Econ 6900: Statistical Problems. Instructor: Yogesh Uppal
|
|
|
- Magdalene Anthony
- 5 years ago
- Views:
Transcription
1 Econ 6900: Statistical Problems Instructor: Yogesh Uppal
2 Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of
3 Statistical Inference The sample results provide only estimates of the values of the population characteristics. With proper sampling methods,, the sample results can provide good estimates of the population characteristics. A parameter is a numerical characteristic of a population.
4 Statistical Inference The purpose of statistical inference is to obtain information about a population from information contained in a sample. A population is the set of all the elements of interest. A sample is a subset of the population.
5 Simple Random Sampling: Finite Population A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.
6 Simple Random Sampling: Finite Population Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often. In large sampling projects, computer-generated random numbers are often used to automate the sample selection process.
7 Point Estimation In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. We refer to as the point estimator of the population mean. s is the point estimator of the population standard deviation. p is the point estimator of the population proportion p.
8 Sampling Error When the epected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased. The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. Sampling error is the result of using a subset of the population (the sample), and not the entire population. Statistical methods can be used to make probability statements about the size of the sampling error.
9 The sampling errors are: Sampling Error for sample mean s for sample standard deviation p p for sample proportion
10 Air Quality Eample Let us suppose that the population of air quality data consists of 191 observations. How would you determine the following population parameters: mean, standard deviation, proportion of cities with good air quality.
11 Air Quality Eample How about picking a random sample from this population representing the air quality? We shall use SPSS to do this random sampling for us. How would you use this sample to provide point estimates of the population parameters?
12 Process of Statistical Inference Population with mean =? A simple random sample of n elements is selected from the population. The value of is used to make inferences about the value of. The sample data provide a value for the sample mean.
13 Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean. Epected Value of where: E( ( ) = = the population mean
14 Sampling Distribution of Standard Deviation of Finite Population Infinite Population ( ) n N N n 1 n A finite population is treated as being infinite if n/n <.05. ( N n )/( N 1 ) is the finite correction factor. is also referred to as the standard error of the mean.
15 The Shape of Sampling Distribution of If the shape of the distribution of in the population is normal, the shape of the sampling distribution of is normal as well. If the shape of the distribution of in the population is approimately normal, the shape of the sampling distribution of is approimately normal as well. If the shape of the population is not approimately normal then If n is small, the shape of the sampling distribution of is unpredictable. If n is large (n 30), the shape of the sampling distribution of can be assumed to be approimately normal.
16 Sampling Distribution of for the air quality eample when the population is (almost) infinite Sampling Distribution of n E( ) 40.9
17 Sampling Distribution of for the air quality eample when the population is finite Sampling Distribution of n Nn N E( ) 40.9
18 Relationship Between the Sample Size and the Sampling Distribution of E( ( ) = regardless of the sample size. In our eample, E( ( ) remains at Whenever the sample size is increased, the standard error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the mean decreases.
19 Relationship Between the Sample Size and the Sampling Distribution of With n With n E( ) 40.9
20 Sampling Distribution of for the air quality Inde when n = 5. What is the probability that a simple random sample of 5 applicants will provide an estimate of the population mean air quality inde that is within +/-2 2 of the actual population mean, μ? In other words, what is the probability that will be between 38.9 and 42.9?
21 Sampling Distribution of for the air quality Inde when n = 100. What is the probability that a simple random sample of 100 applicants will provide an estimate of the population mean air quality inde that is within +/-2 2 of the actual population mean, μ?
22 Relationship Between the Sample Size and the Sampling Distribution of Because the sampling distribution with n = 100 has a smaller standard error, the values of have less variability and tend to be closer to the population mean than the values of with n = 5. Basically, a given interval with smaller standard error (larger n) will cover more area under the normal curve than the same interval with larger standard error (smaller n).
23 Chapter 7, Part B Sampling and Sampling Distributions Sampling Distribution of p
24 Sampling Distribution of p Making Inferences about a Population Proportion Population with proportion p =? A simple random sample of n elements is selected from the population. p The value of is used to make inferences about the value of p. The sample data provide a value for the sample proportion. p
25 Sampling Distribution of p The sampling distribution of p is the probability distribution of all possible values of the sample proportion p. Epected Value of p E ( p ) p where: p = the population proportion
26 Sampling Distribution of p Standard Deviation of Finite Population Infinite Population p ( 1 p ) p n N N n 1 p ( 1 p ) p n is referred to as the standard error of the proportion. p
27 Form of Sampling Distribution of p The sampling distribution of p can be approimated by a normal distribution whenever the sample size is large: Central Limit Theorem (CLT). The sample size is considered large whenever these conditions are satisfied: np > 5 and n(1 p) > 5
28 Chapter 8: Interval Estimation Population Mean: Known Population Mean: Unknown
29 Margin of Error and the Interval Estimate A point estimator cannot be epected to provide the eact value of the population parameter. An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. Point Estimate +/ Margin of Error The purpose of an interval estimate is is to provide information about how close the point estimate is is to the value of the parameter.
30 Margin of Error and the Interval Estimate The general form of an interval estimate of a population mean is is Margin of Error In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation, or the sample standard deviation s These are also Confidence Interval.
31 Interval Estimate of a Population Mean: Known Interval Estimate of z / 2 n where: is the sample mean 1 - is the confidence coefficient z /2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution is the population standard deviation n is the sample size
32 Interval Estimation of a Population Mean: Known There is a 1 probability that the value of a sample mean will provide a margin of error of z or less. Sampling distribution of /2 1 - of all /2 values /2 z /2 z /2
33 Eample: Air Quality Consider our air quality eample. Suppose the population is approimately normal with μ = 40.9 and σ = This is σ known case. If you guys remember, we picked a sample of size 5 (n =5). Given all this information, What is the margin of error at 95% confidence level?
34 Interval Estimation of a Population Mean: Unknown If an estimate of the population standard deviation cannot be developed prior to sampling, we use the sample standard deviation s to estimate. This is the unknown case. In this case, the interval estimate for is based on the t distribution. (We ll assume for now that the population is normally distributed.)
35 t Distribution The t distribution is is a family of similar probability distributions. A specific t distribution depends on a parameter known as the degrees of freedom. Degrees of freedom refer to the number of independent pieces of information that go into the computation of s. s.
36 t Distribution A t distribution with more degrees of freedom has less dispersion. As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.
37 t Distribution Standard normal distribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) 0 z, t
38 t Distribution For more than 100 degrees of freedom, the standard normal z value provides a good approimation to the t value. The standard normal z values can be found in the infinite degrees ( ) row of the t distribution table.
39 t Distribution Degrees Area in Upper Tail of Freedom Standard normal z values
40 Interval Estimation of a Population Mean: Unknown Interval Estimate s t / 2 n where: 1 - = the confidence coefficient t /2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation
41 Eample: Air quality when σ is unknown Now suppose that you did not know what σ is. You can estimate using the sample and then use t-distribution to find the margin of error. What is 95% confidence interval in this case? The sample size n =5. So, the degrees of freedom for the t-distribution is 4. The level of significance ( ) is s = t / 2 s n
42 Summary of Interval Estimation Procedures for a Population Mean Yes Use z z /2 Known Case n Can the population standard deviation be assumed known? Unknown Case No Use the sample standard deviation s to estimate σ Use s t /2 n
43 Interval Estimation of a Population Proportion The general form of an interval estimate of a population proportion is is p Margin of Error
44 Interval Estimation of a Population Proportion Interval Estimate p z / 2 p ( 1 p ) n where: 1 - is the confidence coefficient z /2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution p is the sample proportion
Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to
Lecture Topic 4: Chapter 7 Sampling and Sampling Distributions
Lecture Topic 4: Chapter 7 Sampling and Sampling Distributions Statistical Inference: The aim is to obtain information about a population from information contained in a sample. A population is the set
Chapter 7, Part A Sampling and Sampling Distributions
Slides Prepared by JOHN S. LOUCKS St. Edward s University Slide 1 Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling
Econ 3790: Business and Economic Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economic Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 13, Part A: Analysis of Variance and Experimental Design Introduction to Analysis of Variance Analysis
Business Statistics: A Decision-Making Approach 6 th Edition. Chapter Goals
Chapter 6 Student Lecture Notes 6-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 6 Introduction to Sampling Distributions Chap 6-1 Chapter Goals To use information from the sample
Econ 3790: Statistics Business and Economics. Instructor: Yogesh Uppal
Econ 3790: Statistics Business and Economics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 14 Covariance and Simple Correlation Coefficient Simple Linear Regression Covariance Covariance between
Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal yuppal@ysu.edu Sampling Distribution of b 1 Expected value of b 1 : Variance of b 1 : E(b 1 ) = 1 Var(b 1 ) = σ 2 /SS x Estimate of
Ch. 7 Statistical Intervals Based on a Single Sample
Ch. 7 Statistical Intervals Based on a Single Sample Before discussing the topics in Ch. 7, we need to cover one important concept from Ch. 6. Standard error The standard error is the standard deviation
CHAPTER 7 THE SAMPLING DISTRIBUTION OF THE MEAN. 7.1 Sampling Error; The need for Sampling Distributions
CHAPTER 7 THE SAMPLING DISTRIBUTION OF THE MEAN 7.1 Sampling Error; The need for Sampling Distributions Sampling Error the error resulting from using a sample characteristic (statistic) to estimate a population
Basic methods to solve equations
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here:
Lecture 17: Small-Sample Inferences for Normal Populations. Confidence intervals for µ when σ is unknown
Lecture 17: Small-Sample Inferences for Normal Populations Confidence intervals for µ when σ is unknown If the population distribution is normal, then X µ σ/ n has a standard normal distribution. If σ
Section 7.1 Quadratic Equations
Section 7.1 Quadratic Equations INTRODUCTION In Chapter 2 you learned about solving linear equations. In each of those, the highest power of any variable was 1. We will now take a look at solving quadratic
MATH 108 REVIEW TOPIC 6 Radicals
Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises
One Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x
Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have
Statistic: a that can be from a sample without making use of any unknown. In practice we will use to establish unknown parameters.
Chapter 9: Sampling Distributions 9.1: Sampling Distributions IDEA: How often would a given method of sampling give a correct answer if it was repeated many times? That is, if you took repeated samples
Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.
Chapter 7 Reading 7.1, 7.2 Questions 3.83, 6.11, 6.12, 6.17, 6.25, 6.29, 6.33, 6.35, 6.50, 6.51, 6.53, 6.55, 6.59, 6.60, 6.65, 6.69, 6.70, 6.77, 6.79, 6.89, 6.112 Introduction In Chapter 5 and 6, we emphasized
Row and column spaces
Roberto s Notes on Linear Algebra Chapter 7: Subspaces Section 4 Row and column spaces What you need to know already: What subspaces are. How to identify bases for a subspace. Basic facts about matrices.
Central Limit Theorem and the Law of Large Numbers Class 6, Jeremy Orloff and Jonathan Bloom
Central Limit Theorem and the Law of Large Numbers Class 6, 8.5 Jeremy Orloff and Jonathan Bloom Learning Goals. Understand the statement of the law of large numbers. 2. Understand the statement of the
Beautiful homework # 4 ENGR 323 CESSNA Page 1/5
Beautiful homework # 4 ENGR 33 CESSNA Page 1/5 Problem 3-14 An operator records the time to complete a mechanical assembly to the nearest second with the following results. seconds 30 31 3 33 34 35 36
Estimating a Population Mean. Section 7-3
Estimating a Population Mean Section 7-3 The Mean Age Suppose that we wish to estimate the mean age of residents of Metropolis. Furthermore, suppose we know that the ages of Metropolis residents are normally
Estimators as Random Variables
Estimation Theory Overview Properties Bias, Variance, and Mean Square Error Cramér-Rao lower bound Maimum likelihood Consistency Confidence intervals Properties of the mean estimator Introduction Up until
Inferential statistics
Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,
1 5 π 2. 5 π 3. 5 π π x. 5 π 4. Figure 1: We need calculus to find the area of the shaded region.
. Area In order to quantify the size of a 2-dimensional object, we use area. Since we measure area in square units, we can think of the area of an object as the number of such squares it fills up. Using
Chapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 1 Activity 9A, pp. 486-487 2 We ve just begun a sampling distribution! Strictly speaking, a sampling distribution is: A theoretical distribution of the values of a statistic
Ch. 5 Joint Probability Distributions and Random Samples
Ch. 5 Joint Probability Distributions and Random Samples 5. 1 Jointly Distributed Random Variables In chapters 3 and 4, we learned about probability distributions for a single random variable. However,
Stat 101: Lecture 12. Summer 2006
Stat 101: Lecture 12 Summer 2006 Outline Answer Questions More on the CLT The Finite Population Correction Factor Confidence Intervals Problems More on the CLT Recall the Central Limit Theorem for averages:
Probability & Statistics
MECE 330 MECE 330 Measurements & Instrumentation Probability & tatistics Dr. Isaac Choutapalli Department of Mechanical Engineering University of Teas Pan American MECE 330 Introduction uppose we have
Ch. 7: Estimates and Sample Sizes
Ch. 7: Estimates and Sample Sizes Section Title Notes Pages Introduction to the Chapter 2 2 Estimating p in the Binomial Distribution 2 5 3 Estimating a Population Mean: Sigma Known 6 9 4 Estimating a
14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
CHAPTER TOPICS. Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population
Distribusi Sampel CHAPTER TOPICS Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population 2 3 WHY STUDY SAMPLING DISTRIBUTIONS
Chapter 18: Sampling Distribution Models
Chapter 18: Sampling Distribution Models Suppose I randomly select 100 seniors in Scott County and record each one s GPA. 1.95 1.98 1.86 2.04 2.75 2.72 2.06 3.36 2.09 2.06 2.33 2.56 2.17 1.67 2.75 3.95
Class 15. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 15 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 17 by D.B. Rowe 1 Agenda: Recap Chapter 7.1 7. Lecture Chapter 7. Discussion of Chapters Problem Solving
Taylor Series and Series Convergence (Online)
7in 0in Felder c02_online.te V3 - February 9, 205 9:5 A.M. Page CHAPTER 2 Taylor Series and Series Convergence (Online) 2.8 Asymptotic Epansions In introductory calculus classes the statement this series
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7. Inverse matrices
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7 Inverse matrices What you need to know already: How to add and multiply matrices. What elementary matrices are. What you can learn
Statistical Intervals (One sample) (Chs )
7 Statistical Intervals (One sample) (Chs 8.1-8.3) Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to normally distributed with expected value µ and
Math-2 Lesson 2-4. Radicals
Math- Lesson - Radicals = What number is equivalent to the square root of? Square both sides of the equation ( ) ( ) = = = is an equivalent statement to = 1.7 1.71 1.70 1.701 1.7008... There is no equivalent
SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources
SOLVING QUADRATICS Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS General Form: y a b c Where a, b and c are constants To solve a quadratic equation, the equation
The set of all solutions to the homogeneous equation Ax = 0 is a subspace of R n if A is m n.
0 Subspaces (Now, we are ready to start the course....) Definitions: A linear combination of the vectors v, v,..., v m is any vector of the form c v + c v +... + c m v m, where c,..., c m R. A subset V
Unit 9: Symmetric Functions
Haberman MTH 111 Section I: Functions and Their Graphs Unit 9: Symmetric Functions Some functions have graphs with special types of symmetries, and we can use the reflections we just studied to analyze
Chapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence Section 8.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 8 Estimating with Confidence n 8.1 Confidence Intervals: The Basics n 8.2
Business Statistics. Lecture 5: Confidence Intervals
Business Statistics Lecture 5: Confidence Intervals Goals for this Lecture Confidence intervals The t distribution 2 Welcome to Interval Estimation! Moments Mean 815.0340 Std Dev 0.8923 Std Error Mean
Answers to All Exercises
CAPER 10 CAPER 10 CAPER10 CAPER REFRESING YOUR SKILLS FOR CAPER 10 1a. 5 1 0.5 10 1b. 6 3 0.6 10 5 1c. 0. 10 5 a. 10 36 5 1 0. 7 b. 7 is most likely; probability of 7 is 6 36 1 6 0.1 6. c. 1 1 0.5 36 3a.
CHAPTER 5 Probabilistic Features of the Distributions of Certain Sample Statistics
CHAPTER 5 Probabilistic Features of the Distributions of Certain Sample Statistics Key Words Sampling Distributions Distribution of the Sample Mean Distribution of the difference between Two Sample Means
JUST THE MATHS UNIT NUMBER 7.2. DETERMINANTS 2 (Consistency and third order determinants) A.J.Hobson
JUST THE MATHS UNIT NUMBER 7.2 DETERMINANTS 2 (Consistency and third order determinants) by A.J.Hobson 7.2.1 Consistency for three simultaneous linear equations in two unknowns 7.2.2 The definition of
CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS
CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS A. J. Clark School of Engineering Department of Civil and Environmental Engineering by Dr. Ibrahim A. Assakkaf Spring 1 ENCE 3 - Computation in Civil Engineering
Roberto s Notes on Infinite Series Chapter 1: Sequences and series Section 4. Telescoping series. Clear as mud!
Roberto s Notes on Infinite Series Chapter : Sequences and series Section Telescoping series What you need to now already: The definition and basic properties of series. How to decompose a rational expression
I used college textbooks because they were the only resource available to evaluate measurement uncertainty calculations.
Introduction to Statistics By Rick Hogan Estimating uncertainty in measurement requires a good understanding of Statistics and statistical analysis. While there are many free statistics resources online,
NOTATION: We have a special symbol to use when we wish to find the anti-derivative of a function, called an Integral Symbol,
SECTION 5.: ANTI-DERIVATIVES We eplored taking the Derivative of functions in the previous chapters, but now want to look at the reverse process, which the student should understand is sometimes more complicated.
Steve Smith Tuition: Maths Notes
Maths Notes : Discrete Random Variables Version. Steve Smith Tuition: Maths Notes e iπ + = 0 a + b = c z n+ = z n + c V E + F = Discrete Random Variables Contents Intro The Distribution of Probabilities
Sections 5.1: Areas and Distances
Sections.: Areas and Distances In this section we shall consider problems closely related to the problems we considered at the beginning of the semester (the tangent and velocity problems). Specifically,
The concept of limit
Roberto s Notes on Dierential Calculus Chapter 1: Limits and continuity Section 1 The concept o limit What you need to know already: All basic concepts about unctions. What you can learn here: What limits
Confidence Interval Estimation
Department of Psychology and Human Development Vanderbilt University 1 Introduction 2 3 4 5 Relationship to the 2-Tailed Hypothesis Test Relationship to the 1-Tailed Hypothesis Test 6 7 Introduction In
5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435
![5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435](/thumbs/78/77255700.jpg "5.5 Volumes: Tubes. The Tube Method. = (2π [radius]) (height) ( x k ) = (2πc k ) f (c k ) x k. 5.5 volumes: tubes 435")
5.5 volumes: tubes 45 5.5 Volumes: Tubes In Section 5., we devised the disk method to find the volume swept out when a region is revolved about a line. To find the volume swept out when revolving a region
Learning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.
Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of
CIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8
CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval
AP Statistics Cumulative AP Exam Study Guide
AP Statistics Cumulative AP Eam Study Guide Chapters & 3 - Graphs Statistics the science of collecting, analyzing, and drawing conclusions from data. Descriptive methods of organizing and summarizing statistics
Introduction to Probability and Statistics Twelfth Edition
Introduction to Probability and Statistics Twelfth Edition Robert J. Beaver Barbara M. Beaver William Mendenhall Presentation designed and written by: Barbara M. Beaver Introduction to Probability and
Definition of Statistics Statistics Branches of Statistics Descriptive statistics Inferential statistics
What is Statistics? Definition of Statistics Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make a decision. Branches of Statistics The study of statistics
Number of solutions of a system
Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 7 Number of solutions of a system What you need to know already: How to solve a linear system by using Gauss- Jordan elimination.
Advanced Experimental Design
Advanced Experimental Design Topic Four Hypothesis testing (z and t tests) & Power Agenda Hypothesis testing Sampling distributions/central limit theorem z test (σ known) One sample z & Confidence intervals
Linear Models and Estimation by Least Squares
Linear Models and Estimation by Least Squares Jin-Lung Lin 1 Introduction Causal relation investigation lies in the heart of economics. Effect (Dependent variable) cause (Independent variable) Example:
ACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
Statistics 251: Statistical Methods
Statistics 251: Statistical Methods 1-sample Hypothesis Tests Module 9 2018 Introduction We have learned about estimating parameters by point estimation and interval estimation (specifically confidence
Outline. Limits as x
MS: IT Mathematics Limits & Continuity Limits at Infinity John Carroll School of Mathematical Sciences Dublin City University Introduction So far, we have only considered its as c where c is some finite
Elimination and back substitution
Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 2 Elimination and back substitution What you need to know already: What a (linear) system is. What it means to solve such
COSC 341 Human Computer Interaction. Dr. Bowen Hui University of British Columbia Okanagan
COSC 341 Human Computer Interaction Dr. Bowen Hui University of British Columbia Okanagan 1 Last Topic Distribution of means When it is needed How to build one (from scratch) Determining the characteristics
Unit 4 Probability. Dr Mahmoud Alhussami
Unit 4 Probability Dr Mahmoud Alhussami Probability Probability theory developed from the study of games of chance like dice and cards. A process like flipping a coin, rolling a die or drawing a card from
Chapter 8: Estimating with Confidence
Chapter 8: Estimating with Confidence Section 8.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE The One-Sample z Interval for a Population Mean In Section 8.1, we estimated the
Definition of geometric vectors
Roberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 2 of geometric vectors What you need to know already: The general aims behind the concept of a vector. What you can learn here: The
2x + 5 = x = x = 4
98 CHAPTER 3 Algebra Textbook Reference Section 5.1 3.3 LINEAR EQUATIONS AND INEQUALITIES Student CD Section.5 CLAST OBJECTIVES Solve linear equations and inequalities Solve a system of two linear equations
CENTRAL LIMIT THEOREM (CLT)
CENTRAL LIMIT THEOREM (CLT) A sampling distribution is the probability distribution of the sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic
Stats Review Chapter 14. Mary Stangler Center for Academic Success Revised 8/16
Stats Review Chapter 14 Revised 8/16 Note: This review is meant to highlight basic concepts from the course. It does not cover all concepts presented by your instructor. Refer back to your notes, unit
12.1 The Extrema of a Function
. The Etrema of a Function Question : What is the difference between a relative etremum and an absolute etremum? Question : What is a critical point of a function? Question : How do you find the relative
[Limits at infinity examples] Example. The graph of a function y = f(x) is shown below. Compute lim f(x) and lim f(x).
![[Limits at infinity examples] Example. The graph of a function y = f(x) is shown below. Compute lim f(x) and lim f(x).](/thumbs/87/95751891.jpg "[Limits at infinity examples] Example. The graph of a function y = f(x) is shown below. Compute lim f(x) and lim f(x).")
[Limits at infinity eamples] Eample. The graph of a function y = f() is shown below. Compute f() and f(). y -8 As you go to the far right, the graph approaches y =, so f() =. As you go to the far left,
EE5319R: Problem Set 3 Assigned: 24/08/16, Due: 31/08/16
EE539R: Problem Set 3 Assigned: 24/08/6, Due: 3/08/6. Cover and Thomas: Problem 2.30 (Maimum Entropy): Solution: We are required to maimize H(P X ) over all distributions P X on the non-negative integers
Topic 16 Interval Estimation
Topic 16 Interval Estimation Confidence Intervals for 1 / 14 Outline Overview z intervals t intervals Two Sample t intervals 2 / 14 Overview The quality of an estimator can be evaluated using its bias
Agenda: Questions Chapter 4 so far?
Agenda: Questions Chapter 4 so far? Review Chapter 4. Lecture Go over Homework: Chapt. #, 3 Copyright 6 by D.B. Rowe 4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data
1. Solutions to Systems of Linear Equations. Determine whether the ordered pairs are solutions to the system. x y 6. 3x y 2
78 Chapter Sstems of Linear Equations Section. Concepts. Solutions to Sstems of Linear Equations. Dependent and Inconsistent Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Solving
THE EFFECTS OF MULTICOLLINEARITY IN ORDINARY LEAST SQUARES (OLS) ESTIMATION
THE EFFECTS OF MULTICOLLINEARITY IN ORDINARY LEAST SQUARES (OLS) ESTIMATION Weeraratne N.C. Department of Economics & Statistics SUSL, BelihulOya, Sri Lanka ABSTRACT The explanatory variables are not perfectly
Lab on Taylor Polynomials. This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor.
Lab on Taylor Polynomials This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor. In this Lab we will approimate complicated unctions by simple unctions. The
Testing Independence
Testing Independence Dipankar Bandyopadhyay Department of Biostatistics, Virginia Commonwealth University BIOS 625: Categorical Data & GLM 1/50 Testing Independence Previously, we looked at RR = OR = 1
Chapter 1. Vectors, Matrices, and Linear Spaces
1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Note. In this section we explore the structure of the solution
Confidence Intervals for Population Mean
Confidence Intervals for Population Mean Reading: Sections 7.1, 7.2, 7.3 Learning Objectives: Students should be able to: Understand the meaning and purpose of confidence intervals Calculate a confidence
Eigenvalues and eigenvectors
Roberto s Notes on Linear Algebra Chapter 0: Eigenvalues and diagonalization Section Eigenvalues and eigenvectors What you need to know already: Basic properties of linear transformations. Linear systems
M 340L CS Homework Set 1
M 340L CS Homework Set 1 Solve each system in Problems 1 6 by using elementary row operations on the equations or on the augmented matri. Follow the systematic elimination procedure described in Lay, Section
The t-test: A z-score for a sample mean tells us where in the distribution the particular mean lies
The t-test: So Far: Sampling distribution benefit is that even if the original population is not normal, a sampling distribution based on this population will be normal (for sample size > 30). Benefit
STA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
THS Step By Step Calculus Chapter 1
Name: Class Period: Throughout this packet there will be blanks you are epected to fill in prior to coming to class. This packet follows your Larson Tetbook. Do NOT throw away! Keep in 3 ring binder until
Statistical Concepts. Distributions of Data
Module : Review of Basic Statistical Concepts. Understanding Probability Distributions, Parameters and Statistics A variable that can take on any value in a range is called a continuous variable. Example:
A3. Statistical Inference
Appendi / A3. Statistical Inference / Mean, One Sample-1 A3. Statistical Inference Population Mean μ of a Random Variable with known standard deviation σ, and random sample of size n 1 Before selecting
22. RADICALS. x add 5. multiply by 7
22. RADICALS doing something, then undoing it The concept of doing something and then undoing it is very important in mathematics. Here are some eamples: Take a number. Add 5 to it. How can you get back
Introduction to Probability Theory for Graduate Economics Fall 2008
Introduction to Probability Theory for Graduate Economics Fall 008 Yiğit Sağlam October 10, 008 CHAPTER - RANDOM VARIABLES AND EXPECTATION 1 1 Random Variables A random variable (RV) is a real-valued function
Chapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
ECON 5350 Class Notes Review of Probability and Distribution Theory
ECON 535 Class Notes Review of Probability and Distribution Theory 1 Random Variables Definition. Let c represent an element of the sample space C of a random eperiment, c C. A random variable is a one-to-one
2 3 x = 6 4. (x 1) 6
Solutions to Math 201 Final Eam from spring 2007 p. 1 of 16 (some of these problem solutions are out of order, in the interest of saving paper) 1. given equation: 1 2 ( 1) 1 3 = 4 both sides 6: 6 1 1 (
10.4 Hypothesis Testing: Two Independent Samples Proportion
10.4 Hypothesis Testing: Two Independent Samples Proportion Example 3: Smoking cigarettes has been known to cause cancer and other ailments. One politician believes that a higher tax should be imposed
What Is a Sampling Distribution? DISTINGUISH between a parameter and a statistic
Section 8.1A What Is a Sampling Distribution? Learning Objectives After this section, you should be able to DISTINGUISH between a parameter and a statistic DEFINE sampling distribution DISTINGUISH between
11: Comparing Group Variances. Review of Variance
11: Comparing Group Variances Review of Variance Parametric measures of variability are often based on sum of squares (SS) around e mean: (1) For e data set {3, 4, 5, 8}, = 5 and SS = (3 5) + (4 5) + (5
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Exploring Data: Distributions Look for overall pattern (shape, center, spread) and deviations (outliers). Mean (use a calculator): x = x 1 + x
Chapter 18: Sampling Distributions
Chapter 18: Sampling Distributions All random variables have probability distributions, and as statistics are random variables, they too have distributions. The random phenomenon that produces the statistics