# 15.1 Least Squares as a Maximum Likelihood Estimator

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 15.1 Least Squares as a Maximum Likelihood Estimator 651 should provide (i) parameters, (ii) error estimates on the parameters, and (iii) a statistical measure of goodness-of-fit. When the third item suggests that the model is an unlikely match to the data, then items (i) and (ii) are probably worthless. Unfortunately, many practitioners of parameter estimation never proceed beyond item (i). They deem a fit acceptable if a graph of data and model looks good. This approach is known as chi-by-eye. Luckily, its practitioners get what they deserve. CITED REFERENCES AND FURTHER READING: Bevington, P.R. 1969, Data Reduction and Error Analysis for the Physical Sciences (New York: McGraw-Hill). Brownlee, K.A. 1965, Statistical Theory and Methodology, 2nd ed. (New York: Wiley). Martin, B.R. 1971, Statistics for Physicists (New York: Academic Press). von Mises, R. 1964, Mathematical Theory of Probability and Statistics (New York: Academic Press), Chapter X. Korn, G.A., and Korn, T.M. 1968, Mathematical Handbook for Scientists and Engineers, 2nd ed. (New York: McGraw-Hill), Chapters Least Squares as a Maximum Likelihood Estimator Suppose that we are fitting N data points (x i,y i ),...,N, to a model that has M adjustable parameters a j, j =1,...,M. The model predicts a functional relationship between the measured independent and dependent variables, y(x) =y(x;a 1...a M ) (15.1.1) where the dependence on the parameters is indicated explicitly on the right-hand side. What, exactly, do we want to minimize to get fitted values for the a j s? The first thing that comes to mind is the familiar least-squares fit, minimize over a 1...a M : [y i y(x i ; a 1...a M )] 2 (15.1.2) But where does this come from? What general principles is it based on? The answer to these questions takes us into the subject of maximum likelihood estimators. Given a particular data set of x i s and y i s, we have the intuitive feeling that some parameter sets a 1...a M are very unlikely those for which the model function y(x) looks nothing like the data while others may be very likely those that closely resemble the data. How can we quantify this intuitive feeling? How can we select fitted parameters that are most likely to be correct? It is not meaningful to ask the question, What is the probability that a particular set of fitted parameters a 1...a M is correct? The reason is that there is no statistical universe of models from which the parameters are drawn. There is just one model, the correct one, and a statistical universe of data sets that are drawn from it!

2 652 Chapter 15. Modeling of Data That being the case, we can, however, turn the question around, and ask, Given a particular set of parameters, what is the probability that this data set could have occurred? If the y i s take on continuous values, the probability will always be zero unless we add the phrase,...plus or minus some fixed y on each data point. So let s always take this phrase as understood. If the probability of obtaining the data set is infinitesimally small, then we can conclude that the parameters under consideration are unlikely to be right. Conversely, our intuition tells us that the data set should not be too improbable for the correct choice of parameters. In other words, we identify the probability of the data given the parameters (which is a mathematically computable number), as the likelihood of the parameters given the data. This identification is entirely based on intuition. It has no formal mathematical basis in and of itself; as we already remarked, statistics is not a branch of mathematics! Once we make this intuitive identification, however, it is only a small further step to decide to fit for the parameters a 1...a M precisely by finding those values that maximize the likelihood defined in the above way. This form of parameter estimation is maximum likelihood estimation. We are now ready to make the connection to (15.1.2). Suppose that each data point y i has a measurement error that is independently random and distributed as a normal (Gaussian) distribution around the true model y(x). And suppose that the standard deviations σ of these normal distributions are the same for all points. Then the probability of the data set is the product of the probabilities of each point, { [ N P exp 1 ( ) ] } 2 yi y(x i ) y (15.1.3) 2 σ Notice that there is a factor y in each term in the product. Maximizing (15.1.3) is equivalent to maximizing its logarithm, or minimizing the negative of its logarithm, namely, [ N ] [y i y(x i )] 2 2σ 2 N log y (15.1.4) Since N, σ, and yareall constants, minimizing this equation is equivalent to minimizing (15.1.2). What we see is that least-squares fitting is a maximum likelihood estimation of the fitted parameters if the measurement errors are independent and normally distributed with constant standard deviation. Notice that we made no assumption about the linearity or nonlinearity of the model y(x; a 1...) in its parameters a 1...a M. Just below, we will relax our assumption of constant standard deviations and obtain the very similar formulas for what is called chi-square fitting or weighted least-squares fitting. First, however, let us discuss further our very stringent assumption of a normal distribution. For a hundred years or so, mathematical statisticians have been in love with the fact that the probability distribution of the sum of a very large number of very small random deviations almost always converges to a normal distribution. (For precise statements of this central limit theorem, consult [1] or other standard works on mathematical statistics.) This infatuation tended to focus interest away from the

3 15.1 Least Squares as a Maximum Likelihood Estimator 653 fact that, for real data, the normal distribution is often rather poorly realized, if it is realized at all. We are often taught, rather casually, that, on average, measurements will fall within ±σ of the true value 68 percent of the time, within ±2σ 95 percent of the time, and within ±3σ 99.7 percent of the time. Extending this, one would expect a measurement to be off by ±20σ only one time out of We all know that glitches are much more likely than that! In some instances, the deviations from a normal distribution are easy to understand and quantify. For example, in measurements obtained by counting events, the measurement errors are usually distributed as a Poisson distribution, whose cumulative probability function was already discussed in 6.2. When the number of counts going intoone data point is large, the Poisson distributionconverges towards a Gaussian. However, the convergence is not uniform when measured in fractional accuracy. The more standard deviations out on the tail of the distribution, the larger the number of counts must be before a value close to the Gaussian is realized. The sign of the effect is always the same: The Gaussian predicts that tail events are much less likely than they actually (by Poisson) are. This causes such events, when they occur, to skew a least-squares fit much more than they ought. Other times, the deviations from a normal distribution are not so easy to understand in detail. Experimental points are occasionally just way off. Perhaps the power flickered during a point s measurement, or someone kicked the apparatus, or someone wrote down a wrong number. Points like this are called outliers. They can easily turn a least-squares fit on otherwise adequate data into nonsense. Their probability of occurrence in the assumed Gaussian model is so small that the maximum likelihood estimator is willing to distort the whole curve to try to bring them, mistakenly, into line. The subject of robust statistics deals with cases where the normal or Gaussian model is a bad approximation, or cases where outliers are important. We will discuss robust methods briefly in All the sections between this one and that one assume, one way or the other, a Gaussian model for the measurement errors in the data. It it quite important that you keep the limitations of that model in mind, even as you use the very useful methods that follow from assuming it. Finally, note that our discussion of measurement errors has been limited to statistical errors, the kind that will average away if we only take enough data. Measurements are also susceptible to systematic errors that will not go away with any amount of averaging. For example, the calibration of a metal meter stick might depend on its temperature. If we take all our measurements at the same wrong temperature, then no amount of averaging or numerical processing will correct for this unrecognized systematic error. Chi-Square Fitting We considered the chi-square statistic once before, in Here it arises in a slightly different context. If each data point (x i,y i )has its own, known standard deviation σ i,then equation (15.1.3) is modified only by putting a subscript i on the symbol σ. That subscript also propagates docilely into (15.1.4), so that the maximum likelihood

4 654 Chapter 15. Modeling of Data estimate of the model parameters is obtained by minimizing the quantity χ 2 ( ) 2 yi y(x i ; a 1...a M ) (15.1.5) σ i called the chi-square. To whatever extent the measurement errors actually are normally distributed, the quantity χ 2 is correspondingly a sum of N squares of normally distributed quantities, each normalized to unit variance. Once we have adjusted the a 1...a M to minimize the value of χ 2, the terms in the sum are not all statistically independent. For models that are linear in the a s, however, it turns out that the probability distribution for different values of χ 2 at its minimum can nevertheless be derived analytically, and is the chi-square distribution for N M degrees of freedom. We learned how to compute this probability function using the incomplete gamma function gammq in 6.2. In particular, equation (6.2.18) gives the probability Q that the chi-square should exceed a particular value χ 2 by chance, where ν = N M is the number of degrees of freedom. The quantity Q, or its complement P 1 Q, is frequently tabulated in appendices to statistics books, but we generally find it easier to use gammq and compute our own values: Q = gammq (0.5ν, 0.5χ 2 ). It is quite common, and usually not too wrong, to assume that the chi-square distribution holds even for models that are not strictly linear in the a s. This computed probability gives a quantitative measure for the goodness-of-fit of the model. If Q is a very small probability for some particular data set, then the apparent discrepancies are unlikely to be chance fluctuations. Much more probably either (i) the model is wrong can be statistically rejected, or (ii) someone has lied to you about the size of the measurement errors σ i they are really larger than stated. It is an important point that the chi-square probability Q does not directly measure the credibility of the assumption that the measurement errors are normally distributed. It assumes they are. In most, but not all, cases, however, the effect of nonnormal errors is to create an abundance of outlier points. These decrease the probability Q, so that we can add another possible, though less definitive, conclusion to the above list: (iii) the measurement errors may not be normally distributed. Possibility (iii) is fairly common, and also fairly benign. It is for this reason that reasonable experimenters are often rather tolerant of low probabilities Q. It is not uncommon to deem acceptable on equal terms any models with, say, Q> This is not as sloppy as it sounds: Truly wrong models will often be rejected with vastly smaller values of Q, 10 18, say. However, if day-in and day-out you find yourself accepting models with Q 10 3, you really should track down the cause. If you happen to know the actual distribution law of your measurement errors, then you might wish to Monte Carlo simulate some data sets drawn from a particular model, cf You can then subject these synthetic data sets to your actual fitting procedure, so as to determine both the probability distribution of the χ 2 statistic, and also the accuracy with which your model parameters are reproduced by the fit. We discuss this further in The technique is very general, but it can also be very expensive. At the opposite extreme, it sometimes happens that the probabilityq is too large, too near to 1, literally too good to be true! Nonnormal measurement errors cannot in general produce this disease, since the normal distribution is about as compact

5 15.2 Fitting Data to a Straight Line 655 as a distribution can be. Almost always, the cause of too good a chi-square fit is that the experimenter, in a fit of conservativism, has overestimated his or her measurement errors. Very rarely, too good a chi-square signals actual fraud, data that has been fudged to fit the model. A rule of thumb is that a typical value of χ 2 for a moderately good fit is χ 2 ν. More precise is the statement that the χ 2 statistichas a mean ν and a standard deviation 2ν, and, asymptotically for large ν, becomes normally distributed. In some cases the uncertainties associated with a set of measurements are not known in advance, and considerations related to χ 2 fitting are used to derive a value for σ. If we assume that all measurements have the same standard deviation, σ i = σ, and that the model does fit well, then we can proceed by first assigning an arbitrary constant σ to all points, next fitting for the model parameters by minimizing χ 2, and finally recomputing σ 2 = [y i y(x i )] 2 /(N M) (15.1.6) Obviously, this approach prohibits an independent assessment of goodness-of-fit, a fact occasionally missed by its adherents. When, however, the measurement error is not known, this approach at least allows some kind of error bar to be assigned to the points. If we take the derivative of equation (15.1.5) with respect to the parameters a k, we obtain equations that must hold at the chi-square minimum, 0= ( )( ) yi y(x i ) y(xi ;...a k...) σ 2 i a k k =1,...,M (15.1.7) Equation (15.1.7) is, in general, a set of M nonlinear equations for the M unknown a k. Various of the procedures described subsequently in this chapter derive from (15.1.7) and its specializations. CITED REFERENCES AND FURTHER READING: Bevington, P.R. 1969, Data Reduction and Error Analysis for the Physical Sciences (New York: McGraw-Hill), Chapters 1 4. von Mises, R. 1964, Mathematical Theory of Probability and Statistics (New York: Academic Press), VI.C. [1] 15.2 Fitting Data to a Straight Line A concrete example will make the considerations of the previous section more meaningful. We consider the problem of fitting a set of N data points (x i,y i )to a straight-line model y(x) =y(x;a, b) =a+bx (15.2.1)

### CHI-SQUARE: TESTING FOR GOODNESS OF FIT

CHI-SQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity

### Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

### ERROR ANALYSIS 2: A USER S GUIDE TO LEAST-SQUARES FITTING

ERROR ANALYSIS 2: A USER S GUIDE TO LEAST-SQUARES FITTING INTRODUCTION This activity is a user s guide to least-squares fitting and to determining the goodness of your fits. It doesn t derive many results.

### Measurement and Measurement Error

1 Measurement and Measurement Error PHYS 1301 F99 Prof. T.E. Coan Version: 8 Sep 99 Introduction Physics makes both general and detailed statements about the physical universe. These statements are organized

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### Reading: Analysis of Errors Revised 2/9/13 ANALYSIS OF ERRORS

ANALYSIS OF ERRORS Precision and Accuracy Two terms are commonly associated with any discussion of error: "precision" and "accuracy". Precision refers to the reproducibility of a measurement while accuracy

### 3.6: General Hypothesis Tests

3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

### Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 1: CHI-SQUARE TESTS: When to use a Chi-Square test: Usually in psychological research, we aim to obtain one or more scores

### Unit 21 Student s t Distribution in Hypotheses Testing

Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Experimental Errors and Uncertainty

Experimental Errors and Uncertainty No physical quantity can be measured with perfect certainty; there are always errors in any measurement. This means that if we measure some quantity and, then, repeat

### Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA

Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA Introduction A common question faced by many actuaries when selecting loss development factors is whether to base the selected

### 99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm

Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

### Lecture 3 : Hypothesis testing and model-fitting

Lecture 3 : Hypothesis testing and model-fitting These dark lectures energy puzzle Lecture 1 : basic descriptive statistics Lecture 2 : searching for correlations Lecture 3 : hypothesis testing and model-fitting

### Stocking Strategies: Low or Intermittent Demand

Stocking Strategies: Low or Intermittent Demand Greg Larsen G.A. Larsen Consulting LLC Sponsored by: RockySoft 5/15/2008 Demand for service and other parts is often low volume or intermittent. Commonly

### Linearizing Equations Handout Wilfrid Laurier University

Linearizing Equations Handout Wilfrid Laurier University c Terry Sturtevant 1 2 1 Physics Lab Supervisor 2 This document may be freely copied as long as this page is included. ii Chapter 1 Linearizing

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### TRANSCRIPT: In this lecture, we will talk about both theoretical and applied concepts related to hypothesis testing.

This is Dr. Chumney. The focus of this lecture is hypothesis testing both what it is, how hypothesis tests are used, and how to conduct hypothesis tests. 1 In this lecture, we will talk about both theoretical

### Generalized Linear Model Theory

Appendix B Generalized Linear Model Theory We describe the generalized linear model as formulated by Nelder and Wedderburn (1972), and discuss estimation of the parameters and tests of hypotheses. B.1

### THE STATISTICAL TREATMENT OF EXPERIMENTAL DATA 1

THE STATISTICAL TREATMET OF EXPERIMETAL DATA Introduction The subject of statistical data analysis is regarded as crucial by most scientists, since error-free measurement is impossible in virtually all

### Descriptive Data Summarization

Descriptive Data Summarization (Understanding Data) First: Some data preprocessing problems... 1 Missing Values The approach of the problem of missing values adopted in SQL is based on nulls and three-valued

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### A Summary of Error Propagation

A Summary of Error Propagation Suppose you measure some quantities a, b, c,... with uncertainties δa, δb, δc,.... Now you want to calculate some other quantity Q which depends on a and b and so forth.

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. 2-Sample

### Introduction to Statistics for Computer Science Projects

Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

### (Refer Slide Time: 00:00:56 min)

Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

### 14.2 Do Two Distributions Have the Same Means or Variances?

14.2 Do Two Distributions Have the Same Means or Variances? 615 that this is wasteful, since it yields much more information than just the median (e.g., the upper and lower quartile points, the deciles,

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance

Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance (if the Gauss-Markov

### Errors and Error Estimation

Errors and Error Estimation Errors, precision and accuracy: why study them? People in scientific and technological professions are regularly required to give quantitative answers. How long? How heavy?

### Review of Likelihood Theory

Appendix A Review of Likelihood Theory This is a brief summary of some of the key results we need from likelihood theory. A.1 Maximum Likelihood Estimation Let Y 1,..., Y n be n independent random variables

### Time Series and Forecasting

Chapter 22 Page 1 Time Series and Forecasting A time series is a sequence of observations of a random variable. Hence, it is a stochastic process. Examples include the monthly demand for a product, the

### The factor of safety is a factor of ignorance. If the stress on a part at a critical location (the

Appendix C The Factor of Safety as a Design Variable C.1 INTRODUCTION The factor of safety is a factor of ignorance. If the stress on a part at a critical location (the applied stress) is known precisely,

### Hypothesis Testing. Concept of Hypothesis Testing

Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population

### Chapter 8 CREDIBILITY HOWARD C. MAHLER AND CURTIS GARY DEAN

Chapter 8 CREDIBILITY HOWARD C. MAHLER AND CURTIS GARY DEAN 1. INTRODUCTION Credibility theory provides tools to deal with the randomness of data that is used for predicting future events or costs. For

### Accuracy, Precision and Uncertainty

Accuracy, Precision and Uncertainty How tall are you? How old are you? When you answered these everyday questions, you probably did it in round numbers such as "five foot, six inches" or "nineteen years,

### Distributions: Population, Sample and Sampling Distributions

119 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 9 Distributions: Population, Sample and Sampling Distributions In the three preceding chapters

### Hypothesis Testing for Beginners

Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

### Uncertainties and Error Propagation

Uncertainties and Error Propagation By Vern Lindberg 1. Systematic and random errors. No measurement made is ever exact. The accuracy (correctness) and precision (number of significant figures) of a measurement

### Independent samples t-test. Dr. Tom Pierce Radford University

Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of

### An introduction to Value-at-Risk Learning Curve September 2003

An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk

### Poisson Statistics. lim = R (1)

Poisson Statistics MIT Department of Physics (Dated: August 25, 2013) In this experiment, you will explore the statistics of random, independent events in physical measurements. The random events used

### Chi-Square Tests. In This Chapter BONUS CHAPTER

BONUS CHAPTER Chi-Square Tests In the previous chapters, we explored the wonderful world of hypothesis testing as we compared means and proportions of one, two, three, and more populations, making an educated

### PHYSICS 2150 EXPERIMENTAL MODERN PHYSICS. Lecture 6 Chi Square Test, Poisson Distribution

PHYSICS 2150 EXPERIMENTAL MODERN PHYSICS Lecture 6 Chi Square Test, Poisson Distribution LAB REPORT FORMAT Experiment title Objective : Several sentence description of scientific goal (not to learn about.

### Confidence intervals, t tests, P values

Confidence intervals, t tests, P values Joe Felsenstein Department of Genome Sciences and Department of Biology Confidence intervals, t tests, P values p.1/31 Normality Everybody believes in the normal

### Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to

### 1. χ 2 minimization 2. Fits in case of of systematic errors

Data fitting Volker Blobel University of Hamburg March 2005 1. χ 2 minimization 2. Fits in case of of systematic errors Keys during display: enter = next page; = next page; = previous page; home = first

### COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite

COURSE OUTLINE Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4 Hours: Lecture/Lab/Other 4 Lecture Co- or Pre-requisite MAT151 or MAT149 with a minimum

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### , for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0

Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after Simeon-Denis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will

### STATISTICS AND THE TREATMENT OF EXPERIMENTAL DATA

Adapted from Chapter 4, Techniques for Nuclear and Particle Physics Experiments, by W. R. Leo, Springer-Verlag 1992 STATISTICS AND THE TREATMENT OF EXPERIMENTAL DATA W. R. Leo Statistics plays an essential

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### AP Statistics 2013 Scoring Guidelines

AP Statistics 2013 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the

### Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit

Chi-Square Tests 15 Chapter Chi-Square Test for Independence Chi-Square Tests for Goodness Uniform Goodness- Poisson Goodness- Goodness Test ECDF Tests (Optional) McGraw-Hill/Irwin Copyright 2009 by The

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### Lecture 1: t tests and CLT

Lecture 1: t tests and CLT http://www.stats.ox.ac.uk/ winkel/phs.html Dr Matthias Winkel 1 Outline I. z test for unknown population mean - review II. Limitations of the z test III. t test for unknown population

### 1.0 Mechanical Measurement Systems. 1.1 Definition of a Measurement system

1.0 Mechanical Measurement Systems Measurement of mechanical systems has long been an issue for engineers. Having been trained to deal with mechanical systems, the use of electrical or electronic system

### Bayesian and Classical Inference

Eco517, Part I Fall 2002 C. Sims Bayesian and Classical Inference Probability statements made in Bayesian and classical approaches to inference often look similar, but they carry different meanings. Because

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### Models for Discrete Variables

Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

### Chapter 13. Vehicle Arrival Models : Count Introduction Poisson Distribution

Chapter 13 Vehicle Arrival Models : Count 13.1 Introduction As already noted in the previous chapter that vehicle arrivals can be modelled in two interrelated ways; namely modelling how many vehicle arrive

### Introduction to Regression. Dr. Tom Pierce Radford University

Introduction to Regression Dr. Tom Pierce Radford University In the chapter on correlational techniques we focused on the Pearson R as a tool for learning about the relationship between two variables.

### Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435)

Confidence Intervals for Poisson data For an observation from a Poisson distribution, we have σ 2 = λ. If we observe r events, then our estimate ˆλ = r : N(λ, λ) If r is bigger than 20, we can use this

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### Sampling Distributions and the Central Limit Theorem

135 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 10 Sampling Distributions and the Central Limit Theorem In the previous chapter we explained

### Regression analysis in the Assistant fits a model with one continuous predictor and one continuous response and can fit two types of models:

This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. The simple regression procedure in the

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### Hypothesis Testing. Chapter Introduction

Contents 9 Hypothesis Testing 553 9.1 Introduction............................ 553 9.2 Hypothesis Test for a Mean................... 557 9.2.1 Steps in Hypothesis Testing............... 557 9.2.2 Diagrammatic

### The Random Sampling Road to Reasonableness. Reduce Risk and Cost by Employing a Complete and Integrated Validation Process

The Random Sampling Road to Reasonableness Reduce Risk and Cost by Employing a Complete and Integrated Validation Process By: Michael R. Wade Planet Data Executive Vice President Chief Technology Officer

### CONFIDENCE INTERVALS I

CONFIDENCE INTERVALS I ESTIMATION: the sample mean Gx is an estimate of the population mean µ point of sampling is to obtain estimates of population values Example: for 55 students in Section 105, 45 of

### If you need statistics to analyze your experiment, then you've done the wrong experiment.

When do you need statistical calculations? When analyzing data, your goal is simple: You wish to make the strongest possible conclusion from limited amounts of data. To do this, you need to overcome two

### Treatment and analysis of data Applied statistics Lecture 3: Sampling and descriptive statistics

Treatment and analysis of data Applied statistics Lecture 3: Sampling and descriptive statistics Topics covered: Parameters and statistics Sample mean and sample standard deviation Order statistics and

### PSS 27.2 The Electric Field of a Continuous Distribution of Charge

Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### Introduction to Error Analysis

UNIVERSITÄT BASEL DEPARTEMENT CHEMIE Introduction to Error Analysis Physikalisch Chemisches Praktikum Dr. Nico Bruns, Dr. Katarzyna Kita, Dr. Corinne Vebert 2012 1. Why is error analysis important? First

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements

Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements Blair Hall b.hall@irl.cri.nz Talk given via internet to the 35 th ANAMET Meeting, October 20, 2011.

### BASIC ELECTRONICS PROF. T.S. NATARAJAN DEPT OF PHYSICS IIT MADRAS LECTURE-4 SOME USEFUL LAWS IN BASIC ELECTRONICS

BASIC ELECTRONICS PROF. T.S. NATARAJAN DEPT OF PHYSICS IIT MADRAS LECTURE-4 SOME USEFUL LAWS IN BASIC ELECTRONICS Hello everybody! In a series of lecture on basic electronics, learning by doing, we now

### Monte Carlo Method: Probability

John (ARC/ICAM) Virginia Tech... Math/CS 4414: The Monte Carlo Method: PROBABILITY http://people.sc.fsu.edu/ jburkardt/presentations/ monte carlo probability.pdf... ARC: Advanced Research Computing ICAM:

### Chapter 1 Introduction. 1.1 Introduction

Chapter 1 Introduction 1.1 Introduction 1 1.2 What Is a Monte Carlo Study? 2 1.2.1 Simulating the Rolling of Two Dice 2 1.3 Why Is Monte Carlo Simulation Often Necessary? 4 1.4 What Are Some Typical Situations

### Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

### Error and Uncertainty

Error and Uncertainty All that any experimental procedure can do is to give a for the result that we can say may be near the true. We can never say that we know the true result, only that we have a result

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### 1.2 ERRORS AND UNCERTAINTIES Notes

1.2 ERRORS AND UNCERTAINTIES Notes I. UNCERTAINTY AND ERROR IN MEASUREMENT A. PRECISION AND ACCURACY B. RANDOM AND SYSTEMATIC ERRORS C. REPORTING A SINGLE MEASUREMENT D. REPORTING YOUR BEST ESTIMATE OF

### 1. Chi-Squared Tests

1. Chi-Squared Tests We'll now look at how to test statistical hypotheses concerning nominal data, and specifically when nominal data are summarized as tables of frequencies. The tests we will considered

### MAT X Hypothesis Testing - Part I

MAT 2379 3X Hypothesis Testing - Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating

### 14.1 Moments of a Distribution: Mean, Variance, Skewness, and So Forth. 610 Chapter 14. Statistical Description of Data

610 Chapter 14. Statistical Description of Data In the other category, model-dependent statistics, we lump the whole subject of fitting data to a theory, parameter estimation, least-squares fits, and so

### Scientific Measurements: Significant Figures and Statistical Analysis

Bellevue College CHEM& 161 Scientific Measurements: Significant Figures and Statistical Analysis Purpose The purpose of this activity is to get familiar with the approximate precision of the equipment