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### lecture6

Course: MA 2612, Fall 2008
School: Uni. Worcester
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Word Count: 1100

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6 Lecture Administrative MA 2612 - Applied Statistics II D 2004 1. 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. Today 1. Quick Review and Example of Comparing two population means (PNC 6.8) 2. Comparing two population proportions (PNC 6.9) 3. Power (PNC 6.11) Comparing the means of two independent populations...

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6 Lecture Administrative MA 2612 - Applied Statistics II D 2004 1. 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. Today 1. Quick Review and Example of Comparing two population means (PNC 6.8) 2. Comparing two population proportions (PNC 6.9) 3. Power (PNC 6.11) Comparing the means of two independent populations Statistical model Y1,1 , Y1,2 , . . . , Y1,n1 N(1 , 2 ) 1 Y2,1 , Y2,2 , . . . , Y2,n2 N(2 , 2 ) 2 Statistical hypothesis H0 : 1 2 = 0 Ha : 1 2 < 0 H0 : 1 2 = 0 Ha : 1 2 6= 0 H0 : 1 2 = 0 Ha : 1 2 > 0 For most scientic hypotheses 0 = 0. This corresponds to null hypothesis of no dierence. Test statistic (Y1 Y2 ) 0 s(Y1 Y2 ) is the standard error of (Y1 Y2 ). t = where s(Y1 Y2 ) The standard error is another name for the standard deviation of the sampling distribution. Thus, s(Y1 Y2 ) will depend on any assumptions we make in the statistical model for Y1,1 , . . . , Y1,n1 and Y2,1 , . . . , Y2,n2 . clusions we make about H0 . This will also aect the sampling distribution of t as well as any con- 2 Computation of s(Y1 Y2 ) and the sampling distribution of t Recall the statistical model: Y1,1 , Y1,2 , . . . , Y1,n1 N(1 , 2 ) 1 Y2,1 , Y2,2 , . . . , Y2,n2 N(2 , 2 ) 2 s n 1. With no further assumptions s(Y1 Y2 ) = where 2 S1 2 2 S1 S2 + n1 n2 1 1 X 2 Y1,i Y1 = n1 1 i=1 2 1 X 2 = Y2,i Y2 n2 1 i=1 n 2 S2 (Y1 Y2 ) 0 s(Y1 Y2 ) follows a t distribution where is the largest integer less than or equal to 2 2 2 S1 S2 n1 + n2 t = 2 (S1 /n1 ) n1 1 2 2 + (Sn2/n2 ) 1 2 2 and 2. However, if it is reasonable to assume that 2 = 2 then 1 2 s(Y1 Y2 ) = where 2 Sp s 2 Sp 1 1 + n1 n2 2 2 (n1 1)S1 + (n2 1)S2 = n1 + n2 2 and t follows a tn1 +n2 2 distribution. 3 Example 1: Researchers wish to discover whether the enzyme Glucose 6Phosphate Dehydrogenase (G6PD) is involved in the disease Rheumatoid Arthritis (RA). To study this, they randomly select 14 adults with RA and 17 adults as controls. The control subjects do not have RA or any other arthritic condition. In each adult, the level of G6PD is measured with a blood test. Test results are given in units/g Hgb (Hgb = hemoglobin). The data are summarized in the following table: RA Control mean 17.80 12.30 s.d. 3.20 2.84 Scientic hypothesis Statistical model Statistical hypothesis 4 If we assume 1 = 2 Compute s(Y1 Y2 ) . If we cant assume 1 = 2 Compute t . Find the degrees of freedom. Find the critical value and draw a conclusion about H0 . 5 Comparing two population proportions (PNC 6.9) Examples of where this might be interesting: 1. Compare proportion of subjects in treatment and control group that experience side-eects. 2. Compare cancer rates for people who live close/far from power lines. 3. Compare proportion defective of items from two dierent production lines. Statistical model We observe Y1 , the number out of the n1 units in our sample from the rst population that have some specied property. We observe Y2 , the number out of the n2 units in our sample from the second population that have some specied property. The proportion of units in the rst population that have the specied property is some (unknown) value p1 between 0 and 1. property is some (unknown) value p2 between 0 and 1. The proportion of units in the second population that have the specied Therefore, Y1 bin(n1 , p1 ) and Y2 bin(n2 , p2 ). 6 Statistical hypothesis H0 : p1 p2 = 0 Ha : p1 p2 < 0 H0 : p1 p2 = 0 Ha : p1 p2 6= 0 H0 : p1 p2 = 0 Ha : p1 p2 > 0 For most scientic hypotheses 0 = 0. This corresponds to null hypothesis of no dierence. Test statistic z = q (1 p2 ) 0 p p1 (12 ) p n1 p + p2 (12 ) n2 where p1 = Y1 /n1 and p2 = Y2 /n2 . Sampling distribution When Y1 , Y2 , (n1 Y1 ) and (n2 Y2 ) are all greater than 5, z has approximately a N(0,1) distribution. We use the sampling distribution to make conclusions about H0 in the usual way(s). 7 Example: Two drugs, zidovudine and didanosine, were tested for their effectiveness in preventing progression of HIV in children. In a double-blind clinical trial 276 children with HIV were given zidovudine, 281 were given didanosine. Using the data in the table below, test the scientic hypothesis that zidovudine is less eective than didanosine. Zidovudine Didanosine Total Died 17 7 24 Survived 259 274 533 Total 276 281 557 8 Power (PNC 6.11) Whenever we conduct a hypothesis test, there is the possibility that we will reach the wrong conclusion. Obviou...

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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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Uni. Worcester - OS - 390
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
Uni. Worcester - CS - 07
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
Uni. Worcester - CS - 504
C.S.504H.W. #5Due: April 8/9, 1998Read SECTIONS 7.1, 7.2, 7.3, 7.5 Do EXERCISEs 7.1, 7.2, 7.3, 7.4, 5.15 from GKP. Do not submit your solutions, but check them with the answers from the back of the text. 1. (4 points) Define an r-composition of n
Uni. Worcester - CS - 98
C.S.504H.W. #5Due: April 8/9, 1998Read SECTIONS 7.1, 7.2, 7.3, 7.5 Do EXERCISEs 7.1, 7.2, 7.3, 7.4, 5.15 from GKP. Do not submit your solutions, but check them with the answers from the back of the text. 1. (4 points) Define an r-composition of n
Uni. Worcester - CS - 504
C.S.504 H.W. #3Due: October 1, 19921. (6 points) You should compare three techniques for evaluating the 1 integral 4 1- x 2 d x . For each of the techniques, you should test the 0 rate of convergence by comparing the influence of n upon the accu
Uni. Worcester - CS - 92
C.S.504 H.W. #3Due: October 1, 19921. (6 points) You should compare three techniques for evaluating the 1 integral 4 1- x 2 d x . For each of the techniques, you should test the 0 rate of convergence by comparing the influence of n upon the accu