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problems The for homework#2: 2.6, 2.11 date: Due Monday, 09/08/2008
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USC >> MATH >> 218 (Spring, 2008)
Quiz 1 Last Name First Name Instructions: For this quiz, you may use your own textbook, your own notes, and your own calculator. You may not use a computer or anyone elses notes, calculator, or textbook. [based on Mathematics 218 Final, Spring 20...
NJIT >> PHYS >> 121 (Fall, 2003)
PROBLEM 121P02- 1P: What must be the distance between point charge q1 = 26.0 C and point charge q2 = -47.0 C for the electrostatic force between them to have a magnitude of 5.70 N? Fall 2008 ...
NJIT >> PHYS >> 121 (Fall, 2003)
PROBLEM 121P02 - 2P*: A point charge of +3.00 x 10-6 C is 12.0 cm distant from a second point charge of -1.50 x 10-6 C. Calculate the magnitude of the force on each charge. Fall 2008 ...
NJIT >> PHYS >> 121 (Fall, 2003)
PROBLEM 121P02 - 5P: In the figure , what are the (a) horizontal and (b) vertical components of the net electrostatic force on the charged particle in the lower left corner of the square if q = 1.0x10-7 C and a = 5.0 cm? Fall 2008 ...
NJIT >> PHYS >> 121 (Fall, 2003)
PROBLEM 121P02 - 10P*: Two fixed particles, of charges q1 = +1.0 mC and q2 = -3.0 mC, are 10 cm apart. How far from each should a third charge be located so that no net electrostatic force acts on it? Fall 2008 ...
NJIT >> PHYS >> 121 (Fall, 2003)
PROBLEM 121P02 - 9P: Two free particles (that is, free to move) with charges +q and +4q are a distance L apart. A third charge is placed so that the entire system is in equilibrium. (a) Find the location, magnitude, and sign of the third charge. (b) ...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 1 I. The Basics on Sets and Real Functions 1. Vectors (a) Denote a = (1,1) in the following X-Y space. y x (b) Denote 2 a in the above diagram. Page 1 of 6 ECON 117 Mathematical Economics (c) a ...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 4 Continuous Random Variable Suppose X is a continuous r.v., then P(X = x ) = 0 for < x < . P(a X b ) is the area under the curve of probability density function (p.d.f.) within the interval (a, b). P ( ...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 2 Terminology Experiment: obtaining one observation Sample Space (S): the set of all possible outcomes of an experiment Simple Event: an element of the sample space Event: a subset of the sample space Gener...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 3 Two Independent Variables Let X and Y be two independent discrete random variables, while a is a constant. created by Andrew Yam Var(aX ) = a2Var(X ) E(aX ) = aE(X ) Var(X Y ) = Var( X ) + Var(Y ) E(X ...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 1 Types of Data Qualitative/Categorical data a) Nominal Data - no natural order between the categories, e.g. gender b) Ordinal Data - order exists, e.g. size (S,M,L) Quantitative/Numerical data a) Discrete da...
HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> ISMT >> 111 (Fall, 2008)
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HKUST >> ISMT >> 111 (Fall, 2008)
Topic 8c Linear Regression Testing linearity, CI for mean response, PI for response variable 8.6 Testing of linearity H 0 : 1 = 0 H a : 1 0 Test statistic b t = 1 Sb1 , where S b1 = s S XX 8.7 Confidence Interval for the mean response [y t ...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 8b Linear Regression Measures of Variation 8.3 Assumptions of Linear Regession Model: Linearity (X and mean value of Y have linear relationship) Indepedence of Error (1, 2are independent) Normality of Error (1, 2are normal) Equal variance (...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 8A Linear Regression Basic Concepts 8.1 Linear Regression Model , prediction equation yi=0 + 1 xi + i Y | X = 0 + 1X i Y =b +b X i i i 0 1 i 8.2 Estimation of parameters: S XY = (X i X )(Yi Y ) = X iYi X Y i i n S XX = ( X ...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 7A Hypothesis Testing One sample case 7.1 7.2 7.3 7.4 Errors in Making Decisions Type I Error Reject a true null hypothesis, The probability of Type I Error is Type II Error Fail to reject a false null hypothesis, The probability of Type II...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 7B Hypothesis Testing Two samples case 7.5 7.6 7.7 7.8 Testing of proportions difference H 0 : p1 p2 = 0 H a : p1 p 2 0 case 1 n p +n p p= 1 1 2 2 n1 + n2 Z= p1 p2 p (1 p ) p (1 p ) + n1 n2 case 2: H 0 : p1 p2 = d H a : p1 ...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 6B Confidence Interval Two sample cases 6.4 Confidence Interval for Difference of Population mean Point Estimate (Critical Value)(Standard Error) known, normal population or large sample 1 , 2 X1 X 2 z / 2 12 n1 + 2 2 n2 1 = 2 unk...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 6A Confidence Interval One sample case 6.0 Basic Concepts: Confidence level, Sampling error, Standard Error, Controlling of Sampling Error, 6.1 Confidence Interval for Population mean 6.2 Confidence Interval for population proportion p Z/2...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 5B Sampling Distribution 5.5 Sample Proportion v.s. Population proportion: 5.6 Distribution of sample proportion: Approximated by a normal distribution if p = p np 5, n(1 p) 5 p = p(1 p) n 5.7 Distribution of difference between two samp...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 4 Distribution of Continuous Random Variable 4.1 Uniform distribution: If c and d are numbers on the real line (c < d), the probability curve describing the uniform distribution on [c, d] is f ( x )= 1 d c for c x d The probability that x is...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 5A Sampling Distribution 5.1 Sampling variation: Since the sample is a random subset of the population, the sample mean and population mean will not be the same. Different samples will have different sample means. Sample mean is random variable...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 3 Discrete Probability Distributions Part B 3.7 Combination: the number of ways to choose r objects from a group of n objects without regard to order. n Cr = n! r! (n r)! 3.8 Binomial Distribution 3.8.1 The setup for Binomial Distribution ...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 5 Sampling Distribution of Sample Statistics The sampling distribution of a statistic is the probability distribution of all the possible values of this statistic which are computed from random samples with sam...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 6 Confidence Interval Estimation for Population Mean, For large sample size n, if is unknown, use s instead For small sample size with normal population and known A (1 )100% Confidence Interval for is c...
HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> ISMT >> 111 (Fall, 2008)
Topic 3 Discrete Probability Distributions Part A 3.1 A random variable is a numerical description of the outcome of an experiment. A random variable can be classified as being either discrete or continuous depending on the numerical values it assume...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 L1 BUSINESS STATISTICS TUTORIAL 7 Definition Null Hypothesis ( H 0 ) created by Andrew Yam - a hypothesis we assume to be TRUE, e.g. H 0 : = 0 - assumption we wish to test Alternative Hypothesis ( H a or H 1 ) - a hypothesis we wish to...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 2 Probability 2.1 Probability A probability is a measure of the likelihood that an event in the future will happen. Denote event by A , the probability of the event A by P(A) , 0 P ( A) 1 , when P(A)=0 , we say A is an impossible event, P(A)=...
HKUST >> MATH >> 106 (Spring, 2008)
4 3 z 2 1 0 -2 -1 -2 -1 0 0 y 1 22 1 x ...
HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> MATH >> 106 (Spring, 2008)
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HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 13 3. Optimization with inequality constraints 3.1 The basic idea Q) y = f ( x ), f : x is convex in x Let x solve min f ( x ) Find the F.O.C. for x x1 Q) Let x solve: min f ( x ) x s.t. ...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 11 2. 2.1 f: Ex) Optimization with equality constraints One constraint case n n f , g : continuous & differentiable g: b [OP: Original Program] x1 , x2 , xn max f ( x1 , x2 , , xn ) s.t. g (...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 12 Q) Solve the following program max x1 x2 x1, x2 s.t. x1 + x2 = 1 Page 1 of 11 ECON 117 Mathematical Economics Spring 2008 2.2 Ex) Utility Maximization ( x1 , x2 ) . There are two goods...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 10 1.2 Multiple variable case: y = f (x), f : n Def) f ( x) has a local max at x 0 if f (x0 ) > f (x0 + h x) x 0 n , small h 0 Theorem) f ( x) has a local max at x 0 if f f f Df (x...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 III. Mathematical Programming 1. Maximization and Minimization without Constraint 1.1 One variable case: f : ( f is continuous and continuously differentiable) Q) Find all local max points and a global ...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 9 4.2 Properties of eigenvalues and eigenvectors Let A be symmetric ( nn ) (1) Eigenvalues are real Page 1 of 8 ECON 117 Mathematical Economics Spring 2008 (2) The two eigenvectors x i , x j ...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 8 (Half) 3.6 Cramers Rule Q) x + y + z = 4 x z =1 x y =1 Find x, y, z by using the Cramers Rule. Page 1 of 6 ECON 117 Mathematical Economics Spring 2008 4. Eigenvalue Problems 4.1 Eigenva...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 7 3.4 The Properties of Determinant 1. A = A 2. If B is obtained from A by interchanging any two rows (or columns) of A , then B = A 1 2 1 Ex) A = 2 1 2 0 1 1 2 1 2 B = 1 2 1 0 1 1 ...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 5 Def) Space We call A that is spanned by k (< n) independent n-dimensional vectors has dimension k . A is a subspace of n . 2 Q) Find the space that is spanned by x = 1 2 . Q) Find the sp...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 6 3.1 The Rank of a Matrix A , ( A ) , is the maximum number of rows (or columns) in A that are Def) The rank of matrix linearly independent. Theorem) A , the maximum number of linearly indepe...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 4 Def) Linear combination (broad meaning) Let {xi } = {x1 , , x m } where If y = xi n . x , i =1 i i i m , then we call y is a linear combination of {x i } or y is linearly dependent o...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 3 3.5 Total Derivatives and Partial Derivatives , i.e., f : 2 Def) For z = f ( x, y ) , x, y , z , z (partial derivative of z with respect to x ) is the marginal rate of change of z with resp...
HKUST >> ECON >> 324 (Spring, 2008)
ECON 117 Mathematical Economics Spring 2008 Week 2 3.2 Differentiation of a Real Function Y Def) Real function: f : X Y is a real function if Q) Draw the following real function: f = 2 x2 8x + 2 y = 2x + 2 (1) (2) Def) Differentiation of a...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 0 Basic Concepts of Statistics 0.1 Population :The complete collection of Individuals that are of interest in the study. Sample: A portion, or part, of the population of interest. 0.2 Random sample: A random sample is a sample selected from a ...
HKUST >> ISMT >> 111 (Fall, 2008)
Topic 1 Descriptive Statistics 1.1 Frequency Distribution: a grouping of data into different categories showing the number of observations in each class. Four steps: Decide on the number of classes. Determine the class interval or width. Set the i...
HKUST >> ISMT >> 111 (Fall, 2008)
ISMT 111 BUSINESS STATISTICS TUTORIAL 8 Simple Linear Regression Model Aim: Use X (Independent Variable, or Predictor Variable) to predict Y (Dependent Variable, or Response Variable) after drawing n pairs of observations, (x1 , y 1 ),., (xn , y n...
Cornell >> CHEM >> 2080 (Spring, 2008)
Thermodynamics of Galvanic Cells by Caroline Tabler Lab Instructor: Bob Neuman April 28, 2008 Results and Discussion: For this experiment, a Cu/Ag galvanic cell was created and its temperature measured. A voltimeter was then used to find the cell p...
Cornell >> CHEM >> 2080 (Spring, 2008)
Preparation of a Buffer Solution by Caroline Tabler Lab Instructor: Bob Neuman March 25, 2008 Results and Discussion: For this experiment, a 400 ml buffer was to be prepared using one of three acids with its conjugate base. The buffer needed to ha...
Cornell >> CHEM >> 2080 (Spring, 2008)
Chemical Kinetics: Iodine Clock Reaction by Caroline Tabler Lab Instructor: Bob Neuman February 26, 2008 Results and Discussion: In order to determine the concentration and temperature dependence of the reaction rate between the peroxydisulfate io...
Cornell >> CHEM >> 2080 (Spring, 2008)
pKa of an Unknown Acid-Base Indicator by Caroline Tabler Lab Instructor: Bob Neuman April 13, 2008 Results and Discussion: For this experiment, several tests were made on the bromothymol blue indicator in order to measure its pKa. First, the indic...
Cornell >> CHEM >> 2080 (Spring, 2008)
Hot- and Coldpacks by Caroline Tabler Lab Instructor: Bob Neuman February 12, 2008 Results and Discussion: The heat of solution for one of seven salts (ammonium nitrate) was determined by first calculating the heat capacity of a calorimeter (Table...
Cornell >> CHEM >> 2080 (Spring, 2008)
Identification of an Unknown Weak Acid by Caroline Tabler Lab Instructor: Bob Neuman April 22, 2008 Results and Discussion: For this experiment, the molar mass and the pKa of an unknown acid had to be experimentally calculated. This was done by usi...
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