MA 1125 Lecture 05 - Measures of Spread. Wednesday, September 6, Objectives: Introduce variance, standard deviation, range.
|
|
- George Robertson
- 6 years ago
- Views:
Transcription
1 MA 115 Lecture 05 - Measures of Spread Wednesday, September 6, 017 Objectives: Introduce variance, standard deviation, range. 1. Measures of Spread In Lecture 04, we looked at several measures of central tendency. In an attempt to describe the numbers in a set, we used the mean, the median, the mode, or the midrange, as single numbers that represented the entire data set. Each of these is a single number that represents all of the numbers. For example, if we know that the mean weight of a certain breed of dog is 17.5 pounds, then we know that we re talking about relatively small dogs. The mean doesn t tell us whether a dog of this breed weighing 5 pounds is unusual or not, however. I saw on the news that the median price of a house in the United States just went over $00,000. It s certainly clear that houses cost a lot now, but it does not tell us how many houses go for under $100,000, for example. Given a data set, knowing the mean, the median, the mode, or the midrange, gives us a good start on understanding how big the numbers in the set are. If we would like a little bit more information, knowing how spread out the numbers are would go a long way. Today, we ll look at some measures of spread.. The range I m mostly going to focus on the mean, but I d like to look at one of the other measures of central tendency first as an example alternative. We looked at the midrange last time, and that is the number that is halfway between the smallest and largest values in the data set. Suppose we want to build a garage, but we have no idea about whether we could afford it or not. To check this out, we ask for a number of quotes, and we got (1) $16,000 $,000 $17,000 $19,000 $1,000 $0,000 It s easy to compute the midrange, we just find the average of the smallest and largest numbers. () midrange = $16,000 + $,000 = $19,000. The range is the difference between the smallest and largest numbers. (3) range = $,000 $16, 000 = $6,000. 1
2 If someone were to ask us how much garages cost, we could say, Well according to my research, the midrange is about $19,000 and the range is about $6,000. If you would like this kind of garage, you could plan on spending fairly close to $19, Deviations from the mean The midrange and range are easy to compute, but they can be easily misled. Of the measures of central tendency, the mean lends itself well to mathematical analysis, and there is a lot we can do with it. Most of the rest of the class will be devoted to extending the information given by the mean. Right now, we re interested in measures of spread. Since the mean is, in some sense, in the middle, we re going to look at how far all the other numbers are from the mean, or in fancy statistical language, at the deviations from the mean. If x represents a number in a sample, its deviation from the mean is (4) deviation from the mean = x x. In a population, we have a different symbol for the mean, so the deviation from the mean in a population is (5) deviation from the mean = x µ. If we know the mean for a sample and all the deviations from the mean, we can actually figure out all numbers in the data set. This can be a useful way of looking at the data set, but it s not really much simpler. We d like a single number that describes how big the deviations are. One idea is to take the average of all the deviations from the mean. That is, take the mean of the deviations from the mean. 4. The variance Whenever you take the average of the deviations from the mean, you will always get zero. As a result, the mean deviation from the mean is not a useful measure of spread. We could, if we wanted to, just make all the deviations from the mean positive by using absolute values. We could then take the average of these numbers. That s a great idea, but I ve never seen it used. I m not exactly sure why, but I know that absolute values can be awkward mathematically, and I think the way we re going to solve this problem contains more information.
3 MA 115 Lecture 05 - Measures of Spread 3 We need to get rid of the negative signs in the deviations from the mean somehow, and we re going to do that by squaring them. That may sound odd, but it ends up working quite well. To combine the deviations from the mean into a single number, we re going to do the following. We ll do it for a sample first, then for an entire population. Compute the deviations from the mean (6) x x. Square the deviations to make them positive (or zero) (7) (x x). Then we re going to find the mean for the deviations squared, which means add them up and divide by how many there are () s = (x x) n 1 Two things should look odd. First, the s. This is the symbol for the sample variance. Second, we re dividing by n 1 instead of n. Here s my explanation: The mean, on average, is one of the numbers in the data set, so one of the deviations from the mean is zero, on average, so we re really computing the average deviation for the other numbers. That may or may not make sense, but equation () is the standard formula for a sample variance. Let s compute the variance for the sample in problem 6. I strongly suggest working in a table, as I am going to demonstrate. We ll do this a lot. When we see a Σ in a formula, that will mean that we re going to add up a column in the table. OK. Our table starts off as follows. (9)
4 4 First we compute the mean. This entails computing x, so we ll sum over the first column. After that, we divide by n to get x. (10) x = 50 5 = 10 Once we know the mean, we subtract it from all the x s as formula () tells us. (11) x = 50 5 = 10 Next, we square all the deviations from the mean to make them positive. (1) x = 50 5 = 10
5 MA 115 Lecture 05 - Measures of Spread 5 Finally, we sum over the deviations from the mean and divide by n 1, which is 5 1 = 4 in this case. (13) x = 50 5 = 10 s = 14 4 = 3.5 The variance is s = The standard deviation The variance is a measure of spread. If you get a larger number for s, then this says that the data set is more spread out. More specifically, it s hard to tell what the number means exactly, however. We ll end up using a different number a little more often, but even this other measure of spread needs a lot of mathematical analysis to understand it well. Since we ve squared the deviations to compute the variance, the variance is not quite in line with the sizes of the individual deviations. To compensate, we ll mostly work with something called the standard deviation. The standard deviation, s, is simply the square root of the variance. (14) s = s. This formula looks a bit odd, but we compute the variance first, and then take the square root to get the standard deviation. For the set {,, 1, 11,11 }, we got a variance of s = 3.5. The standard deviation is the square root of this, so (15) s = s = 3.5 = Quiz 05, Part I of I Find the standard deviation for the sample { 3, 5, 6, 6 }.
6 6 7. Population variance and standard deviation In our discussion about the variance and standard deviation, we ve only talked in terms of a sample. The variance and standard deviation for a population are computed in pretty much the same way. These are parameters, of course, and we will use the lowercase Greek letter sigma, σ, instead of the s. The population variance is (16) σ = (x µ). N The population mean µ is used here, but the variance σ is still the average of the deviations from the mean squared. We don t have N 1 in the denominator, just N. In practice, a population size N is going to be a really big number, so subtracting 1 doesn t matter much. The population standard deviation, σ, is again just the square root of the variance. (17) σ = σ.. Homework 05 For problems 1-3, suppose the numbers came out a little differently in our garage survey, and we got (1) $1,000 $5,000 $4,000 $5,000 $4,000 $5, What is the midrange?. What is the range? 3. Does the midrange and range describe the data set in problem 1 very well? (That is, are most of the numbers about the same as the midrange, and are most of the numbers as spread out as the range indicates?) For problems 4 and 5, the questions are general ones about the deviation from the mean. 4. If the deviation from the mean for a number x is positive, is x larger than the mean or smaller? 5. If the deviation from the mean for x is negative, is x larger than the mean or smaller?
7 MA 115 Lecture 05 - Measures of Spread 7 For problems 6-14, work with the sample {,, 6, 5, 7, 4, 3 }. You should put your numbers into a table that looks like (19) Count the number of elements in this set. So n =? 7. Find x.. What is the deviation from the mean for x =? 9. What is the deviation from the mean for x =? 10. What is the deviation from the mean squared (i.e., what is (x x) ) for x =? 11. What is the deviation from the mean squared for x =? 1. What did you get for (x x)? 13. What is s? Round your answer correctly to two decimal places. 14. What is the standard deviation? Round correctly to two decimal places. For problems 15-17, work with the sample {, 5, 4, 5 }. 15. Find x. 16. Find s. 17. Find s. Round your answer correctly to two decimal places. 1. In general, what is σ a symbol for? The population 19. Is σ a statistic or a parameter? 0. Suppose a population variance is What is σ? Round your answer correctly to two decimal places. Answers on next page
8 Quiz Answers (0) x = 0 4 = 5 s = 6 3 = s = 1.41 That is, x = 5, s =, and s = HW Answers 1) $1,000+$5,000 = $1,500. ) $5,000 $1,000 = $13,000. 3) The $1,000 distorts the range and midrange. The prices are mostly $4,000 or $5,000, and they don t vary very much. 4) Positive deviations from the mean go with x s that are larger than x. 5) Negative deviations go with x s that are smaller than x. 6) n = 7. 7) x = 5. ) 3 (corrected /3/14 at 1:30). 9) 3. 10) 9. 11) 9. 1) (x x) =. 13) s = Note: You divide by 6! 14) s = ) x = 4.
9 MA 115 Lecture 05 - Measures of Spread 9 16) s = (Divide by 3.) 17) s = ) σ is the population variance. 19) σ, the population standard deviation is a parameter. 0) σ = 3.05 = 1.75.
Statistics vs. statistics
Statistics vs. statistics Question: What is Statistics (with a capital S)? Definition: Statistics is the science of collecting, organizing, summarizing and interpreting data. Note: There are 2 main ways
More informationMA 1125 Lecture 12 - Mean and Standard Deviation for the Binomial Distribution. Objectives: Mean and standard deviation for the binomial distribution.
MA 5 Lecture - Mean and Standard Deviation for the Binomial Distribution Friday, September 9, 07 Objectives: Mean and standard deviation for the binomial distribution.. Mean and Standard Deviation of the
More informationMA 1125 Lecture 14 - Expected Values. Wednesday, October 4, Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Wednesday, October 4, 27 Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationBiostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras
Biostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras Lecture - 05 Normal Distribution So far we have looked at discrete distributions
More information1. Confidence Intervals (cont.)
Math 1125-Introductory Statistics Lecture 23 11/1/06 1. Confidence Intervals (cont.) Let s review. We re in a situation, where we don t know µ, but we have a number from a normal population, either an
More informationNumerical Descriptive Measures. Measures of Center: Mean and Median
Steve Sawin Statistics Numerical Descriptive Measures Having seen the shape of a distribution by looking at the histogram, the two most obvious questions to ask about the specific distribution is where
More informationChapter 6. y y. Standardizing with z-scores. Standardizing with z-scores (cont.)
Starter Ch. 6: A z-score Analysis Starter Ch. 6 Your Statistics teacher has announced that the lower of your two tests will be dropped. You got a 90 on test 1 and an 85 on test 2. You re all set to drop
More informationBoth the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need.
Both the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need. For exams (MD1, MD2, and Final): You may bring one 8.5 by 11 sheet of
More informationChapter 4 Variability
Chapter 4 Variability PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J Gravetter and Larry B. Wallnau Chapter 4 Learning Outcomes 1 2 3 4 5
More informationMA 1125 Lecture 18 - Normal Approximations to Binomial Distributions. Objectives: Compute probabilities for a binomial as a normal distribution.
MA 25 Lecture 8 - Normal Approximations to Binomial Distributions Friday, October 3, 207 Objectives: Compute probabilities for a binomial as a normal distribution.. Normal Approximations to the Binomial
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationIOP 201-Q (Industrial Psychological Research) Tutorial 5
IOP 201-Q (Industrial Psychological Research) Tutorial 5 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. To establish a cause-and-effect relation between two variables,
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationVertical Asymptotes. We generally see vertical asymptotes in the graph of a function when we divide by zero. For example, in the function
MA 223 Lecture 26 - Behavior Around Vertical Asymptotes Monday, April 9, 208 Objectives: Explore middle behavior around vertical asymptotes. Vertical Asymptotes We generally see vertical asymptotes in
More informationCSC Advanced Scientific Programming, Spring Descriptive Statistics
CSC 223 - Advanced Scientific Programming, Spring 2018 Descriptive Statistics Overview Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions.
More informationMeasures of Variation. Section 2-5. Dotplots of Waiting Times. Waiting Times of Bank Customers at Different Banks in minutes. Bank of Providence
Measures of Variation Section -5 1 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank 6.5 6.6 6.7 6.8 7.1 7.3 7.4 Bank of Providence 4. 5.4 5.8 6. 6.7 8.5 9.3 10.0 Mean
More informationDescriptive Statistics: Measures of Central Tendency and Crosstabulation. 789mct_dispersion_asmp.pdf
789mct_dispersion_asmp.pdf Michael Hallstone, Ph.D. hallston@hawaii.edu Lectures 7-9: Measures of Central Tendency, Dispersion, and Assumptions Lecture 7: Descriptive Statistics: Measures of Central Tendency
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationEvery data set has an average and a standard deviation, given by the following formulas,
Discrete Data Sets A data set is any collection of data. For example, the set of test scores on the class s first test would comprise a data set. If we collect a sample from the population we are interested
More informationDescriptive Statistics (Devore Chapter One)
Descriptive Statistics (Devore Chapter One) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 0 Perspective 1 1 Pictorial and Tabular Descriptions of Data 2 1.1 Stem-and-Leaf
More informationChapter 3 Descriptive Statistics: Numerical Measures Part A
Slides Prepared by JOHN S. LOUCKS St. Edward s University Slide 1 Chapter 3 Descriptive Statistics: Numerical Measures Part A Measures of Location Measures of Variability Slide Measures of Location Mean
More informationECO155L19.doc 1 OKAY SO WHAT WE WANT TO DO IS WE WANT TO DISTINGUISH BETWEEN NOMINAL AND REAL GROSS DOMESTIC PRODUCT. WE SORT OF
ECO155L19.doc 1 OKAY SO WHAT WE WANT TO DO IS WE WANT TO DISTINGUISH BETWEEN NOMINAL AND REAL GROSS DOMESTIC PRODUCT. WE SORT OF GOT A LITTLE BIT OF A MATHEMATICAL CALCULATION TO GO THROUGH HERE. THESE
More informationMidterm Test 1 (Sample) Student Name (PRINT):... Student Signature:... Use pencil, so that you can erase and rewrite if necessary.
MA 180/418 Midterm Test 1 (Sample) Student Name (PRINT):............................................. Student Signature:................................................... Use pencil, so that you can erase
More information19. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE
19. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which
More informationSince his score is positive, he s above average. Since his score is not close to zero, his score is unusual.
Chapter 06: The Standard Deviation as a Ruler and the Normal Model This is the worst chapter title ever! This chapter is about the most important random variable distribution of them all the normal distribution.
More informationL04: Homework Answer Key
L04: Homework Answer Key Instructions: You are encouraged to collaborate with other students on the homework, but it is important that you do your own work. Before working with someone else on the assignment,
More informationWe use probability distributions to represent the distribution of a discrete random variable.
Now we focus on discrete random variables. We will look at these in general, including calculating the mean and standard deviation. Then we will look more in depth at binomial random variables which are
More informationPercents, Explained By Mr. Peralta and the Class of 622 and 623
Percents, Eplained By Mr. Peralta and the Class of 622 and 623 Table of Contents Section 1 Finding the New Amount if You Start With the Original Amount Section 2 Finding the Original Amount if You Start
More informationLecture 18 Section Mon, Feb 16, 2009
The s the Lecture 18 Section 5.3.4 Hampden-Sydney College Mon, Feb 16, 2009 Outline The s the 1 2 3 The 4 s 5 the 6 The s the Exercise 5.12, page 333. The five-number summary for the distribution of income
More informationLecture 18 Section Mon, Sep 29, 2008
The s the Lecture 18 Section 5.3.4 Hampden-Sydney College Mon, Sep 29, 2008 Outline The s the 1 2 3 The 4 s 5 the 6 The s the Exercise 5.12, page 333. The five-number summary for the distribution of income
More informationSynthetic Positions. OptionsUniversity TM. Synthetic Positions
When we talk about the term Synthetic, we have a particular definition in mind. That definition is: to fabricate and combine separate elements to form a coherent whole. When we apply that definition to
More informationWhat s Normal? Chapter 8. Hitting the Curve. In This Chapter
Chapter 8 What s Normal? In This Chapter Meet the normal distribution Standard deviations and the normal distribution Excel s normal distribution-related functions A main job of statisticians is to estimate
More informationFinance 197. Simple One-time Interest
Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for
More informationPart 10: The Binomial Distribution
Part 10: The Binomial Distribution The binomial distribution is an important example of a probability distribution for a discrete random variable. It has wide ranging applications. One readily available
More informationx-intercepts, asymptotes, and end behavior together
MA 2231 Lecture 27 - Sketching Rational Function Graphs Wednesday, April 11, 2018 Objectives: Explore middle behavior around x-intercepts, and the general shapes for rational functions. x-intercepts, asymptotes,
More informationBasic Procedure for Histograms
Basic Procedure for Histograms 1. Compute the range of observations (min. & max. value) 2. Choose an initial # of classes (most likely based on the range of values, try and find a number of classes that
More information2 DESCRIPTIVE STATISTICS
Chapter 2 Descriptive Statistics 47 2 DESCRIPTIVE STATISTICS Figure 2.1 When you have large amounts of data, you will need to organize it in a way that makes sense. These ballots from an election are rolled
More informationThe Standard Deviation as a Ruler and the Normal Model. Copyright 2009 Pearson Education, Inc.
The Standard Deviation as a Ruler and the Normal Mol Copyright 2009 Pearson Education, Inc. The trick in comparing very different-looking values is to use standard viations as our rulers. The standard
More informationA CLEAR UNDERSTANDING OF THE INDUSTRY
A CLEAR UNDERSTANDING OF THE INDUSTRY IS CFA INSTITUTE INVESTMENT FOUNDATIONS RIGHT FOR YOU? Investment Foundations is a certificate program designed to give you a clear understanding of the investment
More informationTHE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management
THE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management BA 386T Tom Shively PROBABILITY CONCEPTS AND NORMAL DISTRIBUTIONS The fundamental idea underlying any statistical
More informationReal Estate Private Equity Case Study 3 Opportunistic Pre-Sold Apartment Development: Waterfall Returns Schedule, Part 1: Tier 1 IRRs and Cash Flows
Real Estate Private Equity Case Study 3 Opportunistic Pre-Sold Apartment Development: Waterfall Returns Schedule, Part 1: Tier 1 IRRs and Cash Flows Welcome to the next lesson in this Real Estate Private
More informationBut suppose we want to find a particular value for y, at which the probability is, say, 0.90? In other words, we want to figure out the following:
More on distributions, and some miscellaneous topics 1. Reverse lookup and the normal distribution. Up until now, we wanted to find probabilities. For example, the probability a Swedish man has a brain
More informationElementary Statistics
Chapter 7 Estimation Goal: To become familiar with how to use Excel 2010 for Estimation of Means. There is one Stat Tool in Excel that is used with estimation of means, T.INV.2T. Open Excel and click on
More informationChapter 3. Numerical Descriptive Measures. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 1
Chapter 3 Numerical Descriptive Measures Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 1 Objectives In this chapter, you learn to: Describe the properties of central tendency, variation, and
More informationTi 83/84. Descriptive Statistics for a List of Numbers
Ti 83/84 Descriptive Statistics for a List of Numbers Quiz scores in a (fictitious) class were 10.5, 13.5, 8, 12, 11.3, 9, 9.5, 5, 15, 2.5, 10.5, 7, 11.5, 10, and 10.5. It s hard to get much of a sense
More informationDavid Tenenbaum GEOG 090 UNC-CH Spring 2005
Simple Descriptive Statistics Review and Examples You will likely make use of all three measures of central tendency (mode, median, and mean), as well as some key measures of dispersion (standard deviation,
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationHPM Module_2_Breakeven_Analysis
HPM Module_2_Breakeven_Analysis Hello, class. This is the tutorial for the breakeven analysis module. And this is module 2. And so we're going to go ahead and work this breakeven analysis. I want to give
More informationLaw of Large Numbers, Central Limit Theorem
November 14, 2017 November 15 18 Ribet in Providence on AMS business. No SLC office hour tomorrow. Thursday s class conducted by Teddy Zhu. November 21 Class on hypothesis testing and p-values December
More informationChapter 5: Summarizing Data: Measures of Variation
Chapter 5: Introduction One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance in statistics.
More informationSTAB22 section 1.3 and Chapter 1 exercises
STAB22 section 1.3 and Chapter 1 exercises 1.101 Go up and down two times the standard deviation from the mean. So 95% of scores will be between 572 (2)(51) = 470 and 572 + (2)(51) = 674. 1.102 Same idea
More informationChapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
More informationMeasures of Center. Mean. 1. Mean 2. Median 3. Mode 4. Midrange (rarely used) Measure of Center. Notation. Mean
Measure of Center Measures of Center The value at the center or middle of a data set 1. Mean 2. Median 3. Mode 4. Midrange (rarely used) 1 2 Mean Notation The measure of center obtained by adding the values
More informationIn this example, we cover how to discuss a sell-side divestiture transaction in investment banking interviews.
Breaking Into Wall Street Investment Banking Interview Guide Sample Deal Discussion #1 Sell-Side Divestiture Transaction Narrator: Hello everyone, and welcome to our first sample deal discussion. In this
More informationMeasure of Variation
Measure of Variation Variation is the spread of a data set. The simplest measure is the range. Range the difference between the maximum and minimum data entries in the set. To find the range, the data
More informationProblem Set 6. I did this with figure; bar3(reshape(mean(rx),5,5) );ylabel( size ); xlabel( value ); mean mo return %
Business 35905 John H. Cochrane Problem Set 6 We re going to replicate and extend Fama and French s basic results, using earlier and extended data. Get the 25 Fama French portfolios and factors from the
More informationNumerical Descriptions of Data
Numerical Descriptions of Data Measures of Center Mean x = x i n Excel: = average ( ) Weighted mean x = (x i w i ) w i x = data values x i = i th data value w i = weight of the i th data value Median =
More informationCHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
More informationEconS Utility. Eric Dunaway. Washington State University September 15, 2015
EconS 305 - Utility Eric Dunaway Washington State University eric.dunaway@wsu.edu September 15, 2015 Eric Dunaway (WSU) EconS 305 - Lecture 10 September 15, 2015 1 / 38 Introduction Last time, we saw how
More informationThe figures in the left (debit) column are all either ASSETS or EXPENSES.
Correction of Errors & Suspense Accounts. 2008 Question 7. Correction of Errors & Suspense Accounts is pretty much the only topic in Leaving Cert Accounting that requires some knowledge of how T Accounts
More informationStat 5303 (Oehlert): Power and Sample Size 1
Stat 5303 (Oehlert): Power and Sample Size 1 Cmd> # The Stat5303 package includes two functions of use here: power.anova.test() and sample.size.anova(). You won t be surprised to learn that the first computes
More informationBasic Data Analysis. Stephen Turnbull Business Administration and Public Policy Lecture 3: April 25, Abstract
Basic Data Analysis Stephen Turnbull Business Administration and Public Policy Lecture 3: April 25, 2013 Abstract Review summary statistics and measures of location. Discuss the placement exam as an exercise
More informationClub Accounts - David Wilson Question 6.
Club Accounts - David Wilson. 2011 Question 6. Anyone familiar with Farm Accounts or Service Firms (notes for both topics are back on the webpage you found this on), will have no trouble with Club Accounts.
More informationP1: TIX/XYZ P2: ABC JWST JWST075-Goos June 6, :57 Printer Name: Yet to Come. A simple comparative experiment
1 A simple comparative experiment 1.1 Key concepts 1. Good experimental designs allow for precise estimation of one or more unknown quantities of interest. An example of such a quantity, or parameter,
More information6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 23
6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 23 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare
More information3.1 Measures of Central Tendency
3.1 Measures of Central Tendency n Summation Notation x i or x Sum observation on the variable that appears to the right of the summation symbol. Example 1 Suppose the variable x i is used to represent
More informationDiscrete Probability Distribution
1 Discrete Probability Distribution Key Definitions Discrete Random Variable: Has a countable number of values. This means that each data point is distinct and separate. Continuous Random Variable: Has
More informationWhen we look at a random variable, such as Y, one of the first things we want to know, is what is it s distribution?
Distributions 1. What are distributions? When we look at a random variable, such as Y, one of the first things we want to know, is what is it s distribution? In other words, if we have a large number of
More informationEconS Constrained Consumer Choice
EconS 305 - Constrained Consumer Choice Eric Dunaway Washington State University eric.dunaway@wsu.edu September 21, 2015 Eric Dunaway (WSU) EconS 305 - Lecture 12 September 21, 2015 1 / 49 Introduction
More informationThe Two-Sample Independent Sample t Test
Department of Psychology and Human Development Vanderbilt University 1 Introduction 2 3 The General Formula The Equal-n Formula 4 5 6 Independence Normality Homogeneity of Variances 7 Non-Normality Unequal
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationClass 11. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 11 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 5.3 continued Lecture 6.1-6.2 Go over Eam 2. 2 5: Probability
More informationProbability and Stochastics for finance-ii Prof. Joydeep Dutta Department of Humanities and Social Sciences Indian Institute of Technology, Kanpur
Probability and Stochastics for finance-ii Prof. Joydeep Dutta Department of Humanities and Social Sciences Indian Institute of Technology, Kanpur Lecture - 07 Mean-Variance Portfolio Optimization (Part-II)
More informationChapter 6 Confidence Intervals
Chapter 6 Confidence Intervals Section 6-1 Confidence Intervals for the Mean (Large Samples) VOCABULARY: Point Estimate A value for a parameter. The most point estimate of the population parameter is the
More informationThe following content is provided under a Creative Commons license. Your support
MITOCW Recitation 6 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make
More informationBINARY OPTIONS: A SMARTER WAY TO TRADE THE WORLD'S MARKETS NADEX.COM
BINARY OPTIONS: A SMARTER WAY TO TRADE THE WORLD'S MARKETS NADEX.COM CONTENTS To Be or Not To Be? That s a Binary Question Who Sets a Binary Option's Price? And How? Price Reflects Probability Actually,
More informationFinance 527: Lecture 31, Options V3
Finance 527: Lecture 31, Options V3 [John Nofsinger]: This is the third video for the options topic. And the final topic is option pricing is what we re gonna talk about. So what is the price of an option?
More informationEstimating parameters 5.3 Confidence Intervals 5.4 Sample Variance
Estimating parameters 5.3 Confidence Intervals 5.4 Sample Variance Prof. Tesler Math 186 Winter 2017 Prof. Tesler Ch. 5: Confidence Intervals, Sample Variance Math 186 / Winter 2017 1 / 29 Estimating parameters
More informationPurchase Price Allocation, Goodwill and Other Intangibles Creation & Asset Write-ups
Purchase Price Allocation, Goodwill and Other Intangibles Creation & Asset Write-ups In this lesson we're going to move into the next stage of our merger model, which is looking at the purchase price allocation
More informationHPM Module_6_Capital_Budgeting_Exercise
HPM Module_6_Capital_Budgeting_Exercise OK, class, welcome back. We are going to do our tutorial on the capital budgeting module. And we've got two worksheets that we're going to look at today. We have
More informationMath 124: Module 8 (Normal Distribution) Normally Distributed Random Variables. Solving Normal Problems with Technology
( ( What we will do today ly Rom Stard ( David Meredith Department of Mathematics San Francisco State University October 6, 2009 ly Rom Stard 1 ly Rom 2 3 Stard 4 ( ( Rom ly Rom Stard A variable is a characteristic
More informationMr M didn t think MBNA had offered enough compensation. He said it hadn t worked out his compensation in the way we d expect it to.
complaint Mr M has complained that he was mis-sold two payment protection insurance ( PPI ) policies alongside two credit cards he had with MBNA Limited ( MBNA ). background Mr M took out two credit cards
More informationPROBABILITY AND STATISTICS CHAPTER 4 NOTES DISCRETE PROBABILITY DISTRIBUTIONS
PROBABILITY AND STATISTICS CHAPTER 4 NOTES DISCRETE PROBABILITY DISTRIBUTIONS I. INTRODUCTION TO RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS A. Random Variables 1. A random variable x represents a value
More informationMultiple regression - a brief introduction
Multiple regression - a brief introduction Multiple regression is an extension to regular (simple) regression. Instead of one X, we now have several. Suppose, for example, that you are trying to predict
More informationBoom & Bust Monthly Insight Video: What the Media Won t Say About the ACA
Boom & Bust Monthly Insight Video: What the Media Won t Say About the ACA Hi, I m Rodney Johnson, co-editor of Boom & Bust and Survive & Prosper. Welcome to the February 2014 educational video. February
More informationStandard Deviation. Lecture 18 Section Robb T. Koether. Hampden-Sydney College. Mon, Sep 26, 2011
Standard Deviation Lecture 18 Section 5.3.4 Robb T. Koether Hampden-Sydney College Mon, Sep 26, 2011 Robb T. Koether (Hampden-Sydney College) Standard Deviation Mon, Sep 26, 2011 1 / 42 Outline 1 Variability
More informationJanuary 29. Annuities
January 29 Annuities An annuity is a repeating payment, typically of a fixed amount, over a period of time. An annuity is like a loan in reverse; rather than paying a loan company, a bank or investment
More informationConfidence Intervals for the Mean. When σ is known
Confidence Intervals for the Mean When σ is known Objective Find the confidence interval for the mean when s is known. Intro Suppose a college president wishes to estimate the average age of students attending
More informationBinomial Random Variable - The count X of successes in a binomial setting
6.3.1 Binomial Settings and Binomial Random Variables What do the following scenarios have in common? Toss a coin 5 times. Count the number of heads. Spin a roulette wheel 8 times. Record how many times
More informationLife Insurance Buyer s Guide
Contents What type of insurance should I buy? How much insurance should I buy? How long should my term life insurance last? How do I compare life insurance quotes? How do I compare quotes from difference
More informationInterest Rates: Inflation and Loans
Interest Rates: Inflation and Loans 23 April 2014 Interest Rates: Inflation and Loans 23 April 2014 1/29 Last Time On Monday we discussed compound interest and saw that money can grow very large given
More informationChapter 12 Module 4. AMIS 310 Foundations of Accounting
Chapter 12, Module 4 AMIS 310: Foundations of Accounting Slide 1 CHAPTER 1 MODULE 1 AMIS 310 Foundations of Accounting Professor Marc Smith Hi everyone welcome back! Let s continue our discussion of cost
More informationThe Assumptions of Bernoulli Trials. 1. Each trial results in one of two possible outcomes, denoted success (S) or failure (F ).
Chapter 2 Bernoulli Trials 2.1 The Binomial Distribution In Chapter 1 we learned about i.i.d. trials. In this chapter, we study a very important special case of these, namely Bernoulli trials (BT). If
More informationMidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
More informationDescriptive Statistics
Chapter 3 Descriptive Statistics Chapter 2 presented graphical techniques for organizing and displaying data. Even though such graphical techniques allow the researcher to make some general observations
More informationPre-Algebra, Unit 7: Percents Notes
Pre-Algebra, Unit 7: Percents Notes Percents are special fractions whose denominators are 100. The number in front of the percent symbol (%) is the numerator. The denominator is not written, but understood
More informationSTA Module 3B Discrete Random Variables
STA 2023 Module 3B Discrete Random Variables Learning Objectives Upon completing this module, you should be able to 1. Determine the probability distribution of a discrete random variable. 2. Construct
More informationappstats5.notebook September 07, 2016 Chapter 5
Chapter 5 Describing Distributions Numerically Chapter 5 Objective: Students will be able to use statistics appropriate to the shape of the data distribution to compare of two or more different data sets.
More informationWhen we look at a random variable, such as Y, one of the first things we want to know, is what is it s distribution?
Distributions 1. What are distributions? When we look at a random variable, such as Y, one of the first things we want to know, is what is it s distribution? In other words, if we have a large number of
More informationFor personal use only
Media Release For Release: 21 July 2015 Transcript of interview with ANZ CFO Shayne Elliott ANZ today released the transcript of an interview conducted by its digital newsroom BlueNotes with the bank s
More information