Class 15. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
|
|
- Vanessa Parsons
- 5 years ago
- Views:
Transcription
1 Class 15 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 17 by D.B. Rowe 1
2 Agenda: Recap Chapter Lecture Chapter 7. Discussion of Chapters Problem Solving Session. Go Over Eam 3.
3 Recap Chapter
4 7. The Sampling Distribution of Sample Means When we take a random sample 1,, n from a population, one of the things that we do is compute the sample mean. The value of is not μ. Each time we take a random sample of size n, we get a different set of values 1,, n and a different value for. There is a distribution of possible s. 4
5 7. The Sampling Distribution of Sample Means Sampling Distribution of a sample statistic: The distribution of values for a sample statistic obtained from repeated samples, all of the same size and all drawn from the same population. If we take a random sample of size n and compute sample mean,. The collection of all possible means is called the sampling distribution of the sample mean (SDSM). 5
6 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. Population data values:,, 4, 6, P( ) 1/ 5 1/ 5 4 1/ 5 6 1/ 5 8 1/ 5 P() possible values [ P( )] 4 [( ) P( )] 8 8 6
7 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n= with replacement. Population data values: ,, 4, 6, 8. (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) 5 possible samples 7
8 5 possible samples. 7. The Sampling Distribution of Sample Means (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) Eample: N=5, values:,, 4, 6, 8, n= (with replacement). Prob. of each samples mean = 1/5 =
9 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). Prob. of each samples mean = 1/5 = , one time 1, two times, three times 3, four times 4, five times 5, four times 6, three times 7, two times 8, one time 5 possible samples. (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) 9
10 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). Prob. of each samples mean = 1/5 = , one time 1, two times, three times 3, four times 4, five times 5, four times 6, three times 7, two times 8, one time 5 possible samples. (,) (,) (4,) (6,) (8,) (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 1
11 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 Represent this distribution function with a histogram. note that intermediate values of 1,3,5,7 are now possible. Figure from Johnson & Kuby, 1. 11
12 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n=1 or (with replacement). P() from n=1 distribution from n= distribution Figure from Johnson & Kuby, 1. 1
13 7. The Sampling Distribution of Sample Means Sample distribution of sample means (SDSM): If all possible random samples, each of size n, are taken from any population with mean μ and standard deviation σ, then the sampling distribution of sample means will have the following: 1. A mean equal to μ. A standard deviation equal to n Furthermore, if the sampled population has a normal distribution, then the sampling distribution of will also be normal for all samples of all sizes. Discuss Later: What if the sampled population does not have a normal distribution? 13
14 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). P( ) (1/ 5) 1( / 5) (3 / 5) 3(4 / 5) 4(5 / 5) 5(4 / 5) 6(3 / 5) 7( / 5) 8(1/ 5) 4 Eactly as from formula! 4 P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 14
15 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n= (with replacement). ( 4) (1/ 5) (1 4) ( / 5) 4 ( 4) (3 / 5) (3 4) (4 / 5) (4 4) (5 / 5) (5 4) (4 / 5) ( ) P( ) (6 4) (3 / 5) (7 4) ( / 5) (8 4) (1/ 5) Eactly as from formula! n P ( ) 1/ 5 P ( 1) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) 1/ 5 15
16 7. The Sampling Distribution of Sample Means Eample: N=5, values:,, 4, 6, 8, n=1 or (with replacement). P(). from n=1 distribution 4, 8 from n= distribution 4, n Figure from Johnson & Kuby, 1. 16
17 Questions? Homework: Chapter 7 # 6, 1, 3, 9, 33, 35 17
18 Lecture Chapter 7. continued 18
19 7. The Sampling Distribution of Sample Means Sample distribution of sample means (SDSM): If all possible random samples, each of size n, are taken from any population with mean μ and standard deviation σ, then the sampling distribution of sample means will have the following: 1. A mean equal to μ. A standard deviation equal to n Furthermore, if the sampled population has a normal distribution, then the sampling distribution of will also be normal for all samples of all sizes. Discuss Later: What if the sampled population does not have a normal distribution? 19
20 7. The Sampling Distribution of Sample Means P()..16 from n=1 distribution 4, 8 from n= distribution 4, ? from n=3 distribution 4, 8 3 Looks like? Figure from Johnson & Kuby, 1. from n=4 distribution 4, 8 4? Looks like? n large? 4 8 Looks like? n
21 7. The Sampling Distribution of Sample Means We have a couple of definitions. Standard error of the mean ( ): The standard deviation of the sampling distribution of sample means. Central Limit Theorem (CLT): The sampling distribution of sample means will more closely resemble the normal distribution as the sample size increases. The CLT is etremely important in Statistics. 1
22 7. The Sampling Distribution of Sample Means The Central Limit Theorem: Assume that we have a population (arbitrary distribution) with mean μ and standard deviation σ. If we take random samples of size n (with replacement), then for large n, the distribution of the sample means, the s, is approimately normally distributed with, n where in general n 3 is sufficiently large, but can be as small as15 or as big as 5 depending upon the shape of distribution!
23 7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: For a discrete distribution P(), we found the mean μ and variance σ from P( ) and ( ) P( ). 3
24 7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: For a discrete distribution P(), we found the mean μ and variance σ from P( ) and ( ) P( ). For a continuous distribution f(), we find the mean μ and variance σ from f ( ) d and ( ) f ( ) d. 4
25 7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: Uniform(a,b) Normal(μ,σ ) f( ) 1 b a f( ) e 1 a μ b μ 5
26 7. The Sampling Distribution of Sample Means AKA Don t need to remember. Aside: Uniform(a,b) Normal(μ,σ ) f( ) 1 b a f( ) e 1 b b a f ( ) d a b ( b a) ( ) f ( ) d 1 a a μ b f ( ) d ( ) f ( ) d μ 6
27 7. The Sampling Distribution of Sample Means b a ( b a) 1 Eample: The random observation comes from either a Uniform(,) population or Normal population μ=1,σ=57.7. Uniform(a=,b=) Normal(μ=1,σ = /1). Generate data,..., 1 n, for n=1,, 3, 4, 5, and 15. Calculate, and repeat one million times. 7
28 7. The Sampling Distribution of Sample Means Eample: The random observation comes from either a Uniform(,) population or Normal population μ=1,σ=57.7. Uniform(a=,b=) Normal(μ=1,σ = /1) f( ) 1.16 b a e f( ) b a ( b a) Generate data,..., 1 n, for n=1,, 3, 4, 5, and 15. Calculate, and repeat one million times. 8 limited range
29 7. The Sampling Distribution of Sample Means According to SDSM, if we had a sample of size n=1,, 3, 4, 5, and 15 from Uniform(a=,b=), then n Sample Size, Mean, SD, ,..., 1 n For Uniform a to b b a ( b a) 1 Theoretical values from the SDSM n 9
30 7. The Sampling Distribution of Sample Means According to SDSM, if we had a sample of size n=1,, 3, 4, 5, and 15 from Normal(μ=1,σ =(57.7) ), then n Sample Size, Mean, SD, ,..., 1 n Theoretical values from the SDSM For Normal b a ( b a) 1 n 3
31 7. The Sampling Distribution of Sample Means I wrote a computer program to generate 1 million random observations,..., 1 1 from the Uniform(a=,b=) and also from Normal(μ=1,σ =(57.7) ) distributions. Show histograms! 31
32 7. The Sampling Distribution of Sample Means Epanded program to generate n million random observations 1,..., n 1 6 from the Uniform(a=,b=) and also from the Normal(μ=1,σ =(57.7) ) distributions, for each of n=1,, 3, 4, 5, and data sets of Uniform and Normal random observations 3
33 7. The Sampling Distribution of Sample Means n= observations scale changes observations are uniform 6 5 observations are normal limited range
34 7. The Sampling Distribution of Sample Means n= 1 6 observations scale changes observations are uniform 1. 1 observations are normal limited range
35 7. The Sampling Distribution of Sample Means n= observations scale changes observations are uniform 1.5 observations are normal 1 1 limited range
36 7. The Sampling Distribution of Sample Means n= observations scale changes observations are uniform.5 observations are normal limited range
37 7. The Sampling Distribution of Sample Means n= observations scale changes observations are uniform 3.5 observations are normal limited range
38 7. The Sampling Distribution of Sample Means n= observations scale changes observations are uniform observations are normal 5 5 limited range
39 7. The Sampling Distribution of Sample Means Sample means and standard deviations from each of the n million observations from the Uniform(a=,b=) and Normal(μ=1,σ =(57.7) ) distributions. 6 data sets of Uniform and Normal 39
40 7. The Sampling Distribution of Sample Means Sample means and standard deviations from each of the n million observations from the Uniform(a=,b=) and Normal(μ=1,σ =(57.7) ) distributions. i.e. n=5, Groups of n=5 Mean of groups 1 1,, 3, 4, data sets of Uniform and Normal , 7 8, 9, ,.,.,., s Histogram of 's 4
41 7. The Sampling Distribution of Sample Means Computed sample means and standard deviations from the one million means,..., for n=1,, 3, 4, 5, and Sample Size n Mean Mean U Mean N SD SD U SD N s s 41
42 7. The Sampling Distribution of Sample Means n= means scale same Histogram of means from uniform Histogram of means from normal s 1 s limited range same classes same classes 4
43 7. The Sampling Distribution of Sample Means n= means scale same Histogram of means from uniform Histogram of means from normal s 1 s limited range same classes same classes 43
44 7. The Sampling Distribution of Sample Means n= means scale same Histogram of means from uniform Histogram of means from normal s 1 s limited range same classes same classes 44
45 7. The Sampling Distribution of Sample Means n= means scale same Histogram of means from uniform Histogram of means from normal 1 1 s s same classes same classes 45
46 7. The Sampling Distribution of Sample Means n= means scale same Histogram of means from uniform Histogram of means from normal 1 1 s s same classes same classes 46
47 7. The Sampling Distribution of Sample Means n= means scale same Histogram of means from uniform Histogram of means from normal 1 1 s s same classes same classes 47
48 7. The Sampling Distribution of Sample Means With a population mean μ and standard deviation σ. Random samples of size n with replacement, for large n, the distribution of the sample means quickly becomes normally distributed with, n Generally n 3 is sufficiently large, but can be as small as15 or as big as 5 depending upon the shape of distribution! 48
49 Lecture Chapter
50 3: Descriptive Analysis and Bivariate Data 3.1 Bivariate Data: Scatter Diagram Our data. Height vs. Weight Recall TuTh 1 n=48 responses Find yourself. If not here then removed () or you did not respond. 5
51 TuTh Application of the Sampling Distribution of Sample Means Assume that this is a population of data. Does this population look normally distributed? Put normal with same μ and σ. Height 34
52 7.3 Application of the Sampling Distribution of Sample Means If I m interested in the mean of a sample of size n=15, then by SDSM and, in addition, n by the CLT is (hopefully) normally distributed. I wrote a computer program to take a sample of n=1,3,5,15 from the population of N heights with replacement 1 6 times. 35
53 7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and Histogram of the 1 million means n=1 TuTh 1 N=48 values Put normal with same and. 36
54 7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and.4. 3 Histogram of the 1 million means n=3 TuTh 1 N=48 values Put normal with same and. 37
55 TuTh Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and Histogram of the 1 million means n=5 N=48 values Put normal with same and. 38
56 7.3 Application of the Sampling Distribution of Sample Means 4. By SDSM, 68.4 and Histogram of the 1 million means n=15 TuTh 1 N=48 values Put normal with same and. By CLT becomes normal 39
57 7.3 Application of the Sampling Distribution of Sample Means Now that we believe that the mean from a sample of n=15 is normally distributed with mean and standard deviation n, we can find probabilities. a b b a P( a b) P( b ) P( a) 4
58 7.3 Application of the Sampling Distribution of Sample Means To find these probabilities, we first convert to z scores a b b a P( a b) P( b ) P( c z d) same area P( d z) same area P( a) P( z c) same area z a b, c, d and use the table in book. 58
59 7.3 Application of the Sampling Distribution of Sample Means Eample: What is probability that sample mean from a random sample of n=15 heights is greater than 7 when 68.4 and 4.? 59
60 Questions? Homework: Chapter 7 # 6, 1, 3, 9, 33, 35 64
61 Discussion: Chapters 65
62 Discussion on Course We re moving into a new phase of the course Part III on Inferential Statistics. Parts I and II were all foundational material for Part III. Before we discussed 66
63 Discussion on Course Part I: Descriptive Statistics Chapter 1: Statistics Background material. Definitions. Chapter : Descriptive Analysis and Presentation of single variable data Graphs, Central Tendency, Dispersion, Position Chapter 3: Descriptive Analysis and Presentation of bivariate data Scatter plot, Correlation, Regression 67
64 Discussion on Course Part II: Probability Chapter 4: Probability Conditional, Rules, Mutually Eclusive, Independent Chapter 5: Probability Distributions (Discrete) Random variables, Probability Distributions, Mean & Variance, Binomial Distribution with Mean & Variance Chapter 6: Probability Distributions (Continuous) Normal Distribution, Standard Normal, Applications, Notation Chapter Sampling Distributions, SDSM, CLT 68
65 Discussion on Course Part III: Inferential Statistics Chapter 8: Introduction to Statistical Inferences Confidence Intervals, Hypothesis testing Chapter 9: Inferences Involving One Population Mean μ (σ unknown), proportion p, variance σ Chapter 1: Inferences Involving Two Populations Difference in means μ 1 -μ, proportions p 1 -p, variances Part IV: More Inferential Statistics Chapter 11: Applications of Chi-Square Chi-square statistics.. We will discuss later. / 1 69
66 Discussion on Course Net Lecture will be on Chapter 8. Chapter 8: Introduction to Statistical Inferences Confidence Intervals and Hypothesis testing 7
67 Go over Eam 3. 71
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
More informationClass 19. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 19 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 8.3-8.4 Lecture Chapter 8.5 Go over Exam. Problem Solving
More informationClass 24. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 4 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 9. and 9.3 Lecture Chapter 10.1-10.3 Review Exam 6 Problem Solving
More informationMarquette University MATH 1700 Class 5 Copyright 2017 by D.B. Rowe
Class 5 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 3.2-3.3 Lecture Chapter 4.1-4.2 Review Chapter 1 3.1 (Exam
More informationClass 26: review for final exam 18.05, Spring 2014
Probability Class 26: review for final eam 8.05, Spring 204 Counting Sets Inclusion-eclusion principle Rule of product (multiplication rule) Permutation and combinations Basics Outcome, sample space, event
More informationIntroduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p.
Preface p. xi Introduction and Descriptive Statistics p. 1 Introduction to Statistics p. 3 Statistics, Science, and Observations p. 5 Populations and Samples p. 6 The Scientific Method and the Design of
More informationFrom Practical Data Analysis with JMP, Second Edition. Full book available for purchase here. About This Book... xiii About The Author...
From Practical Data Analysis with JMP, Second Edition. Full book available for purchase here. Contents About This Book... xiii About The Author... xxiii Chapter 1 Getting Started: Data Analysis with JMP...
More informationAP 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
More informationLecture 30. DATA 8 Summer Regression Inference
DATA 8 Summer 2018 Lecture 30 Regression Inference Slides created by John DeNero (denero@berkeley.edu) and Ani Adhikari (adhikari@berkeley.edu) Contributions by Fahad Kamran (fhdkmrn@berkeley.edu) and
More informationTransition Passage to Descriptive Statistics 28
viii Preface xiv chapter 1 Introduction 1 Disciplines That Use Quantitative Data 5 What Do You Mean, Statistics? 6 Statistics: A Dynamic Discipline 8 Some Terminology 9 Problems and Answers 12 Scales of
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More information14.30 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.
More informationChapter 6: Large Random Samples Sections
Chapter 6: Large Random Samples Sections 6.1: Introduction 6.2: The Law of Large Numbers Skip p. 356-358 Skip p. 366-368 Skip 6.4: The correction for continuity Remember: The Midterm is October 25th in
More informationStatistics and Quantitative Analysis U4320. Segment 5: Sampling and inference Prof. Sharyn O Halloran
Statistics and Quantitative Analysis U4320 Segment 5: Sampling and inference Prof. Sharyn O Halloran Sampling A. Basics 1. Ways to Describe Data Histograms Frequency Tables, etc. 2. Ways to Characterize
More informationExam details. Final Review Session. Things to Review
Exam details Final Review Session Short answer, similar to book problems Formulae and tables will be given You CAN use a calculator Date and Time: Dec. 7, 006, 1-1:30 pm Location: Osborne Centre, Unit
More informationCentral 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
More informationWeek 11 Sample Means, CLT, Correlation
Week 11 Sample Means, CLT, Correlation Slides by Suraj Rampure Fall 2017 Administrative Notes Complete the mid semester survey on Piazza by Nov. 8! If 85% of the class fills it out, everyone will get a
More informationChapter 18. Sampling Distribution Models /51
Chapter 18 Sampling Distribution Models 1 /51 Homework p432 2, 4, 6, 8, 10, 16, 17, 20, 30, 36, 41 2 /51 3 /51 Objective Students calculate values of central 4 /51 The Central Limit Theorem for Sample
More informationDetermining the Spread of a Distribution
Determining the Spread of a Distribution 1.3-1.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Department of Mathematics University of Houston Lecture 3-2311 Lecture 3-2311 1 / 58 Outline 1 Describing Quantitative
More informationDover- Sherborn High School Mathematics Curriculum Probability and Statistics
Mathematics Curriculum A. DESCRIPTION This is a full year courses designed to introduce students to the basic elements of statistics and probability. Emphasis is placed on understanding terminology and
More informationDetermining the Spread of a Distribution
Determining the Spread of a Distribution 1.3-1.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Department of Mathematics University of Houston Lecture 3-2311 Lecture 3-2311 1 / 58 Outline 1 Describing Quantitative
More informationCOMP6053 lecture: Sampling and the central limit theorem. Markus Brede,
COMP6053 lecture: Sampling and the central limit theorem Markus Brede, mb8@ecs.soton.ac.uk Populations: long-run distributions Two kinds of distributions: populations and samples. A population is the set
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 informationFRANKLIN UNIVERSITY PROFICIENCY EXAM (FUPE) STUDY GUIDE
FRANKLIN UNIVERSITY PROFICIENCY EXAM (FUPE) STUDY GUIDE Course Title: Probability and Statistics (MATH 80) Recommended Textbook(s): Number & Type of Questions: Probability and Statistics for Engineers
More informationStatistic: 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
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
More informationChapter 2: Elements of Probability Continued
MSCS6 Chapter : Elements of Probability Continued Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 7 by MSCS6 Agenda.7 Chebyshev s Inequality and the law of Large
More informationBusiness Statistics. Lecture 9: Simple Regression
Business Statistics Lecture 9: Simple Regression 1 On to Model Building! Up to now, class was about descriptive and inferential statistics Numerical and graphical summaries of data Confidence intervals
More informationEcon 6900: Statistical Problems. Instructor: Yogesh Uppal
Econ 6900: Statistical Problems Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 7, Part A Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions
More informationChapter # classifications of unlikely, likely, or very likely to describe possible buying of a product?
A. Attribute data B. Numerical data C. Quantitative data D. Sample data E. Qualitative data F. Statistic G. Parameter Chapter #1 Match the following descriptions with the best term or classification given
More informationBusiness Statistics. Lecture 10: Course Review
Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,
More informationApproximate Linear Relationships
Approximate Linear Relationships In the real world, rarely do things follow trends perfectly. When the trend is expected to behave linearly, or when inspection suggests the trend is behaving linearly,
More information5.6 Asymptotes; Checking Behavior at Infinity
5.6 Asymptotes; Checking Behavior at Infinity checking behavior at infinity DEFINITION asymptote In this section, the notion of checking behavior at infinity is made precise, by discussing both asymptotes
More information20 Hypothesis Testing, Part I
20 Hypothesis Testing, Part I Bob has told Alice that the average hourly rate for a lawyer in Virginia is $200 with a standard deviation of $50, but Alice wants to test this claim. If Bob is right, she
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 informationOne-sample categorical data: approximate inference
One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution
More informationChapter 9 Regression. 9.1 Simple linear regression Linear models Least squares Predictions and residuals.
9.1 Simple linear regression 9.1.1 Linear models Response and eplanatory variables Chapter 9 Regression With bivariate data, it is often useful to predict the value of one variable (the response variable,
More informationHomework #1. Denote the sum we are interested in as To find we subtract the sum to find that
Homework #1 CMSC351 - Spring 2013 PRINT Name : Due: Feb 12 th at the start of class o Grades depend on neatness and clarity. o Write your answers with enough detail about your approach and concepts used,
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 informationSTATISTICS ANCILLARY SYLLABUS. (W.E.F. the session ) Semester Paper Code Marks Credits Topic
STATISTICS ANCILLARY SYLLABUS (W.E.F. the session 2014-15) Semester Paper Code Marks Credits Topic 1 ST21012T 70 4 Descriptive Statistics 1 & Probability Theory 1 ST21012P 30 1 Practical- Using Minitab
More informationIV. The Normal Distribution
IV. The Normal Distribution The normal distribution (a.k.a., a the Gaussian distribution or bell curve ) is the by far the best known random distribution. It s discovery has had such a far-reaching impact
More informationLecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 9.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola Copyright 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9.1-1 Chapter 9 Inferences
More information16.400/453J Human Factors Engineering. Design of Experiments II
J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential
More informationStat 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:
More informationCh. 1: Data and Distributions
Ch. 1: Data and Distributions Populations vs. Samples How to graphically display data Histograms, dot plots, stem plots, etc Helps to show how samples are distributed Distributions of both continuous and
More informationDETAILED CONTENTS PART I INTRODUCTION AND DESCRIPTIVE STATISTICS. 1. Introduction to Statistics
DETAILED CONTENTS About the Author Preface to the Instructor To the Student How to Use SPSS With This Book PART I INTRODUCTION AND DESCRIPTIVE STATISTICS 1. Introduction to Statistics 1.1 Descriptive and
More informationStatistical 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
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 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 informationLecture 7: Hypothesis Testing and ANOVA
Lecture 7: Hypothesis Testing and ANOVA Goals Overview of key elements of hypothesis testing Review of common one and two sample tests Introduction to ANOVA Hypothesis Testing The intent of hypothesis
More informationExample. If 4 tickets are drawn with replacement from ,
Example. If 4 tickets are drawn with replacement from 1 2 2 4 6, what are the chances that we observe exactly two 2 s? Exactly two 2 s in a sequence of four draws can occur in many ways. For example, (
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 informationCourse. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.
Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series
More informationBusiness 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
More informationStatistics and Data Analysis in Geology
Statistics and Data Analysis in Geology 6. Normal Distribution probability plots central limits theorem Dr. Franz J Meyer Earth and Planetary Remote Sensing, University of Alaska Fairbanks 1 2 An Enormously
More informationLast Lecture. Distinguish Populations from Samples. Knowing different Sampling Techniques. Distinguish Parameters from Statistics
Last Lecture Distinguish Populations from Samples Importance of identifying a population and well chosen sample Knowing different Sampling Techniques Distinguish Parameters from Statistics Knowing different
More informationCHAPTER 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
More informationDescriptive Statistics
Descriptive Statistics Summarizing a Single Variable Reference Material: - Prob-stats-review.doc (see Sections 1 & 2) P. Hammett - Lecture Eercise: desc-stats.ls 1 Topics I. Discrete and Continuous Measurements
More informationMATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Ramesh Yapalparvi Week 2 Chapter 4 Bivariate Data Data with two/paired variables, Pearson correlation coefficient and its properties, general variance sum law Chapter 6
More informationCHAPTER 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
More informationLecture 8 Sampling Theory
Lecture 8 Sampling Theory Thais Paiva STA 111 - Summer 2013 Term II July 11, 2013 1 / 25 Thais Paiva STA 111 - Summer 2013 Term II Lecture 8, 07/11/2013 Lecture Plan 1 Sampling Distributions 2 Law of Large
More informationBackground to Statistics
FACT SHEET Background to Statistics Introduction Statistics include a broad range of methods for manipulating, presenting and interpreting data. Professional scientists of all kinds need to be proficient
More informationCOMP6053 lecture: Sampling and the central limit theorem. Jason Noble,
COMP6053 lecture: Sampling and the central limit theorem Jason Noble, jn2@ecs.soton.ac.uk Populations: long-run distributions Two kinds of distributions: populations and samples. A population is the set
More informationBNG 495 Capstone Design. Descriptive Statistics
BNG 495 Capstone Design Descriptive Statistics Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential statistical methods, with a focus
More informationIV. The Normal Distribution
IV. The Normal Distribution The normal distribution (a.k.a., the Gaussian distribution or bell curve ) is the by far the best known random distribution. It s discovery has had such a far-reaching impact
More informationIntroduction to Statistics and Error Analysis II
Introduction to Statistics and Error Analysis II Physics116C, 4/14/06 D. Pellett References: Data Reduction and Error Analysis for the Physical Sciences by Bevington and Robinson Particle Data Group notes
More informationChapter 24. Comparing Means. Copyright 2010 Pearson Education, Inc.
Chapter 24 Comparing Means Copyright 2010 Pearson Education, Inc. Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side-by-side. For example:
More informationSampling Distributions of Statistics Corresponds to Chapter 5 of Tamhane and Dunlop
Sampling Distributions of Statistics Corresponds to Chapter 5 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT), with some slides by Jacqueline Telford (Johns Hopkins University) 1 Sampling
More informationOverview. Overview. Overview. Specific Examples. General Examples. Bivariate Regression & Correlation
Bivariate Regression & Correlation Overview The Scatter Diagram Two Examples: Education & Prestige Correlation Coefficient Bivariate Linear Regression Line SPSS Output Interpretation Covariance ou already
More informationUNIT 4 MATHEMATICAL METHODS SAMPLE REFERENCE MATERIALS
UNIT 4 MATHEMATICAL METHODS SAMPLE REFERENCE MATERIALS EXTRACTS FROM THE ESSENTIALS EXAM REVISION LECTURES NOTES THAT ARE ISSUED TO STUDENTS Students attending our mathematics Essentials Year & Eam Revision
More informationA3. 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
More informationDistribution-Free Procedures (Devore Chapter Fifteen)
Distribution-Free Procedures (Devore Chapter Fifteen) MATH-5-01: Probability and Statistics II Spring 018 Contents 1 Nonparametric Hypothesis Tests 1 1.1 The Wilcoxon Rank Sum Test........... 1 1. Normal
More informationStatistics Introductory Correlation
Statistics Introductory Correlation Session 10 oscardavid.barrerarodriguez@sciencespo.fr April 9, 2018 Outline 1 Statistics are not used only to describe central tendency and variability for a single variable.
More informationMFin Econometrics I Session 4: t-distribution, Simple Linear Regression, OLS assumptions and properties of OLS estimators
MFin Econometrics I Session 4: t-distribution, Simple Linear Regression, OLS assumptions and properties of OLS estimators Thilo Klein University of Cambridge Judge Business School Session 4: Linear regression,
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 informationElementary Statistics Triola, Elementary Statistics 11/e Unit 17 The Basics of Hypotheses Testing
(Section 8-2) Hypotheses testing is not all that different from confidence intervals, so let s do a quick review of the theory behind the latter. If it s our goal to estimate the mean of a population,
More informationCorrelation and regression
NST 1B Experimental Psychology Statistics practical 1 Correlation and regression Rudolf Cardinal & Mike Aitken 11 / 12 November 2003 Department of Experimental Psychology University of Cambridge Handouts:
More informationØ Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.
Statistical Tools in Evaluation HPS 41 Fall 213 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific
More informationDefinition 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
More informationBayesian Updating with Continuous Priors Class 13, Jeremy Orloff and Jonathan Bloom
Bayesian Updating with Continuous Priors Class 3, 8.05 Jeremy Orloff and Jonathan Bloom Learning Goals. Understand a parameterized family of distributions as representing a continuous range of hypotheses
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 informationRegression Analysis. BUS 735: Business Decision Making and Research
Regression Analysis BUS 735: Business Decision Making and Research 1 Goals and Agenda Goals of this section Specific goals Learn how to detect relationships between ordinal and categorical variables. Learn
More informationMultiple Linear Regression estimation, testing and checking assumptions
Multiple Linear Regression estimation, testing and checking assumptions Lecture No. 07 Example 1 The president of a large chain of fast-food restaurants has randomly selected 10 franchises and recorded
More information1 Descriptive statistics. 2 Scores and probability distributions. 3 Hypothesis testing and one-sample t-test. 4 More on t-tests
Overall Overview INFOWO Statistics lecture S3: Hypothesis testing Peter de Waal Department of Information and Computing Sciences Faculty of Science, Universiteit Utrecht 1 Descriptive statistics 2 Scores
More informationReview. One-way ANOVA, I. What s coming up. Multiple comparisons
Review One-way ANOVA, I 9.07 /15/00 Earlier in this class, we talked about twosample z- and t-tests for the difference between two conditions of an independent variable Does a trial drug work better than
More informationAn inferential procedure to use sample data to understand a population Procedures
Hypothesis Test An inferential procedure to use sample data to understand a population Procedures Hypotheses, the alpha value, the critical region (z-scores), statistics, conclusion Two types of errors
More information5.1 Introduction. # of successes # of trials. 5.2 Part 1: Maximum Likelihood. MTH 452 Mathematical Statistics
MTH 452 Mathematical Statistics Instructor: Orlando Merino University of Rhode Island Spring Semester, 2006 5.1 Introduction An Experiment: In 10 consecutive trips to the free throw line, a professional
More informationThe Normal Distribution. MDM4U Unit 6 Lesson 2
The Normal Distribution MDM4U Unit 6 Lesson 2 Normal Distributions Many data sets display similar characteristics The normal distribution is a way of describing a certain kind of "ideal" data set Although
More informationPsych Jan. 5, 2005
Psych 124 1 Wee 1: Introductory Notes on Variables and Probability Distributions (1/5/05) (Reading: Aron & Aron, Chaps. 1, 14, and this Handout.) All handouts are available outside Mija s office. Lecture
More informationOne-factor analysis of variance (ANOVA)
One-factor analysis of variance (ANOVA) March 1, 2017 psych10.stanford.edu Announcements / Action Items Schedule update: final R lab moved to Week 10 Optional Survey 5 coming soon, due on Saturday Last
More informationMath Review. Daniel B. Rowe, Ph.D. Professor Department of Mathematics, Statistics, and Computer Science. Copyright 2018 by D.B.
Math Review Daniel B. Rowe, Ph.D. Professor Department of Mathematics, Statistics, and Computer Science Copyright 2018 by 1 Outline Differentiation Definition Analytic Approach Numerical Approach Integration
More informationBusiness 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
More information1/11/2011. Chapter 4: Variability. Overview
Chapter 4: Variability Overview In statistics, our goal is to measure the amount of variability for a particular set of scores, a distribution. In simple terms, if the scores in a distribution are all
More informationECE 510 Lecture 6 Confidence Limits. Scott Johnson Glenn Shirley
ECE 510 Lecture 6 Confidence Limits Scott Johnson Glenn Shirley Concepts 28 Jan 2013 S.C.Johnson, C.G.Shirley 2 Statistical Inference Population True ( population ) value = parameter Sample Sample value
More information6.6 General Form of the Equation for a Linear Relation
6.6 General Form of the Equation for a Linear Relation FOCUS Relate the graph of a line to its equation in general form. We can write an equation in different forms. y 0 6 5 y 10 = 0 An equation for this
More informationSection - 9 GRAPHS. (a) y f x (b) y f x. (c) y f x (d) y f x. (e) y f x (f) y f x k. (g) y f x k (h) y kf x. (i) y f kx. [a] y f x to y f x
44 Section - 9 GRAPHS In this section, we will discuss graphs and graph-plotting in more detail. Detailed graph plotting also requires a knowledge of derivatives. Here, we will be discussing some general
More informationGlossary. The ISI glossary of statistical terms provides definitions in a number of different languages:
Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the
More informationThe t-distribution. Patrick Breheny. October 13. z tests The χ 2 -distribution The t-distribution Summary
Patrick Breheny October 13 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction Introduction What s wrong with z-tests? So far we ve (thoroughly!) discussed how to carry out hypothesis
More informationMath 20 Spring 2005 Final Exam Practice Problems (Set 2)
Math 2 Spring 2 Final Eam Practice Problems (Set 2) 1. Find the etreme values of f(, ) = 2 2 + 3 2 4 on the region {(, ) 2 + 2 16}. 2. Allocation of Funds: A new editor has been allotted $6, to spend on
More informationThe Normal Distribution
The Mary Lindstrom (Adapted from notes provided by Professor Bret Larget) February 10, 2004 Statistics 371 Last modified: February 11, 2004 The The (AKA Gaussian Distribution) is our first distribution
More information