple choice questions on concepts Example i Xi 1 5 2 7 3 2 4 7 5 9 Yi 12 11 12 9 6 2...

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13 Lecture Administrative 1. Lab 3 tomorrow. MA 2612 - Applied Statistics II D 2004 2. Exam 1 on Monday. Today 1. Example using SLR model. 2. Inference about regression parameters. Exam 1: Monday 11:00am (60 minutes, no extra time) Chapters 6 and 9 Open book/notes Some problems similar to quizzes Some true/false or multiple choice questions on concepts Example i Xi 1 5 2 7 3 2 4 7 5 9 Yi 12 11 12 9 6 2 Example i Xi 1 5 2 7 3 2 4 7 5 9 Sum 30 SS Yi (Xi X) (Yi Y ) (Xi X)(Yi Y ) Xi Yi ei Yi 12 -1 2 -2 60 10.79 1.209 11 1 1 1 77 9.22 1.780 12 -4 2 -8 24 13.15 -1.149 9 1 -1 -1 63 9.22 -0.220 6 3 -4 -12 54 7.65 -1.649 50 0 0 -22 278 28 26 8.714 In lecture 12 we saw that Pn Xi X Yi Y 22 i=1 1 = = = 0.79 2 Pn 28 i=1 Xi X 50 22 30 + = 14.74 0 = Y 1X = 5 28 5 Therefore, Yi = 0 + 1 Xi = 14.74 0.79Xi We can also ll in the ANOVA table for this model. Source Regression Error Total df SS MS 3 Inference about the parameters of the SLR model Condence intervals Recall from MA 2611: A condence interval is a range of likely values for the true value of the parameter that we compute based on the data. Whenever we estimate a parameter it is more informative to give a con- dence interval than to simply state the point estimate. This is because our estimate is subject to sampling variability. In situations where there is a lot of sampling variability, it is more di- cult to accurately estimate the parameter and so the condence interval is wider. Similarly, when there is very little sampling variability, it is more likely that our estimate is close to the true value of the parameter and so the condence interval is narrower. The form of most condence intervals we encounter is ESTIMATE MARGIN OF ERROR where margin the of error has two components 1. The estimated standard error of the estimate (of the parameter). 2. A constant that is related to the sampling distribution of the esti- mate. For example, in MA 2611 you learned that 95% condence interval for the mean of a normal population based on n observations is SY Y tn1,0.975 n 4 Condence intervals for 0 and 1 We have already seen that our estimates of 0 and 1 are 0 = Y 1X 1 Pn Xi X Yi Y i=1 = Pn 2 Xi X i=1 The estimated standard errors for these estimates are v ! u u 2 X 1 + Pn ( 0 ) = tMSE 2 n i=1 Xi X s MSE ( 1 ) = Pn 2 Xi X i=1 And the sampling distributions of both 0 0 ( 0 ) is the tn2 distribution. Therefore, the 100(1 )% condence intervals for 0 and 1 are: 0 tn2,1/2 ( 0 ) 1 tn2,1/2 ( 1 ) and 1 1 ( 1 ) 5 Hypothesis tests 0 and 1 Scientic hypothesis Statistical model We observe (X1 , Y1 ), (X2 , Y2 ), . . . (Xn , Yn ) which we assume are described by Yi = 0 + 1 Xi + i where 1, 2, . . . n N(0, 2 ) or equivalently, Y...

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Uni. Worcester - MA - 590
Lecture 13 Administrative1. Lab 3 tomorrow.MA 2612 - Applied Statistics IID 20042. Exam 1 on Monday.Today1. Example using SLR model. 2. Inference about regression parameters.Exam 1: Monday 11:00am (60 minutes, no extra time) Chapters 6 a
Uni. Worcester - MA - 2
Uni. Worcester - MA - 2051
Uni. Worcester - MA - 2612
Lecture 6 AdministrativeMA 2612 - Applied Statistics IID 20041. Lab 1 Handout is due today. 2. This is the last lecture for Chapter 6. 3. Homework for Chapter 6 is due on Tuesday. 4. Quiz 2 is tomorrow.Today1. Quick Review and Example of Com
Uni. Worcester - MA - 590
Lecture 6 AdministrativeMA 2612 - Applied Statistics IID 20041. Lab 1 Handout is due today. 2. This is the last lecture for Chapter 6. 3. Homework for Chapter 6 is due on Tuesday. 4. Quiz 2 is tomorrow.Today1. Quick Review and Example of Com
Uni. Worcester - MA - 2051
Uni. Worcester - OS - 390
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143CHAPTER9Data RepresentationRepresentation of Numeric Variables 143 Representation of Floating-Point Numbers 143 Representation of Integers 145 Using the LENGTH Statement to Save Storage Space How Character Values Are Stored 146145Represe
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Operating Systems WPI CS3013February 5, 20067 Midterm ExamName _ Answer the following SIX questions. Theres lots of partial credit, so put down what you know. You are to use NO notes or papers for this exam.Problem 1: Short Answers:a) What is a
Uni. Worcester - CS - 3013
Operating Systems WPI CS3013February 5, 20067 Midterm ExamName _ Answer the following SIX questions. Theres lots of partial credit, so put down what you know. You are to use NO notes or papers for this exam.Problem 1: Short Answers:a) What is a
Uni. Worcester - CS - 07
UNIX Input/Output BueringWhen a C/C+ program begins execution, the operating system environment is responsible for opening three les and providing le pointers to them: stdoutstandard output stderrstandard error stdinstandard input For C programs
Uni. Worcester - CS - 3013
UNIX Input/Output BueringWhen a C/C+ program begins execution, the operating system environment is responsible for opening three les and providing le pointers to them: stdoutstandard output stderrstandard error stdinstandard input For C programs
Uni. Worcester - CS - 04
Homework 3: Numeric PredictionsAbraao Lourenco aln@WPI.EDU BS-MS October 1st, 2004Contents3 Homework 3: Numeric Predictions 3.1 Linear Regression . . . . . . . . . . . . . . . . . . . . . 3.1.1 Regression Equation . . . . . . . . . . . . . . . 3.
Uni. Worcester - CS - 4445
Homework 3: Numeric PredictionsAbraao Lourenco aln@WPI.EDU BS-MS October 1st, 2004Contents3 Homework 3: Numeric Predictions 3.1 Linear Regression . . . . . . . . . . . . . . . . . . . . . 3.1.1 Regression Equation . . . . . . . . . . . . . . . 3.
Uni. Worcester - CS - 04
Candidate GenerationSize 1 C1 Initially all possible attributes and their possible values are candidates. Min Song Candidate Attribute cap-surface = fibrous cap-surface = grooves cap-surface = scaly cap-surface = smooth bruises? = bruises bruises? =
Uni. Worcester - CS - 4445
Candidate GenerationSize 1 C1 Initially all possible attributes and their possible values are candidates. Min Song Candidate Attribute cap-surface = fibrous cap-surface = grooves cap-surface = scaly cap-surface = smooth bruises? = bruises bruises? =
Uni. Worcester - CS - 4
James Martineau CS 4445 Homework 4 I. Instance-based Learning A. Non-Normalized Nearest Neighbors 1. Non-weighted Nearest Neighbors MPG ? 18 27 34 24 28 32 26 29 44 10 28 16 MPG ? 18 27 34 24 28 32 26 29 44 10 28 16 MPG ? 18 27 34 24 28 32 26 29 # C
Uni. Worcester - CS - 4445
James Martineau CS 4445 Homework 4 I. Instance-based Learning A. Non-Normalized Nearest Neighbors 1. Non-weighted Nearest Neighbors MPG ? 18 27 34 24 28 32 26 29 44 10 28 16 MPG ? 18 27 34 24 28 32 26 29 44 10 28 16 MPG ? 18 27 34 24 28 32 26 29 # C
Uni. Worcester - CS - 04
PRISM Using P/T Min Song Rule 1IF ? Then Poisonous = edible APPLICABLE INSTANCES Instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 cap-surface scaly smooth fibrous scaly scaly scaly smooth scaly scaly smooth fibrous fibrous smooth fibrous
Uni. Worcester - CS - 4445
PRISM Using P/T Min Song Rule 1IF ? Then Poisonous = edible APPLICABLE INSTANCES Instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 cap-surface scaly smooth fibrous scaly scaly scaly smooth scaly scaly smooth fibrous fibrous smooth fibrous
Uni. Worcester - CS - 504
CS504: Analysis of Computations and Systems Spring 1999Sample problems, use for exam? March 3, 19991. A basketball team has 12 players. In our version of the rules a team is allowed up to 5 exchanges during the entire game, and an exchanged play
Uni. Worcester - CS - 504
EX: Worst-case # probes for binary search in sorted table of n entries 1, if n = 1 f ( n) = f ( n / 2) +1,if n &gt; 1 DEF : x - largest integer xTry unfolding the recurrence. f ( n) = f ( n / 2) +1= f n / 2 / 2 +1+1 Actually, this is a special cas
Uni. Worcester - CS - 504
Read: GKP 9.1, 9.2 Asymptoticsvoid InsertionSort(A) int A[]; { int i, j, temp; for (j=1; j&lt;Length; j+) /* A[0.j -1] sorted */ for (i=j-1; i&gt;=0 &amp; A[i]&gt;A[i+1]; i-){ temp=A[i]; A[i]=A[i+1]; A[i+1]=temp; } } n=2000, 62 seconds A[j] copied up many times
Uni. Worcester - CS - 504
Binomial Coefficients In how many ways can we choose k elements from an n element set? There are n choices for the first element, n -1 for the second,., down to n-k +1 for the k th , yielding n*(n-1)*.*(n-k+1). So there are 4*3=12 ways to choose 2 el
Uni. Worcester - CS - 504
1+2+3+.+(n -1)+n What is the .? Actually 1+2+.+n suffices. How about 1+2+41.7 ? For a1+a2+.+an, Lagrange(1772) introduced notation.k =1 ak or ak .1kn P(k)nIn general we write akto denote the sum of all ak s.t.:-k integer, -P(k) for p
Uni. Worcester - CS - 504
Read: GKP 8.1,8.2 Quicksort is the sorting technique of choice although its worst-case behavior is inferior to that of many others. pWhy? What does average case mean? Seeking a phone number (which exists) in a phone book sees &quot;on average&quot; half the el
Uni. Worcester - CS - 504
Ex :In how many ways can we give money using 2 coins? (z + z + z + z ) (z + z + z + z ) No! Why not? [z ]( z + z + z + z )( z + z + z + z ) = 25 10 25 5 10 25 6 5 10 25 5 10 25In how many ways can we give money using 1 coin? (z + z + z + z ) [z
Uni. Worcester - CS - 504
Problem : Arithmetic on integers a,b, two n-digit numbers, for large n. Note : Naive addition takes linear time, multiplication quadratic. Can we speed up multiplication at the expense of more additions? Let a = 2,345 = 23*102+45 = a1*102+a2b = 6,
Uni. Worcester - CS - 504
1+2+3+.+(n -1)+n What is the .? Actually 1+2+.+n suffices. How about 1+2+41.7 ? For a1+a2+.+an, Lagrange(1772) introduced notation.k =1 ak or ak .1kn P(k)nIn general we write akto denote the sum of all ak s.t.:-k integer, -P(k) for pr
Uni. Worcester - CS - 504
Ex : Bucket Sorting (Gonnet, 4.2.3, pg. 176) (1-pass version) Assume you know the distribution of a set of keys to be sorted. That is, assume we have M buckets that partition the key space, M keys, and Pr{key belongs in bucket bm} = 1/M , 1m M . for
Uni. Worcester - CS - 504
Read : GKP 8.1,8.2 Quicksort is the sorting technique of choice although its worst-case behavior is inferior to that of many others. Why? What does average case mean? Seeking a phone number (which exists) in a phone book sees &quot;on average&quot; half the el
Uni. Worcester - CS - 504
Given set S, assume each s S has a weight |s|. A GF of S is F ( z) = S an = {s S: s = n} , F ( z) = an z n S NOTE: Since S = a n , S = FS (1) .n ns Szs. LettingEx: In how many ways can we give money using 1 coin? (z + z 5 + z 10 + z 25
Uni. Worcester - CS - 504
CS504: Analysis of Computations and Systems Spring 1999Homework VIDue: March 8 or 10, 1999One more recurrence1. Solve the second-order linear recurrence xn 1 xn 312 xn 32nn2x0a x1bGenerating Functions2. Here is a different w
Uni. Worcester - CS - 99
CS504: Analysis of Computations and Systems Spring 1999Homework VIDue: March 8 or 10, 1999One more recurrence1. Solve the second-order linear recurrence xn 1 xn 312 xn 32nn2x0a x1bGenerating Functions2. Here is a different w
Uni. Worcester - CS - 9906
CS504: Analysis of Computations and Systems Spring 1999Homework VIDue: March 8 or 10, 1999One more recurrence1. Solve the second-order linear recurrence xn 1 xn 312 xn 32nn2x0a x1bGenerating Functions2. Here is a different w
Uni. Worcester - CS - 504
C.S.504 H.W. #7Due: November 5, 1991 1. (3 points) Use the method of characteristic roots to solve the tn = tn -2 + 4n , n 2 recurrence: t 0 = 1, t 1 = 4 2. (3 points) Use the method of characteristic roots to solve the recurrence: tn = 2*tn -1 + (n
Uni. Worcester - CS - 91
C.S.504 H.W. #7Due: November 5, 1991 1. (3 points) Use the method of characteristic roots to solve the tn = tn -2 + 4n , n 2 recurrence: t 0 = 1, t 1 = 4 2. (3 points) Use the method of characteristic roots to solve the recurrence: tn = 2*tn -1 + (n
Uni. Worcester - CS - 504
C.S.504 H.W. #4Due: October 8, 19921. (1 point) Evaluate1 2 n2 +10n2 5 2. (1 point) Evaluate0m kn 3. (4 points) Assume that you're sequentially seeking a card in a perfectly shuffled deck of 106 distinct cards. The card is in the deck. A)W
Uni. Worcester - CS - 92
C.S.504 H.W. #4Due: October 8, 19921. (1 point) Evaluate1 2 n2 +10n2 5 2. (1 point) Evaluate0m kn 3. (4 points) Assume that you're sequentially seeking a card in a perfectly shuffled deck of 106 distinct cards. The card is in the deck. A)W
Uni. Worcester - CS - 504
C.S.504 H.W. #1Due: September 23, 19931. (2 points) What does the following algorithm compute? function f(n,m : integer) : integer; X m; K 0; while Xn do X X*m; K K+1; return(K); 2. (9 points) We want to find the maximum and the minimum element
Uni. Worcester - CS - 93
C.S.504 H.W. #1Due: September 23, 19931. (2 points) What does the following algorithm compute? function f(n,m : integer) : integer; X m; K 0; while Xn do X X*m; K K+1; return(K); 2. (9 points) We want to find the maximum and the minimum element
Uni. Worcester - CS - 504
C.S.504 H.W. #5Due: October 15, 1991 1.(5 points) Prove the equality on page 72 of our text that for direct chaining hashing,s 2 ( A n' ) =2. (3 points) Given two sorted lists,n ( m -1) m2a 1 ,., a n a b 1,.,b nbshow that any algorithm whi
Uni. Worcester - CS - 91
C.S.504 H.W. #5Due: October 15, 1991 1.(5 points) Prove the equality on page 72 of our text that for direct chaining hashing,s 2 ( A n' ) =2. (3 points) Given two sorted lists,n ( m -1) m2a 1 ,., a n a b 1,.,b nbshow that any algorithm whi
Uni. Worcester - CS - 504
C.S.504H.W. #4Due: March 25/26, 1998Read SECTIONS 5.1, 5.2, 5.3 (Trick 1) Do EXERCISEs 5.1, 5.2, 5.4, 5.15 from GKP. Do not submit your solutions, but check them with the answers from the back of the text. 1. (3 points) Find a closed form for coe
Uni. Worcester - CS - 98
C.S.504H.W. #4Due: March 25/26, 1998Read SECTIONS 5.1, 5.2, 5.3 (Trick 1) Do EXERCISEs 5.1, 5.2, 5.4, 5.15 from GKP. Do not submit your solutions, but check them with the answers from the back of the text. 1. (3 points) Find a closed form for coe
Uni. Worcester - CS - 504
C.S.504 H.W. #1Due: September 17, 1991 Assume that you're sequentially seeking a card in a perfectly shuffled deck of 106 distinct cards. Use Chebyshev's inequality to give a bound on the probability that you'll have to examine at least 900,000 card
Uni. Worcester - CS - 91
C.S.504 H.W. #1Due: September 17, 1991 Assume that you're sequentially seeking a card in a perfectly shuffled deck of 106 distinct cards. Use Chebyshev's inequality to give a bound on the probability that you'll have to examine at least 900,000 card
Uni. Worcester - CS - 504
C.S.504SOLUTION FOR H.W. #1k n n n n( n +1) n( n - 1) n 1 - 1 = ( n - k) = n - k = n 2 = 1. 1 = 2 2 k =1 j =k+ 1 k =1 j =1 j =1 k =1 k =1 k =1 n n n2. In fact, Gn = (-1)n+1Fn.0, if n = 0 Gn = if n = 1 1, G - G , if n &gt;1 n -2 n -13.
Uni. Worcester - CS - 98
C.S.504SOLUTION FOR H.W. #1k n n n n( n +1) n( n - 1) n 1 - 1 = ( n - k) = n - k = n 2 = 1. 1 = 2 2 k =1 j =k+ 1 k =1 j =1 j =1 k =1 k =1 k =1 n n n2. In fact, Gn = (-1)n+1Fn.0, if n = 0 Gn = if n = 1 1, G - G , if n &gt;1 n -2 n -13.
Uni. Worcester - CS - 504
C.S.504 H.W. #2Due: September 24, 1992(6 points) Consider the following procedure void InsertionSort(int A, int n) {int i, j,temp; for (i=1; i&lt;n; i+) { /* A[0.i-1] already sorted */ temp=A[i]; for (j=i-1; j&gt;=0 &amp; temp&lt;A[j]; j-) A[j+1]=A[j]; /*&lt;-*/ A
Uni. Worcester - CS - 92
C.S.504 H.W. #2Due: September 24, 1992(6 points) Consider the following procedure void InsertionSort(int A, int n) {int i, j,temp; for (i=1; i&lt;n; i+) { /* A[0.i-1] already sorted */ temp=A[i]; for (j=i-1; j&gt;=0 &amp; temp&lt;A[j]; j-) A[j+1]=A[j]; /*&lt;-*/ A
Uni. Worcester - CS - 504
C.S.504 H.W. #3Due: October 21, 1993 1. (5 points) We define an Isaac tree In recursively by: -I0 consists of a single node, -the Isaac tree In, n1, consists of two Isaac trees In-1 such that the root of one is the rightmost child of the root of the
Uni. Worcester - CS - 93
C.S.504 H.W. #3Due: October 21, 1993 1. (5 points) We define an Isaac tree In recursively by: -I0 consists of a single node, -the Isaac tree In, n1, consists of two Isaac trees In-1 such that the root of one is the rightmost child of the root of the
Uni. Worcester - CS - 504
C.S.504 H.W. #10Due: Tuesday, December 15, 1992 1. (8 points) One way to estimate the size of a set X ={ x 1 ,., x n } is to sample the elements of X from a uniform distribution with replacement until an element is sampled twice. The number of eleme
Uni. Worcester - CS - 92
C.S.504 H.W. #10Due: Tuesday, December 15, 1992 1. (8 points) One way to estimate the size of a set X ={ x 1 ,., x n } is to sample the elements of X from a uniform distribution with replacement until an element is sampled twice. The number of eleme
Uni. Worcester - CS - 504
C.S.504SOLUTION FOR H.W. #5 1. (A) If we restrict the composition to have one part (k=1), then the GF is z + + z r . For 1 an arbitrary number of parts (no restrictions on k), the GF is . 1- z - - z r 1 (B) fn ,2 = z n F2 ( z) = z n . Recognizing, y
Uni. Worcester - CS - 98
C.S.504SOLUTION FOR H.W. #5 1. (A) If we restrict the composition to have one part (k=1), then the GF is z + + z r . For 1 an arbitrary number of parts (no restrictions on k), the GF is . 1- z - - z r 1 (B) fn ,2 = z n F2 ( z) = z n . Recognizing, y
Uni. Worcester - CS - 504
C.S.504H.W. #2Due: February 11/12, 19981. (6 points) Find a closed form solution for the linear first-order nonhomogeneous recurrence with nonconstant coefficients 0,if n = 0 xn = n + 3 x + n + 3,if n &gt; 0 n +1 n -1 2 The first three terms are
Uni. Worcester - CS - 98
C.S.504H.W. #2Due: February 11/12, 19981. (6 points) Find a closed form solution for the linear first-order nonhomogeneous recurrence with nonconstant coefficients 0,if n = 0 xn = n + 3 x + n + 3,if n &gt; 0 n +1 n -1 2 The first three terms are
Uni. Worcester - CS - 504
C.S.504 H.W. #3Due: October 1,1991 1.(2 points) In Section 3.1.1 of our text, what is E[An ] when Pr{An =i } = if 1i n -1 then (1/2)i else if i =n then (1/2)n -1. 2. (3 points) Prove that0 k &lt;n 0k &lt;n for n 0. 3. (2 points) Suppose a program has
Uni. Worcester - CS - 91
C.S.504 H.W. #3Due: October 1,1991 1.(2 points) In Section 3.1.1 of our text, what is E[An ] when Pr{An =i } = if 1i n -1 then (1/2)i else if i =n then (1/2)n -1. 2. (3 points) Prove that0 k &lt;n 0k &lt;n for n 0. 3. (2 points) Suppose a program has