Lecture 3. The Population Variance. The population variance, denoted σ 2, is the sum. of the squared deviations about the population
|
|
- Job Greer
- 5 years ago
- Views:
Transcription
1 Lecture 5 1 Lecture 3 The Population Variance The population variance, denoted σ 2, is the sum of the squared deviations about the population mean divided by the number of observations in the population, N: σ 2 = (xi µ) 2 N = (x 1 µ) 2 + (x 2 µ) 2 + (x N µ) 2 N. σ 2 = x 2 i N Another alternative formula is: ( xi N ) 2 = x 2 i N µ2. REMARK: To avoid round-off errors, which accumulate quickly in these formulas, do not round until the last computation, and use as
2 Lecture 5 2 many decimal places as allowed in your calculator.
3 Lecture 5 3 The Sample Variance When the population is large, we approximate the population mean µ with the sample mean, x. Similarly, we approximate the population variance σ 2 by the sample variance, denoted s 2 : s 2 = (xi x) 2 n 1 = (x 1 x) 2 + (x 2 x) (x n x) 2 n 1. The alternative form is: s 2 = (xi x) 2 n 1 ( x i ) 2 n(n 1). REMARK: Notice that we divide by the sample size minus one (this is different from the formula for the population variance).
4 Lecture 5 4 Informally, we say: a sample of size n has n degrees of freedom; one degree of freedom is used up in computing x, so there are only n 1 degrees of freedom available for the sample variance.
5 Lecture 5 5 The Standard Deviation For both cases (the population or the sample), the standard deviation is the square root of the corresponding variance: The population standard deviation is denoted by σ: σ = σ 2. The sample standard deviation is denoted by s: s = s 2. Advantage of the (population or sample) standard
6 Lecture 5 6 deviation: it is given in the same units as the observations. Advantage of the (population or sample) variance: it is easier to manipulate algebraically, in some cases. Both the standard deviations and variances are interpreted as follows: the larger they are, the more spread is the distribution (if they equal 0, the smallest possible value, then all observations must be equal). Remark 1. Standard deviation measures spread about the mean and should be used only when the mean is chosen as the measure of center. Remark 2. Standard deviation is not robust.
7 Lecture 5 7 Remark 3. The sum of the deviations of the observations from their mean will always be zero.
8 Lecture 5 8 Density curves Histograms are approximations to an exact variable distribution. Increasing the number of classes in a histogram makes each rectangle less wide and as the number of rectangles approaches infinity, the graph becomes a curve, called density curve. Properties of the density curve 1. The curve is always above the x-axis: the function f(x) describing the curve is nonnegative (could be zero) for all x 2. The total area underneath the curve and above the x-axis equal 1.
9 Lecture 5 9 Density curves, as we saw, have mean, medians and modes as well as standard deviation. the notations are similar to the one for the population mean and standard deviation (why?). Most of the time we use software to estimate density curves. Many times we assume that data follows a certain density curve.
10 Lecture 5 10 The normal distribution Often called Gaussian curve, the normal curve was introduced by Carl Friedrich Gauss in 1809 as an error curve of least square regression, about which we will talk next time. There are other symmetric bell-shaped density curves that are not normal. Remark 4. The curve is described completely by 2 parameters: µ-the mean and σ-the standard deviation.
11 Lecture 5 11 The Empirical Rule If the distribution is approximately bell shaped (not only normal), then: 1. Approximately 68% of the data will lie within one standard deviation of the mean. That is, about 68% of the data will be between µ σ and µ + σ. 2. Approximately 95% of the data will lie within two standard deviations of the mean. 3. Approximately 99.7% of the data will lie within three standard deviations of the mean. For exact values, we need to integrate to find the area between two points.
12 Lecture 5 12 In general, for any distribution, not only the normal distribution, Chebyshev s rule could be applied: The proportion of values from a data set that will fall within k standard deviations of the mean will be at least (1 1 k 2 )100% where k > 1. his rule could be applied to samples too.
13 Lecture 5 13 Finding the area under the normal density curve is not an easy task. It requires a lot of calculus. One way of avoiding this is to use tables that give us these areas (probabilities). But for each µ and σ we would need a new table. How can we avoid this? By transforming somehow all these curves into a standard one. Choose µ = 0 and σ 2 = 1 Standardizing Convert other values to standard units or z-scores, by subtracting the mean and dividing by standard deviation z = x µ σ
14 Lecture 5 14 Example: Standardize x = 3 with µ = 2 and σ = 4. What z-score range corresponds to (8, 17) with µ = 12 and σ 2 = 9?
15 Lecture 5 15 Interpretation: z is the number of standard deviations that x is away from the mean. The z-score is unit free. We can use it to compare observations from different sources ( apples to oranges ). Notation The standard normal distribution is denoted by N(0, 1) and any other normal distribution with mean µ and variance σ 2 by N(µ, σ).
16 Lecture 5 16 Relations between variables. Scatter diagrams In practice statisticians are interested in multiple variable relationships. For 2 variables, the pairs of data points match forming an observation. Sometimes we use the value of one variable in order to predict another variable.the response variable is the variable whose value can be explained by, or is determined by, the value of the explanatory variable. The response variable measures the outcome of a study. An explanatory variable explains or causes changes in the response variable. Example:
17 Lecture 5 17 The relationship between two variables could be represented by crosstabulation, side by side or clustered bar graphs, and scatterplots.
18 Lecture 5 18 Definition 5. A scatter diagram is a graph that shows the relationship between two quantitative variables measured on the same individual. How to draw a scatter diagram: The explanatory variable is plotted on the horizontal axis and the response variable is plotted on the vertical axis. Each individual in the data set is represented by a point in the scatter diagram. Do not connect the points when drawing a scatter diagram.
19 Lecture 5 19 How we interpret a scatter diagram Scatter diagrams imply a linear relationship nonlinear relationship no relation Definition 6. Two variables that are linearly related are said to be positively associated if, whenever the values of the predictor variable
20 Lecture 5 20 increase, the values of the response variable also increase, and it is said to be negatively associated if, whenever the values of the predictor variable increase, the value of the response variable decrease.
21 Lecture 5 21 Be careful!! Do not conclude causation through association.
22 Lecture 5 22 Definition 7. The linear correlation coefficient is a measure of the strength of linear relation between two quantitative variables. The sample correlation correlation coefficient is computed by: r = n i=1 ( x i x s x )( y i y s y ) n 1 where x is the sample mean of the predictor variable s x is the sample standard deviation of the predictor variable. y is the sample mean of the response variable s x is the sample standard deviation of the response variable. n is the number of individuals in the sample.
23 Lecture 5 23 The population correlation coefficient is denoted by ρ Example: (0, 0)(1, 2)(2, 2)(3, 5)(4, 6)
24 Lecture 5 24 Interpretation and properties of r 1 r 1 If r = 1 there is a perfect positive linear relation between the 2 variables. If r = 1 there is a perfect negative linear relation between the 2 variables. The closer r is to 1 the stronger the evidence of a positive linear relation and the closer to -1 the stronger the evidence of negative association between the two variables. If r is close to 0 there is evidence of no linear relation between the 2 variables. This does not mean no relation, just no linear relation.
25 Lecture 5 25 r is a untiles measure of association. r is not resistant. It is strongly affected by outlier. Both variables should be quantitative.
Lecture 2 and Lecture 3
Lecture 2 and Lecture 3 1 Lecture 2 and Lecture 3 We can describe distributions using 3 characteristics: shape, center and spread. These characteristics have been discussed since the foundation of statistics.
More informationChapter 7. Scatterplots, Association, and Correlation
Chapter 7 Scatterplots, Association, and Correlation Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 29 Objective In this chapter, we study relationships! Instead, we investigate
More informationSTP 420 INTRODUCTION TO APPLIED STATISTICS NOTES
INTRODUCTION TO APPLIED STATISTICS NOTES PART - DATA CHAPTER LOOKING AT DATA - DISTRIBUTIONS Individuals objects described by a set of data (people, animals, things) - all the data for one individual make
More informationMATH 1150 Chapter 2 Notation and Terminology
MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the
More informationChapter 5. Understanding and Comparing. Distributions
STAT 141 Introduction to Statistics Chapter 5 Understanding and Comparing Distributions Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 27 Boxplots How to create a boxplot? Assume
More informationTHE PEARSON CORRELATION COEFFICIENT
CORRELATION Two variables are said to have a relation if knowing the value of one variable gives you information about the likely value of the second variable this is known as a bivariate relation There
More informationMA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.
MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at
More informationSection 3.4 Normal Distribution MDM4U Jensen
Section 3.4 Normal Distribution MDM4U Jensen Part 1: Dice Rolling Activity a) Roll two 6- sided number cubes 18 times. Record a tally mark next to the appropriate number after each roll. After rolling
More informationSociology 6Z03 Review I
Sociology 6Z03 Review I John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review I Fall 2016 1 / 19 Outline: Review I Introduction Displaying Distributions Describing
More informationScatterplots and Correlations
Scatterplots and Correlations Section 4.1 1 New Definitions Explanatory Variable: (independent, x variable): attempts to explain observed outcome. Response Variable: (dependent, y variable): measures outcome
More informationDescriptive Univariate Statistics and Bivariate Correlation
ESC 100 Exploring Engineering Descriptive Univariate Statistics and Bivariate Correlation Instructor: Sudhir Khetan, Ph.D. Wednesday/Friday, October 17/19, 2012 The Central Dogma of Statistics used to
More informationMath 223 Lecture Notes 3/15/04 From The Basic Practice of Statistics, bymoore
Math 223 Lecture Notes 3/15/04 From The Basic Practice of Statistics, bymoore Chapter 3 continued Describing distributions with numbers Measuring spread of data: Quartiles Definition 1: The interquartile
More informationChapter 4 Describing the Relation between Two Variables
Chapter 4 Describing the Relation between Two Variables 4.1 Scatter Diagrams and Correlation The is the variable whose value can be explained by the value of the or. A is a graph that shows the relationship
More informationChapter 3: Examining Relationships
Chapter 3: Examining Relationships Most statistical studies involve more than one variable. Often in the AP Statistics exam, you will be asked to compare two data sets by using side by side boxplots or
More informationUnit 2. Describing Data: Numerical
Unit 2 Describing Data: Numerical Describing Data Numerically Describing Data Numerically Central Tendency Arithmetic Mean Median Mode Variation Range Interquartile Range Variance Standard Deviation Coefficient
More informationAMS 7 Correlation and Regression Lecture 8
AMS 7 Correlation and Regression Lecture 8 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Suumer 2014 1 / 18 Correlation pairs of continuous observations. Correlation
More information2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table
2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations
More informationChapter 12 - Part I: Correlation Analysis
ST coursework due Friday, April - Chapter - Part I: Correlation Analysis Textbook Assignment Page - # Page - #, Page - # Lab Assignment # (available on ST webpage) GOALS When you have completed this lecture,
More informationSection 3. Measures of Variation
Section 3 Measures of Variation Range Range = (maximum value) (minimum value) It is very sensitive to extreme values; therefore not as useful as other measures of variation. Sample Standard Deviation The
More informationChapter 6: Exploring Data: Relationships Lesson Plan
Chapter 6: Exploring Data: Relationships Lesson Plan For All Practical Purposes Displaying Relationships: Scatterplots Mathematical Literacy in Today s World, 9th ed. Making Predictions: Regression Line
More informationWhy Is It There? Attribute Data Describe with statistics Analyze with hypothesis testing Spatial Data Describe with maps Analyze with spatial analysis
6 Why Is It There? Why Is It There? Getting Started with Geographic Information Systems Chapter 6 6.1 Describing Attributes 6.2 Statistical Analysis 6.3 Spatial Description 6.4 Spatial Analysis 6.5 Searching
More informationCHAPTER 3 Describing Relationships
CHAPTER 3 Describing Relationships 3.1 Scatterplots and Correlation The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Scatterplots and Correlation Learning
More informationSTA 218: Statistics for Management
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Problem How much do people with a bachelor s degree
More informationA company recorded the commuting distance in miles and number of absences in days for a group of its employees over the course of a year.
Paired Data(bivariate data) and Scatterplots: When data consists of pairs of values, it s sometimes useful to plot them as points called a scatterplot. A company recorded the commuting distance in miles
More informationLecture 2. Descriptive Statistics: Measures of Center
Lecture 2. Descriptive Statistics: Measures of Center Descriptive Statistics summarize or describe the important characteristics of a known set of data Inferential Statistics use sample data to make inferences
More informationDensity Curves and the Normal Distributions. Histogram: 10 groups
Density Curves and the Normal Distributions MATH 2300 Chapter 6 Histogram: 10 groups 1 Histogram: 20 groups Histogram: 40 groups 2 Histogram: 80 groups Histogram: 160 groups 3 Density Curve Density Curves
More informationBusiness Analytics and Data Mining Modeling Using R Prof. Gaurav Dixit Department of Management Studies Indian Institute of Technology, Roorkee
Business Analytics and Data Mining Modeling Using R Prof. Gaurav Dixit Department of Management Studies Indian Institute of Technology, Roorkee Lecture - 04 Basic Statistics Part-1 (Refer Slide Time: 00:33)
More informationLesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table
Lesson Plan Answer Questions Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 1 2. Summary Statistics Given a collection of data, one needs to find representations
More informationStatistics for Managers using Microsoft Excel 6 th Edition
Statistics for Managers using Microsoft Excel 6 th Edition Chapter 3 Numerical Descriptive Measures 3-1 Learning Objectives In this chapter, you learn: To describe the properties of central tendency, variation,
More informationChapter 10. Regression. Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania
Chapter 10 Regression Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania Scatter Diagrams A graph in which pairs of points, (x, y), are
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationBusiness Statistics. Lecture 10: Correlation and Linear Regression
Business Statistics Lecture 10: Correlation and Linear Regression Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. It displays the Form
More informationMAT Mathematics in Today's World
MAT 1000 Mathematics in Today's World Last Time 1. Three keys to summarize a collection of data: shape, center, spread. 2. Can measure spread with the fivenumber summary. 3. The five-number summary can
More informationLooking at Data Relationships. 2.1 Scatterplots W. H. Freeman and Company
Looking at Data Relationships 2.1 Scatterplots 2012 W. H. Freeman and Company Here, we have two quantitative variables for each of 16 students. 1) How many beers they drank, and 2) Their blood alcohol
More informationLecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 3.1- #
Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures
More informationLecture 3: Chapter 3
Lecture 3: Chapter 3 C C Moxley UAB Mathematics 26 January 16 3.2 Measurements of Center Statistics involves describing data sets and inferring things about them. The first step in understanding a set
More informationWeek 8: Correlation and Regression
Health Sciences M.Sc. Programme Applied Biostatistics Week 8: Correlation and Regression The correlation coefficient Correlation coefficients are used to measure the strength of the relationship or association
More informationChapter 2: Tools for Exploring Univariate Data
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 2: Tools for Exploring Univariate Data Section 2.1: Introduction What is
More informationAP Final Review II Exploring Data (20% 30%)
AP Final Review II Exploring Data (20% 30%) Quantitative vs Categorical Variables Quantitative variables are numerical values for which arithmetic operations such as means make sense. It is usually a measure
More informationP8130: Biostatistical Methods I
P8130: Biostatistical Methods I Lecture 2: Descriptive Statistics Cody Chiuzan, PhD Department of Biostatistics Mailman School of Public Health (MSPH) Lecture 1: Recap Intro to Biostatistics Types of Data
More informationMath 1710 Class 20. V2u. Last Time. Graphs and Association. Correlation. Regression. Association, Correlation, Regression Dr. Back. Oct.
,, Dr. Back Oct. 14, 2009 Son s Heights from Their Fathers Galton s Original 1886 Data If you know a father s height, what can you say about his son s? Son s Heights from Their Fathers Galton s Original
More informationReview of Statistics 101
Review of Statistics 101 We review some important themes from the course 1. Introduction Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods
More informationLecture 6. Probability events. Definition 1. The sample space, S, of a. probability experiment is the collection of all
Lecture 6 1 Lecture 6 Probability events Definition 1. The sample space, S, of a probability experiment is the collection of all possible outcomes of an experiment. One such outcome is called a simple
More informationContents. Acknowledgments. xix
Table of Preface Acknowledgments page xv xix 1 Introduction 1 The Role of the Computer in Data Analysis 1 Statistics: Descriptive and Inferential 2 Variables and Constants 3 The Measurement of Variables
More informationObjective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode.
Chapter 3 Numerically Summarizing Data Chapter 3.1 Measures of Central Tendency Objective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode. A1. Mean The
More informationBig Data Analysis with Apache Spark UC#BERKELEY
Big Data Analysis with Apache Spark UC#BERKELEY This Lecture: Relation between Variables An association A trend» Positive association or Negative association A pattern» Could be any discernible shape»
More informationChapter 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
More information3.1 Measures of Central Tendency: Mode, Median and Mean. Average a single number that is used to describe the entire sample or population
. Measures of Central Tendency: Mode, Median and Mean Average a single number that is used to describe the entire sample or population. Mode a. Easiest to compute, but not too stable i. Changing just one
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Overview 3-2 Measures
More informationMeasures of. U4 C 1.2 Dot plot and Histogram 2 January 15 16, 2015
U4 C 1. Dot plot and Histogram January 15 16, 015 U 4 : C 1.1 CCSS. 9 1.S ID.1 Dot Plots and Histograms Objective: We will be able to represent data with plots on the real number line, using: Dot Plots
More informationReview of Multiple Regression
Ronald H. Heck 1 Let s begin with a little review of multiple regression this week. Linear models [e.g., correlation, t-tests, analysis of variance (ANOVA), multiple regression, path analysis, multivariate
More informationStats Review Chapter 3. Mary Stangler Center for Academic Success Revised 8/16
Stats Review Chapter Revised 8/16 Note: This review is composed of questions similar to those found in the chapter review and/or chapter test. This review is meant to highlight basic concepts from the
More informationMathematics for Economics MA course
Mathematics for Economics MA course Simple Linear Regression Dr. Seetha Bandara Simple Regression Simple linear regression is a statistical method that allows us to summarize and study relationships between
More informationappstats8.notebook October 11, 2016
Chapter 8 Linear Regression Objective: Students will construct and analyze a linear model for a given set of data. Fat Versus Protein: An Example pg 168 The following is a scatterplot of total fat versus
More informationClass 11 Maths Chapter 15. Statistics
1 P a g e Class 11 Maths Chapter 15. Statistics Statistics is the Science of collection, organization, presentation, analysis and interpretation of the numerical data. Useful Terms 1. Limit of the Class
More informationThe response variable depends on the explanatory variable.
A response variable measures an outcome of study. > dependent variables An explanatory variable attempts to explain the observed outcomes. > independent variables The response variable depends on the explanatory
More informationOverview. 4.1 Tables and Graphs for the Relationship Between Two Variables. 4.2 Introduction to Correlation. 4.3 Introduction to Regression 3.
3.1-1 Overview 4.1 Tables and Graphs for the Relationship Between Two Variables 4.2 Introduction to Correlation 4.3 Introduction to Regression 3.1-2 4.1 Tables and Graphs for the Relationship Between Two
More informationM 225 Test 1 B Name SHOW YOUR WORK FOR FULL CREDIT! Problem Max. Points Your Points Total 75
M 225 Test 1 B Name SHOW YOUR WORK FOR FULL CREDIT! Problem Max. Points Your Points 1-13 13 14 3 15 8 16 4 17 10 18 9 19 7 20 3 21 16 22 2 Total 75 1 Multiple choice questions (1 point each) 1. Look at
More informationLinear Regression. Linear Regression. Linear Regression. Did You Mean Association Or Correlation?
Did You Mean Association Or Correlation? AP Statistics Chapter 8 Be careful not to use the word correlation when you really mean association. Often times people will incorrectly use the word correlation
More informationDr. Allen Back. Sep. 23, 2016
Dr. Allen Back Sep. 23, 2016 Look at All the Data Graphically A Famous Example: The Challenger Tragedy Look at All the Data Graphically A Famous Example: The Challenger Tragedy Type of Data Looked at the
More informationElementary Statistics
Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q:
More informationChapter. Numerically Summarizing Data. Copyright 2013, 2010 and 2007 Pearson Education, Inc.
Chapter 3 Numerically Summarizing Data Section 3.1 Measures of Central Tendency Objectives 1. Determine the arithmetic mean of a variable from raw data 2. Determine the median of a variable from raw data
More informationReview. Midterm Exam. Midterm Review. May 6th, 2015 AMS-UCSC. Spring Session 1 (Midterm Review) AMS-5 May 6th, / 24
Midterm Exam Midterm Review AMS-UCSC May 6th, 2015 Spring 2015. Session 1 (Midterm Review) AMS-5 May 6th, 2015 1 / 24 Topics Topics We will talk about... 1 Review Spring 2015. Session 1 (Midterm Review)
More informationChapter 7 Linear Regression
Chapter 7 Linear Regression 1 7.1 Least Squares: The Line of Best Fit 2 The Linear Model Fat and Protein at Burger King The correlation is 0.76. This indicates a strong linear fit, but what line? The line
More informationAP Statistics. Chapter 6 Scatterplots, Association, and Correlation
AP Statistics Chapter 6 Scatterplots, Association, and Correlation Objectives: Scatterplots Association Outliers Response Variable Explanatory Variable Correlation Correlation Coefficient Lurking Variables
More informationArvind Borde / MAT , Week 5: Relationships I
Arvind Borde / MAT 19.001, Week 5: Relationships I 1 Review of Standard Deviation Population (N observations) Sample (sample size n) (xi µ) σ = (xi x) s = N n 1 µ = mean x = mean Where are most of the
More informationSimple Linear Regression. Material from Devore s book (Ed 8), and Cengagebrain.com
12 Simple Linear Regression Material from Devore s book (Ed 8), and Cengagebrain.com The Simple Linear Regression Model The simplest deterministic mathematical relationship between two variables x and
More informationLecture 18: Simple Linear Regression
Lecture 18: Simple Linear Regression BIOS 553 Department of Biostatistics University of Michigan Fall 2004 The Correlation Coefficient: r The correlation coefficient (r) is a number that measures the strength
More informationChapter 8. Linear Regression. Copyright 2010 Pearson Education, Inc.
Chapter 8 Linear Regression Copyright 2010 Pearson Education, Inc. Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: Copyright
More information3.1 Scatterplots and Correlation
3.1 Scatterplots and Correlation Most statistical studies examine data on more than one variable. In many of these settings, the two variables play different roles. Explanatory variable (independent) predicts
More informationF78SC2 Notes 2 RJRC. If the interest rate is 5%, we substitute x = 0.05 in the formula. This gives
F78SC2 Notes 2 RJRC Algebra It is useful to use letters to represent numbers. We can use the rules of arithmetic to manipulate the formula and just substitute in the numbers at the end. Example: 100 invested
More informationThe Normal Distribuions
The Normal Distribuions Sections 5.4 & 5.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 15-3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationUnit 6 - Introduction to linear regression
Unit 6 - Introduction to linear regression Suggested reading: OpenIntro Statistics, Chapter 7 Suggested exercises: Part 1 - Relationship between two numerical variables: 7.7, 7.9, 7.11, 7.13, 7.15, 7.25,
More informationIndex I-1. in one variable, solution set of, 474 solving by factoring, 473 cubic function definition, 394 graphs of, 394 x-intercepts on, 474
Index A Absolute value explanation of, 40, 81 82 of slope of lines, 453 addition applications involving, 43 associative law for, 506 508, 570 commutative law for, 238, 505 509, 570 English phrases for,
More informationUnit 6 - Simple linear regression
Sta 101: Data Analysis and Statistical Inference Dr. Çetinkaya-Rundel Unit 6 - Simple linear regression LO 1. Define the explanatory variable as the independent variable (predictor), and the response variable
More informationChapter 5 Least Squares Regression
Chapter 5 Least Squares Regression A Royal Bengal tiger wandered out of a reserve forest. We tranquilized him and want to take him back to the forest. We need an idea of his weight, but have no scale!
More informationCorrelation & Simple Regression
Chapter 11 Correlation & Simple Regression The previous chapter dealt with inference for two categorical variables. In this chapter, we would like to examine the relationship between two quantitative variables.
More informationSimple Linear Regression. (Chs 12.1, 12.2, 12.4, 12.5)
10 Simple Linear Regression (Chs 12.1, 12.2, 12.4, 12.5) Simple Linear Regression Rating 20 40 60 80 0 5 10 15 Sugar 2 Simple Linear Regression Rating 20 40 60 80 0 5 10 15 Sugar 3 Simple Linear Regression
More informationLAB 3 INSTRUCTIONS SIMPLE LINEAR REGRESSION
LAB 3 INSTRUCTIONS SIMPLE LINEAR REGRESSION In this lab you will first learn how to display the relationship between two quantitative variables with a scatterplot and also how to measure the strength of
More informationADMS2320.com. We Make Stats Easy. Chapter 4. ADMS2320.com Tutorials Past Tests. Tutorial Length 1 Hour 45 Minutes
We Make Stats Easy. Chapter 4 Tutorial Length 1 Hour 45 Minutes Tutorials Past Tests Chapter 4 Page 1 Chapter 4 Note The following topics will be covered in this chapter: Measures of central location Measures
More informationMath 120 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency
Math 1 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency The word average: is very ambiguous and can actually refer to the mean, median, mode or midrange. Notation:
More informationES-2 Lecture: More Least-squares Fitting. Spring 2017
ES-2 Lecture: More Least-squares Fitting Spring 2017 Outline Quick review of least-squares line fitting (also called `linear regression ) How can we find the best-fit line? (Brute-force method is not efficient)
More informationProbability Distributions
CONDENSED LESSON 13.1 Probability Distributions In this lesson, you Sketch the graph of the probability distribution for a continuous random variable Find probabilities by finding or approximating areas
More informationHow spread out is the data? Are all the numbers fairly close to General Education Statistics
How spread out is the data? Are all the numbers fairly close to General Education Statistics each other or not? So what? Class Notes Measures of Dispersion: Range, Standard Deviation, and Variance (Section
More informationSCATTERPLOTS. We can talk about the correlation or relationship or association between two variables and mean the same thing.
SCATTERPLOTS When we want to know if there is some sort of relationship between 2 numerical variables, we can use a scatterplot. It gives a visual display of the relationship between the 2 variables. Graphing
More informationSTAB22 Statistics I. Lecture 7
STAB22 Statistics I Lecture 7 1 Example Newborn babies weight follows Normal distr. w/ mean 3500 grams & SD 500 grams. A baby is defined as high birth weight if it is in the top 2% of birth weights. What
More informationStatistics 1. Edexcel Notes S1. Mathematical Model. A mathematical model is a simplification of a real world problem.
Statistics 1 Mathematical Model A mathematical model is a simplification of a real world problem. 1. A real world problem is observed. 2. A mathematical model is thought up. 3. The model is used to make
More informationThe empirical ( ) rule
The empirical (68-95-99.7) rule With a bell shaped distribution, about 68% of the data fall within a distance of 1 standard deviation from the mean. 95% fall within 2 standard deviations of the mean. 99.7%
More informationUnit 2: Numerical Descriptive Measures
Unit 2: Numerical Descriptive Measures Summation Notation Measures of Central Tendency Measures of Dispersion Chebyshev's Rule Empirical Rule Measures of Relative Standing Box Plots z scores Jan 28 10:48
More informationAccess Algebra Scope and Sequence
Access Algebra Scope and Sequence Unit 1 Represent data with plots on the real number line (dot plots and histograms). Use statistics appropriate to the shape of the data distribution to compare center
More informationChapter 10 Correlation and Regression
Chapter 10 Correlation and Regression 10-1 Review and Preview 10-2 Correlation 10-3 Regression 10-4 Variation and Prediction Intervals 10-5 Multiple Regression 10-6 Modeling Copyright 2010, 2007, 2004
More informationMatrices, Row Reduction of Matrices
Matrices, Row Reduction of Matrices October 9, 014 1 Row Reduction and Echelon Forms In the previous section, we saw a procedure for solving systems of equations It is simple in that it consists of only
More informationChapter 3 Statistics for Describing, Exploring, and Comparing Data. Section 3-1: Overview. 3-2 Measures of Center. Definition. Key Concept.
Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Overview 3- Measures of Center 3-3 Measures of Variation Section 3-1: Overview Descriptive Statistics summarize or describe the important
More informationSection 5.4. Ken Ueda
Section 5.4 Ken Ueda Students seem to think that being graded on a curve is a positive thing. I took lasers 101 at Cornell and got a 92 on the exam. The average was a 93. I ended up with a C on the test.
More informationSTT 315 This lecture is based on Chapter 2 of the textbook.
STT 315 This lecture is based on Chapter 2 of the textbook. Acknowledgement: Author is thankful to Dr. Ashok Sinha, Dr. Jennifer Kaplan and Dr. Parthanil Roy for allowing him to use/edit some of their
More informationACMS Statistics for Life Sciences. Chapter 11: The Normal Distributions
ACMS 20340 Statistics for Life Sciences Chapter 11: The Normal Distributions Introducing the Normal Distributions The class of Normal distributions is the most widely used variety of continuous probability
More informationScatterplots. STAT22000 Autumn 2013 Lecture 4. What to Look in a Scatter Plot? Form of an Association
Scatterplots STAT22000 Autumn 2013 Lecture 4 Yibi Huang October 7, 2013 21 Scatterplots 22 Correlation (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) (x n, y n ) A scatter plot shows the relationship between two
More informationequal to the of the. Sample variance: Population variance: **The sample variance is an unbiased estimator of the
DEFINITION The variance (aka dispersion aka spread) of a set of values is a measure of equal to the of the. Sample variance: s Population variance: **The sample variance is an unbiased estimator of the
More informationHUDM4122 Probability and Statistical Inference. February 2, 2015
HUDM4122 Probability and Statistical Inference February 2, 2015 Special Session on SPSS Thursday, April 23 4pm-6pm As of when I closed the poll, every student except one could make it to this I am happy
More informationChapter 8. Linear Regression /71
Chapter 8 Linear Regression 1 /71 Homework p192 1, 2, 3, 5, 7, 13, 15, 21, 27, 28, 29, 32, 35, 37 2 /71 3 /71 Objectives Determine Least Squares Regression Line (LSRL) describing the association of two
More information