tree parity P 7. Partial Order Dimension NP-complete 8. Precedence constrained 3-processor schedu open 9. Linear Programming P **10**. Total Unimodularity P 11. Composite number open (P) 12. Minimum length triangulation open Integer Programming 2011 NP-complete in the/strongly NP-complete) if is in NP and there exists a **polynomial** function p **for** which p is NP-complete. It is the problem **class** **for** which there might not exist even pseudo-**polynomial** time algorithms. If a problem is not a number problem and /

O(n) (**polynomial** in the input size). **Class** P and **Class** NP PNP If we can design a **polynomial** time algorithm **for** problem A, then A is in P. However, if we have not been able to design a **polynomial** time algorithm **for** A, then two possibilities: No **polynomial** time algorithm **for** A exists /: (Directly connect two nodes in R based on the visited order of the traversal.) 7 1 2 3 4 5 6 8 9 **10** From triangle inequality, w(ST)≤w(traversal) ≤2OPT(I). ..........(2) Inequality(2) says that the cost of ST is no more than/

is an input **for** R; x’ is an input **for** S (R S) x x’ Algorithm **for** S T Yes or No answer Algorithm **for** R P NPC NP NP: Non-deterministic **Polynomial** P: **Polynomial** ? NP: Non-deterministic **Polynomial** P: **Polynomial** NPC: Non-deterministic **Polynomial** Complete NP : the **class** of decision problem /5) x(2)=7 v i≠2 v FAILURE (6) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (**10**) The NP searching algorithm reduces to SAT Satisfiable at the following assignment : i=1 satisfying (1) i≠2 satisfying (2), (4) and (6) /

of other “completeness” theories; 9.Helps explain why verifying proofs seems to be easier than constructing them; **10**.Illuminates the fundamental nature of algorithms and computation; NP Completeness Benefits 11.Gave rise to new and novel / “Karp” reduction (transformation) **Polynomial**-time Turing reduction: solves A by **polynomially**-many calls to “oracle” **for** B. “Cook” reduction B A Open: do **polynomial**-time-bounded many-one and Turing reductions yield the same complexity **classes**? (NP, co-NP, NP-complete/

lecture, during the last **10** minutes of **class**. CLOSE Please YOUR LAPTOPS, and get out your note-taking materials. and turn off and put away your cell phones, and get out your note-taking materials. Section 6.1 Introduction to Factoring **Polynomials**: Greatest Common Factor (GCF/ that on quizzes and tests!) How would you check this? Example Factor x3 + 4x + x2 + 4 by grouping. SOLUTION: First, look **for** a GCF. (Always do this first!) There isn’t one, so now separate the four terms into two groups of two: x3 + 4x /

**polynomial** p(x) such that, **for** any w L, there is a certificate c with |c| p(|w|) All problems from the previous slide have a **polynomial** time verifier Computability and Complexity 13-5 The **Class** NP Definition The **class** of languages that have **polynomial** time verifiers is called NP Definition The **class** of languages that have **polynomial**/ **for** whites in a checker game on n n board? Checkers Computability and Complexity 13-**10** Non-deterministic Machines We can get an alternative definition of the **class** /

s t CS 311 Fall 2008 9 Hamiltonian paths HAMPATH = { | ∃ Hamiltonian path from s to t} s t CS 311 Fall 2008 **10** Checking **for** Hamiltonian paths Brute force method E = "On input : 1.Generate all orderings, p 1, p 2,..., p n, of the nodes of / **for** some string c} –A **polynomial** time verifier runs in **polynomial** time in the length of w –A language A is **polynomially** verifiable if it has a **polynomial** time verifier CS 311 Fall 2008 15 The **class** NP Definition 7.19: NP is the **class** of languages that have **polynomial** /

solved in **polynomial** time by using a reasonable sequential model of computation if and only if it can be solved in **polynomial** time by a Turing machine. 2 The complexity **class** P P := the **class** of decision problems (languages) decided by a Turing machine so that **for** some **polynomial** p and all/resulting in languages L 1 and L 2 we have L 1 2 P iff L 2 2 P. **10** Terminology When we say, “Problem X can be solved in **polynomial** time”, we mean L binary X 2 P, i.e., we assume binary representation of integers of input/

“an O(T(n)) running-time algorithm.” 3 The **class** P uIf a DTM M is T(n)-time bounded **for** some **polynomial** T(n), then we say M is **polynomial**-time (“polytime ”) bounded. uAnd L(M) is said to be in the **class** P. uImportant point: when we talk of P, it doesn/uEach table needs only space O(m) to represent all the positive and negative differences we could achieve. uEach table can be constructed in time O(n). **10** Knapsack – (4) uSince n < m, we can build the final table in O(m 2 ) time. uFrom that table, we can see /

Kharitonov **Polynomials**. **For** continuous-time interval **polynomials** we have an even simpler method available: An interval **polynomial** of invariant degree (with real coefficients) is known to be stable if and only if just its four “extreme” **polynomials** (called the Kharitonov **polynomials**) are stable. **For** the interval **polynomial** of Example 3 the Kharitonov **polynomials** are computed by [stability,K1,K2,K3,K4] = kharit(pminus,pplus) Polytopes of **polynomials** A more general **class**/

of computation if and only if it can be solved in **polynomial** time by a Turing machine. P is the **class** of these decision problems 2 Search Problems: NP L is in NP iff there is a language L’ in P and a **polynomial** p so that: 3 Intuition The y-strings are the /-hard is called NP-complete. NPC := the **class** of NP-complete problems. Proposition: L 2 NPC ) [L 2 P iff P=NP]. **10** L is in NP means: There is a language L’ in P and a **polynomial** p so that L 1 · L 2 means: **For** some **polynomial** time computable map r : 8 x: x /

: Chapter **10** Sections 1, 2, 3, and 4 only. NP-Complete Problems Problems in Computer Science are classified into –Tractable: There exists a **polynomial** time algorithm that solves the problem O(n k ) –Intractable: Unlikely **for** a **polynomial** time algorithm/algorithm? The **Class** NP Definition: The **class** of decision problems NP consists of those decision problems **for** which there exists a nondeterministic algorithm that runs in **polynomial** time Example Show that the coloring problem belongs to the **class** of NP /

can solve a problem of size k (i.e., compute 2k operations) in a hour/week/month. If the new computer is **10** times faster than ours, then the new computer can solve the problem of size k+3. The improvement is very little. Story All/ the first NP-complete problem. S. A. Cook in 1971 Won Turing prize **for** this work. Significance: If Satisfiability problem can be solved in **polynomial** time, then ALL problems in **class** NP can be solved in **polynomial** time. If you want to solve PNP, then you should work on NPC/

program... while (!optimal) { a = generator(i); if (is_optimal(a)) optimal = true; } ....and its been running **for** **10** years? It could still be **polynomial** (imagine O(n10) **for** n = 1,000) and halt(p,i) might be just about to finish and return true. But you will never know/exist according to its own rules. **For** the halting problem, you can also agree that the wtf program can’t exist according to its own rules. But the wtf program does exist, feel free to implement it after **class**. The real show stopper is that/

is the first NP-complete problem. –S. A. Cook in 1971 –Won Turing prize **for** this work. Significance: –If Satisfiability problem can be solved in **polynomial** time, then ALL problems in **class** NP can be solved in **polynomial** time. –If you want to solve P NP, then you should work on NPC/: (Directly connect two nodes in R based on the visited order of the traversal.) 7 1 2 3 4 5 6 8 9 **10** From triangle inequality, w(ST)≤w(traversal) ≤2OPT(I)...........(2) 37 Inequality(2) says that the cost of the spanning tree ST/

a Partner. Use the guiding questions to focus your work. Time: **10** minutes 4 Fraction Considerations from CTS Need experience before alogrithms Use of /when multiplying) of two **polynomials** will always be. 18 18 I can define “closure” as it applies to **polynomials**. I can tell what it means **for** **polynomials** to be closed under/… Anticipation of success (11G projects) Zeros Flexible due dates Retesting Confidence and in-**class** behavior (Cody) Routine/mental ease (Tianna & Morgan) Encouraging risk-taking (11G)/

following function. c(x) = 2x 4 – 4x 3 – 7x 2 – 13x – **10** 4 2 –4 –7 –13 –**10** 8 16 36 92 2 4 9 23 82 Time **for** **Class** work Time **for** **Class** work Evaluate each function at the given value. The Factor Theorem The binomial (x – a) is a factor of the **polynomial** f(x) if and only if f(a) = 0. The Factor Theorem/

**for** all values of the variable (x+1) 2 = x 2 +2x+1 Equation holds true **for** all real x _H001 **Polynomial** identity If a **polynomial** equation of degree n satisfies **for**/Find the roots of the equation x 2 -10x+22=0 Here a=1, b=-**10**, c=22 Apply the general solution form _H003 Illustrative Problem Solution: Ans: Roots are /quadratic equation. Solution: As irrational roots are not in conjugate form. Co-efficient are not rational. **Class** Exercise6 If the sum of the roots of is zero, then prove that product of the roots is/

similar in nature to thermodynamic entropy, which you may have discussed in a physics or chemistry **class** The more entropy, the less we can compress, and the less entropy the more we/way we learn to multiply in school –However, we are using base 2 rather than base **10** –See example on board Run-time of algorithm? How to implement? –We need to /your CS research WWhat should you do? T ry to find an efficient solution (**polynomial**) **for** the problem I f unsuccessful (or if you think it likely that it is not possible/

**for** running nondeterministic algorithms do not exist and they would never exist in reality. (They can only be made by allowing unbounded parallelism in computation.) (Quantum computation may be an approximation.) Nondeterministic algorithms are useful only because they will help us define a **class** of problems: NP problems 5 NP (Nondeterministic **Polynomial**/ we are addressing the lower bound of a problem and D ∝ O **10** To express Nondeterministic Algorithm by three special commands Choice(S) : arbitrarily chooses/

BCH Code a larger **class** of powerful random error-correcting cyclic codes a remarkable generalization of the Hamming codes **for** multiple-error correction Def : Let q and m be given and let α be any element of GF(q m ) of order n. Then **for** any positive integer / Peterson-Gorenstein-Zierler decoder. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU23 Eg: BCH(15,5) triple-error-correcting code. g(X) = X **10** + X 8 + X 5 + X 4 + X 2 + X + 1 received code **polynomial** V(X) = X 7 + X 2 In GF(2 4 ) S 1 = α 7 + α 2 = α 12 S 2/

-Wesley Section 5.2 Addition and Subtraction of **Polynomials** Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives Monomials and **Polynomials** Addition of **Polynomials** Subtraction of **Polynomials** Evaluating **Polynomial** Expressions Monomials and **Polynomials** A monomial is a number, a variable, or/5 + 2x− 5 2x – 5 −5x 2 + 0 + **10** The quotient is Try Q: 41 pg 349 Due **for** this week… Homework 1 (on MyMathLab – via the Materials Link) The fifth night after **class** at 11:59pm. Read Chapter 6.1-6.4 Do /

into bit strings The corresponding language contains all the strings of “YES” instances 5 **Class** P V.S. **Class** NP P stands **for** what? **Polynomial** ! **Class** P: Problems solvable in deterministic **polynomial** time What about NP? No Problem? Not **Polynomial** (i.e. **polynomial** time unsolvable)? Nondeterministic **Polynomial** ! **Class** NP: Problems solvable in nondeterministic **polynomial** time 6 Deterministic/Nondeterministic **Polynomial** time? Where do these terms come from? They’re based on different computation/

Vanishing **Polynomials** **for** Equivalence Verification of Fixed-Size Arithmetic Datapaths GIEE, NTU ALCom Lab Presenter: 陳炳元 Outline Introduction Modeling Vanishing **Polynomials** Algorithm Conclusions and Future Work Outline Introduction Modeling Vanishing **Polynomials** Algorithm / c a,b,c R Introduction to Rings Example: 我們在 上定義一個 equivalence relation, 若 (a,b) R a b mod n 對應之 equivalence **class** set n ={[0],[1],…,[n-1]}. 在 n 上定義兩個 binary operation +, by [a]+[b]=[a+b mod n] 與 [a] [b]=[ab mod n/

the size of an instance is n. With a problem , we associate the co- problem co- The set of positive instances **for** is denoted I+( ,) cdlh 2007 cdlh, Barcelona, July 2007 108 **10** Complexity **Classes** P : deterministic **polynomial** time NP : non deterministic **polynomial** time cdlh 2007 cdlh, Barcelona, July 2007 109 Turing machines Only one tape Alphabet of 2 symbols An input of length n/

TSP NP. Complexity **Classes** NP and co-NP ․ Is **class** NP closed under complement? ․ **Class** co-NP: **class** of problems whose complement problems/**polynomial** time. Significance of Reduction ․ Significance of L 1 P L 2 : **polynomial**-time algorithm **for** L 2 **polynomial**-time algorithm **for** L 1 (L 2 P L 1 P). **polynomial**-time algorithm **for** L 1 **polynomial**- time algorithm **for**/with the conjunction ( ) of clauses describing the operation of each gate, e.g., = x **10** (x 4 x 3 ) (x 5 (x 1 x 2 )) /

MATLAB-Tutorial 2 **Class** ECES-304 Presented by : Shubham Bhat % Nise, N.S. % Control /. "tf" needs two MATLAB-vectors, one containing the coefficients of the numerator **polynomial** - taken in descending orders - and one **for** the denominator **polynomial** of the transfer function. As an example we will create the transfer function H0/=zpk(Gtf); [r,p,k]=residue(numg,deng); Homework Problems % **10**. Write the transfer function in phase variable form ‘a’ Num=100; Den=[1 20 **10** 7 100]; G=tf(num,den) [Acc,Bcc,Ccc,Dcc]=tf2ss(/

**classes** Def: A Decision problem **for** a language L {0,1} * is to decide whether a given string x belongs to the language L. Def: P is the **class** of languages (decision problems) that can be recognized by a deterministic **polynomial** time Turing machine. Def: NP is the **class**/, finding a solution can only be harder than just deciding whether it exists. Note: NP = { L(R) : R is **polynomially**-verifiable } **10** Problem: 3-coloring graphs Instance: An undirected graph G=(V,E) Corresponding relation: R 3COL = { (G, ) :/

similar in nature to thermodynamic entropy, which you may have discussed in a physics or chemistry **class** The more entropy, the less we can compress, and the less entropy the more we/way we learn to multiply in school –However, we are using base 2 rather than base **10** –See example on board Run-time of algorithm? How to implement? –We need to /your CS research WWhat should you do? T ry to find an efficient solution (**polynomial**) **for** the problem I f unsuccessful (or if you think it likely that it is not possible/

has power of todays supercomputers... And each processor works **for** the life of the universe... Will not help solve 1,000 city TSP problem via brute force. 1000! >> **10**^1000 >> **10**^{79+13+17} Exponential growth (Source from Wikipedia) Sessa/ in **polynomial** time. (Tractable) **For** some problems, we still cannot find **polynomial** time solutions. (Intractable) Decision problems Turing machine Where are we? Introduction to intractability **Class** P and NP **Class** NPC **Class** P Definition: ClassP A problem π is in **class** P /

minimum number (G) of colors needed to color a graph in such a way that no two adjacent vertices have the same color. **10** Traveling Salesman Given a finite set C={c 1,...,c m } of cities, a distance function d(c i, c j ) of nonnegative/leads to an answer of no **for** a certain guess, that it does not accept it? 18 The **Class** NP Definition: The **class** of decision problems NP consists of those decision problems **for** which there exists a nondeterministic algorithm that runs in **polynomial** time. –The running time of /

of ( + ) **10** **for** some values of and. By The Binomial Theorem:Binomial Theorem **10** +**10** 9 +45 82 +120 73 +210 64 +252 55 +210 46 +120 3 7 +45 28 +**10** 9 + **10** The trick: we can evaluate the dot product between degree **10** **polynomials** without explicitly forming them./lies on the margin, that is i ( T i + ) = 1 a training sample i with ( α i ≠0) is called a support vector The **class** of is h() = sign( T + ), substituting gives: We only need to remember the few training samples where ( α i ≠0). 47 Recovering the bias/

interesting **class** of problems whose status is unknown No **polynomial**-time algorithm has been discovered **for** an NP-Complete problem No one has been able to prove that no-**polynomial** algorithm exists **for** any of them No super-**polynomial** (e.g. exponential) low bound has been shown **for** /problem cannot be found with the same greedy strategy Greedy strategy: take in order of dollars/pound Example: 3 items weighing **10**, 20, and 30 pounds, knapsack can hold 50 pounds Suppose item 2 is worth $100. Assign values to the/

y) 2 X £ {0,1} f* = argmin f 2F P [f(x) y] 3/15 n n nn Set X n µ R n, F n **class** of functions f: X n ! {0,1}. Efficient Agnostic Learner w.h.p. n h: X n ! {0,1} n, poly(n,1/ ) samples/free **for** halfspaces Useful properties of logconcave dists: projection is logconcave, …, q(x) ¼ I (x ¸ 0) degree d=**10** q(w ¢ x) ¼ I (w ¢ x ¸ 0) degree d=**10** 14/15 Approximating I(x ¸ ) (1 dimension) Bound min deg(q) · d E [(q(x) – I (x ¸ )) 2 ] Continuous distributions: orthogonal **polynomials** Normal: Hermite **polynomials** Logconcave /

If f(w)HAMCYCLE then wHAMPATH easy to verify Complexity ©D.Moshkovitz Closeness Under Reductions Definition: A complexity **class** C is closed under reductions if, whenever L is reducible to L’ and L’C, then L is also in C. C / “black box” (procedure) that decides B, to construct a **polynomial**-time machine which decides A. **polynomial** time algorithm **for** B **polynomial** time algorithm **for** A Complexity ©D.Moshkovitz Cook reduction of HAMCYCLE to HAMPATH. **For** each edge (u,v)E, if there’s a Hamiltonian path/

Not closed under edge deletion Not closed under edge contraction 9 TRIANGLE PACKING AND H-MATCHING Structure theory and kernels **10** Triangle Packing Input:Graph G, integer k Question:Are there k vertex-disjoint triangles in G? Parameter:k NP/ a subset of what another red vertex sees Same **for** the blue vertices Structural theorems show that in such colored graphs the sizes of the color **classes** are **polynomially** related So size of the largest **class** is **polynomial** in the size of smallest **class** Hence |V| = |R| + |B|/

6 (8.9/**10**), 7 (8.6/**10**) and 8 (8.1/**10**) We’ve learned how to show some problems are undecidable –Harder (average on question 4 = 7.7) Today: reasoning about the complexity of some problems 21 April 2004CS 200 Spring 20043 Complexity **Classes** **Class** P: problems that can be solved in **polynomial** time by a deterministic TM. O (n k ) **for** some constant k/

] = Pr[(V P)(w,r) = accept], where r is a randomly selected string of length p(n). The **class** IP **10**.4.f Definition **10**.28 Say that a language A is in IP if some **polynomial** time function V and arbitrary function P exist, where **for** every function P’ and string w: 1. w A implies Pr[V P accepts w] > 2/3, and/

, precalculus, calculus) **for** graduation. These students take MATH 1530, Probability and Statistics, as their core math **class**. About 90% of/**10**.1 Radicals and Radical Functions Appendices D. An Introduction to Using a Graphing Utility D. An Introduction to Using a Graphing Utility G. Mean, Median, and Mode G. Mean, Median, and Mode New DSPM 0850 5. Exponents and **Polynomials** 5. Exponents and **Polynomials** 5.1 Exponents 5.1 Exponents 5.2 **Polynomial** Functions and Adding and Subtracting **Polynomials** 5.2 **Polynomial**/

i.e. a path which visits every vertex exactly once? HAMILTONIAN CYCLE HAMILTONIAN PATH **10** NP Two equivalent definitions: Given a solution, we can check in **polynomial** time whether it is correct. There is a non-deterministic algorithm (a magic /if it is correct in **polynomial** time. P is the **class** of problems that we can find a solution in **polynomial** time. NP (Non-deterministic **polynomial** time): A **class** of decision problems whose solutions can be “verified” in **polynomial** time. **For** each “yes” instance, there/

**Class** “P” of Decision Problems What is an efficient algorithm? **polynomial** time O(n c ) **for** some constant c non-**polynomial** time Is an O(n) algorithm efficient? How about O(n log n)? O(n 2 ) ? O(n **10** ) ? O(n log n ) ? O(2 n ) ? O(n!) ? We consider non-**polynomial** time algorithms to be inefficient. And hence a necessary condition **for**/ HAM NP The **Class** NP The **class** of sets L **for** which there exist “short” proofs of membership (of **polynomial** length) that can be “quickly” verified (in **polynomial** time). Recall: A /

e.g. side constraints –New query policies New **polynomially** elicitable valuation **classes** Using models of how costly it is to answer different queries [Hudson & S. AMEC-02] Decision-theoretic elicitation using priors Elicitors **for** markets beyond combinatorial auctions –(Combinatorial) reverse auctions & /queries with order queries Integrated as before –Computationally more expensive agents items 345678 9 2 **10** 23456 queries 1 **10** 100 1000 40 80 120 160 queries Full revelation Total cost Value cost Order cost /

13 Semantic tree In a semantic tree, each path from the root to a leaf node represents a **class** of assignments. If each leaf node is attached with a clause, then it is unsatisfiable. 88-/(6) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (**10**) 88- 24 Satisfiable at the following assignment : i=1 satisfying (1) i≠2 satisfying (2), /NP-hard: Every NP algorithm **for** problem A can be transformed in **polynomial** time to SAT [Horowitz 1998] such that SAT is satisfiable if and only if the answer **for** A is “ YES ”. /

Jan Maluszynski - HT 200710.2 Outline Lecture 9-**10** : NP-completeness (Sipser 7.4 – 7.5) 1.Motivation P vs. NP 2.NP-complete problems definition & intuition 3.3-SAT 4.**Polynomial** time reducibility 5.Cook-Levin theorem 6.A survey / Importance of time-complexity analysis **Polynomial**-time complexity preserved by realistic models of computations. P is a **class** of problems that can be realistically solved on a computer. The notion of NP-completeness based on P; relevant **for** practical computing. CS Master –/

**for** fixed k (Graph Coloring) Because it is compositional (Long Path) The natural parameter might not admit **polynomial** kernels Change the parameter instead of the **class** of inputs Alternative direction to kernels **for** restricted graph **classes** Guide the search **for** reduction/vertices on the outer face Outerplanar Deletion number Distance to planar with all vertices on the outer face **10** Vertex Cover Distance to linear forest Distance to Cograph Distance to Chordal Treewidth Chromatic Number Odd Cycle Transversal/

most c p(n) steps on input of length n, **for** some constant c and **polynomial** p. uSay L(M) is in the **class** EP. uIf M is an NTM instead, say L(M) is in the **class** NEP (nondeterministic exponential polytime ). 6 More **Class** Relationships uP NP PS EP, and at least/t p(n). wIf M accepts, it does so without repeating an ID. uThus, M’ is exponential-polytime bounded, proving L(M) is in EP. **10** Savitch’s Theorem: PS = NPS uKey Idea: a polyspace NTM has “only” c p(n) different ID’s it can enter. Implement a /

**Polynomial** P: **Polynomial** NPC: Non-deterministic **Polynomial** Complete P=NP? X = P 3 P: the **class** of problems which can be solved by a deterministic **polynomial** algorithm. NP : the **class** of decision problem which can be solved by a non-deterministic **polynomial** algorithm. NP-hard: the **class**/ as not in P No known deterministic algorithms that are **polynomially** bounded **for** many problems in NP So, “ does P = NP/ (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (**10**) 30 Satisfiable with the following assignment: i=1 satisfying (1) i≠2/

use commutative and associative properties to rearrange and regroup the factors so as to group the “a” factors and powers of **10** separately and to use rules of exponents to end up with an answer in scientific notation It is also common to round/ 1 – 55, 59 – 69, 73 – 77 MyMathLab Section 4.4 **for** practice MyMathLab Homework Quiz 4.4 is due **for** a grade on the date of our next **class** meeting Multiplying **Polynomials** To multiply **polynomials**: –Get rid of parentheses by multiplying every term of the first by every /

**for** squaring a binomial: Problem 3 Perform the indicated operation: Answer: Dividing a **Polynomial** by a **Polynomial** First write each **polynomial** /**polynomials** as being prime We will also completely factor **polynomials** by writing them as a product of prime **polynomials** Importance of Factoring If you don’t learn to factor **polynomials** you can’t pass college algebra or more advanced math **classes**/ and bottom: Divide out common factors to get: Problem **10** Reduce to lowest terms: Answer: Finding the Least Common /

EL = 0 When Hi EL = 1, Two regression lines: one **for** each Hi STR group. **Class** size reduction is estimated to have a larger effect when the percent of English learners / the sample, especially large values of income. Question #1: Investigate by considering a **polynomial** in STR(column 5) Interpreting the regression function via plots (preceding regression is labeled/H 1 : nonzero interaction t = -0.58/0.50 = -1.17 not significant at the **10**% level. H 0 : both coeffs involving STR = 0 vs. H 1 : at least /

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