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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 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 factor on more recent data or to use a longer period of experience. If there are changes in the development patterns over time, a shorter period of experience may be more appropriate. If the experience is volatile, a longer period of experience may be more appropriate. Rarely is the case purely one or the other, but usually some combination of these influences. Most actuaries would agree on these basic principles, but the practical application of these principles often lacks analytical structure. This paper will suggest two systematic techniques intended to aid the actuary with this problem. Change or Random Fluctuation? At the center of this issue is the basic question of whether observed differences between current and past observed development factors are due to an underlying change or are due to random fluctuation (i.e. changes that are ephemeral in nature and not predictive of the future). Consider the following data for loss development factors from development age 1 to development age 2 for a number of accident years: Year Age 1 Loss Age 2 Loss Development Factor

2 Dev Factor Accident Year Looking at the data points, there appears to be a trend and that we might be likely to expect a development factor for accident year 10 that is less than the long term average. But is this impression of changing development factors credible? One statistical test we can perform is a run test. If the data points are in fact randomly distributed from the same distribution, each point will be as likely to be above the median value as below it (regardless of the distribution). In data with a trend, there will be a tendency for the values that are above the median to be more likely at either end of the data series and fewer runs would result. Adding the median value to the data for this example we have the following:

3 Dev Factor Median Accident Year We can label the series as (++ + +). The third point, which is exactly at the median, is discarded. This constitutes five runs above and below the median. The probability distribution under randomness (i.e. zero trend) given nine data points is as follows: Runs Probability Cumulative Probability % 2.86% % 11.43% % 37.14% % 62.86% % 88.57% % 97.14% % % % % So according to this test, the observed pattern of data points is highly likely under the scenario where there is actually no underlying trend, despite the appearance to the contrary. A larger table with varying number of data points is included at the end of this paper. Another test we could run is to perform a regression and test the slope parameter for significance. Using simple linear regression we arrive at the following parameters:

4 estimate standard error significance level slope intercept This analysis suggests that there is a non zero trend. Why do these two tests provide inconsistent results? The run test, since it is based purely on whether observations are above or below the median is very robust (i.e. resistant to outliers) and is free of assumptions about the form of the probability distribution. The t test of the regression parameter on the other hand assumes that the residuals from the linear relationship are independent random variables drawn from the same normal distribution N(0,). If the assumptions made by the regression test are correct then the observations are inconsistent with zero trend in the factors. The amount by which the individual data points deviate from the mean has an important impact. The run test does not consider the size of these deviations. The amount by which they are less or more than the median does not matter only that they are less or more than the median, because the run test has no distributional framework within which to place them. Consider another example: Development Year Age 1 Loss Age 2 Loss Factor

5 Dev Factor Median Accident Year In this example, the number of runs is 2 (,++++), which is unlikely under the randomness assumption. The parameters of a linear regression are: estimate standard error significance level slope intercept In this example the two tests give consistent indications regarding changes in the development factor. The Observed Data To this point, no mention has been made about the creation of the data points for these two examples. Both of these examples were created from the same underlying simulated random process with the same parameters. Specifically the following random process was simulated: 100 claim payments all of identical size Each claim is paid at a random age of development according to an exponential distribution (i.e. the lag from the beginning of the accident year to the claim payment is exponentially distributed) The parameter for the exponential distribution varies from accident year to accident year

6 Accident Year Implied 1 2 Development Factor Expected Lag The implied (true) development factor is calculated as the ratio CDF(2)/CDF(1) = (1 e 2/lag )/(1 e 1/lag ). The simulated claim payments were summarized by accident year and development period, and the observed age to age factors calculated. Practical Application of Significance Tests As a practical reality we as actuaries do not have the luxury of saying, The data is inconclusive. and moving on to the next academic study. We are required to use our best judgment to predict the future. This judgment includes looking at data, but also reflects knowledge of changes in claims practices, underlying tort environment, etc. Unfortunately, however, this judgment can be biased by the data itself. An apparent change causes us to look for reasons for the change, justifying an impression that may be misleading. Statistical testing can help us to recognize whether or not a change is truly significant, but we are left with many choices regarding the nature of tests and the assumptions made by these tests. This paper will now present two methods to assist with this problem, each with flexibility to allow the actuary to adjust judgmentally the amount of stability or responsiveness to observed fluctuations, but still within general framework of statistics. Method 1 Run Test Data Reduction This method is simply built on the run test described above, eliminating data until the randomness hypothesis is not rejected. Step 1 Determine a significance parameter between 0 and 1. (0=least responsive, most stable, 1 = most responsive, least stable). Step2 Calculate the median of all accident years factors Step3 Perform a run test. Reject the randomness hypothesis if the cumulative probability of observing this number of runs is below the significance level selected in step 1. Step4 If the randomness hypothesis was rejected in Step 3, eliminate the oldest observation and repeat Step 2 Step4 until the hypothesis is not rejected. Step5 If the randomness hypothesis was rejected in Step 3, calculate the average (volume weighted or simple) of the remaining observations. Method 2 Stabilized Regression

7 This method uses regression, but expands upon the simple linear regression described above. It is volume weighted and uses logarithms to adjust for the typical skewness of observed age to age factors. It reflects trend in development factors only to the extent the trend is considered statistically significant. The amount of trend that is used tempered based on this significance test. It also defaults to the long term weighted average factor when the trend is not considered statistically significant. It minimizes weighted squared residuals, subject to a constraint of balance (i.e. the total projected loss across all historical periods is equal to the total actual loss). Let X ij denote the cumulative loss for accident year i at development period j. Step 1 Determine a significance parameter between 0 and 1. (0=least responsive, most stable, 1 = most responsive, least stable). Step 2 Calculate the weighted average development factor,, Step 3 Calculate the Average Accident Year,, Step 4 Calculate Raw Slope 1,,,, Step 5 Calculate Raw Intercept 1 1 Step 6 Calculate Standard Error of the Raw Slope,, 1 1,,,, Step 7 Calculate Stability Threshold for Slope 1/2 Step 8 If S1 > T and S1 > 0 then set tempered slope S = S1 T. If S1 > T and S1 < 0 then set tempered slope S = S1 + T. If S1 < T then set tempered slope S = 0.

8 Step 9 Calculate Intercept consistent with slope S. Step 10 Calculate the projected development factor for Accident Year i as: Applying each of these methods to the two examples shown above to estimate the development factor to use for Accident Year 10 gives the following table: Projected Development Factor for Accident Year 10 Example 1 Example 2 Significance Parameter Method 1 Method 2 Method 1 Method The true factor is Both of the methods return the loss weighted average factor (all data points) when the significance parameter is set to 0. These are only two examples. Simulating 10,000 times we have the following results:

9 Relative Frequency Run Test Data Reduction Significance Parameter True Estimated Development Factor Relative Frequency Stabilized Regression Significance Parameter True Estimated Development Factor Run Test Data Reduction Stabilized Regression Significance Parameter Average Bias RMSE Average Bias RMSE The full triangle weighted average (significance parameter =0 for both methods) is of course negatively biased for this example of increasing development lags. Not surprisingly the run test data reduction method never is able to eliminate its negative bias in this scenario, since it will never extrapolate. The stabilized regression method, since it does extrapolate, eventually eliminates the negative bias, but at a cost of a significantly increased root mean squared error.

10 These results are specific to the simulated claims process that was used. The results of each of these methods would obviously vary depending on the actual underlying claim process, which of course is unknown, including non monotonic shifts, etc. The actual choice of a significance parameter to use in either method could be based on the perceived potential for changes in claims handling, mix of business, legal environment, etc.

11 Cumulative Probability of Number of Runs for a Particular Number of Observations Number of Observations Number of Runs

### An Alternative Route to Performance Hypothesis Testing

EDHEC-Risk Institute 393-400 promenade des Anglais 06202 Nice Cedex 3 Tel.: +33 (0)4 93 18 32 53 E-mail: research@edhec-risk.com Web: www.edhec-risk.com An Alternative Route to Performance Hypothesis Testing

### Statistics 2014 Scoring Guidelines

AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

### Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

### Module 5: Multiple Regression Analysis

Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Chapter 7 Development Technique

Sec Description Pages 1 Introduction 84 2 Key Assumptions 84 3 Common Uses of the Development Technique 84-85 4 Mechanics of the Development Technique 85-92 5 Unpaid Claim Estimate Based on the Development

### South Carolina College- and Career-Ready (SCCCR) Probability and Statistics

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)

### Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### 4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4

4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression

### 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

### Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance

2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

### 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

### 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

### Hypothesis Testing with z Tests

CHAPTER SEVEN Hypothesis Testing with z Tests NOTE TO INSTRUCTOR This chapter is critical to an understanding of hypothesis testing, which students will use frequently in the coming chapters. Some of the

### the number of organisms in the squares of a haemocytometer? the number of goals scored by a football team in a match?

Poisson Random Variables (Rees: 6.8 6.14) Examples: What is the distribution of: the number of organisms in the squares of a haemocytometer? the number of hits on a web site in one hour? the number of

BIOSTATISTICS QUIZ ANSWERS 1. When you read scientific literature, do you know whether the statistical tests that were used were appropriate and why they were used? a. Always b. Mostly c. Rarely d. Never

### A Review of Statistical Outlier Methods

Page 1 of 5 A Review of Statistical Outlier Methods Nov 2, 2006 By: Steven Walfish Pharmaceutical Technology Statistical outlier detection has become a popular topic as a result of the US Food and Drug

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### 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

### Nonlinear Regression Functions. SW Ch 8 1/54/

Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### A Primer on Forecasting Business Performance

A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### Employment, unemployment and real economic growth. Ivan Kitov Institute for the Dynamics of the Geopsheres, Russian Academy of Sciences

Employment, unemployment and real economic growth Ivan Kitov Institute for the Dynamics of the Geopsheres, Russian Academy of Sciences Oleg Kitov Department of Economics, University of Oxford Abstract

### Chapter 7. Estimates and Sample Size

Chapter 7. Estimates and Sample Size Chapter Problem: How do we interpret a poll about global warming? Pew Research Center Poll: From what you ve read and heard, is there a solid evidence that the average

### CURVE FITTING LEAST SQUARES APPROXIMATION

CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

### 15.1 Least Squares as a Maximum Likelihood Estimator

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

### Lean Six Sigma Analyze Phase Introduction. TECH 50800 QUALITY and PRODUCTIVITY in INDUSTRY and TECHNOLOGY

TECH 50800 QUALITY and PRODUCTIVITY in INDUSTRY and TECHNOLOGY Before we begin: Turn on the sound on your computer. There is audio to accompany this presentation. Audio will accompany most of the online

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### Credibility and Pooling Applications to Group Life and Group Disability Insurance

Credibility and Pooling Applications to Group Life and Group Disability Insurance Presented by Paul L. Correia Consulting Actuary paul.correia@milliman.com (207) 771-1204 May 20, 2014 What I plan to cover

### How To Use A Monte Carlo Study To Decide On Sample Size and Determine Power

How To Use A Monte Carlo Study To Decide On Sample Size and Determine Power Linda K. Muthén Muthén & Muthén 11965 Venice Blvd., Suite 407 Los Angeles, CA 90066 Telephone: (310) 391-9971 Fax: (310) 391-8971

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

### Unit 24 Hypothesis Tests about Means

Unit 24 Hypothesis Tests about Means Objectives: To recognize the difference between a paired t test and a two-sample t test To perform a paired t test To perform a two-sample t test A measure of the amount

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### Data Analysis and Sampling

Data Analysis and Sampling About This Course Course Description In order to perform successful internal audits, you must know how to reduce a large data set down to critical subsets based on risk or importance,

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### Penalized regression: Introduction

Penalized regression: Introduction Patrick Breheny August 30 Patrick Breheny BST 764: Applied Statistical Modeling 1/19 Maximum likelihood Much of 20th-century statistics dealt with maximum likelihood

### Recent developments in surveys of exchange rate forecasts

By Sally Harrison and Caroline Mogford of the Bank s Foreign Exchange Division. Expectations of future exchange rates can influence moves in the current exchange rate. This article summarises recent developments

### TESTING THE ASSUMPTIONS OF AGE-TO-AGE FACTORS. Abstract

TESTING THE ASSUMPTIONS OF AGE-TO-AGE FACTORS GARY G. VENTER Abstract The use of age-to-age factors applied to cumulative losses has been shown to produce least-squares optimal reserve estimates when certain

### 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 15 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately

### 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

### Non Parametric Inference

Maura Department of Economics and Finance Università Tor Vergata Outline 1 2 3 Inverse distribution function Theorem: Let U be a uniform random variable on (0, 1). Let X be a continuous random variable

### Data Transforms: Natural Logarithms and Square Roots

Data Transforms: atural Log and Square Roots 1 Data Transforms: atural Logarithms and Square Roots Parametric statistics in general are more powerful than non-parametric statistics as the former are based

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D.

Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D. In biological science, investigators often collect biological

### AP Statistics 2012 Scoring Guidelines

AP Statistics 2012 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

### AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

### Imputing Missing Data using SAS

ABSTRACT Paper 3295-2015 Imputing Missing Data using SAS Christopher Yim, California Polytechnic State University, San Luis Obispo Missing data is an unfortunate reality of statistics. However, there are

### 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

### Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

### Data. ECON 251 Research Methods. 1. Data and Descriptive Statistics (Review) Cross-Sectional and Time-Series Data. Population vs.

ECO 51 Research Methods 1. Data and Descriptive Statistics (Review) Data A variable - a characteristic of population or sample that is of interest for us. Data - the actual values of variables Quantitative

### Sample Size Issues for Conjoint Analysis

Chapter 7 Sample Size Issues for Conjoint Analysis I m about to conduct a conjoint analysis study. How large a sample size do I need? What will be the margin of error of my estimates if I use a sample

### 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

### 1 The Pareto Distribution

Estimating the Parameters of a Pareto Distribution Introducing a Quantile Regression Method Joseph Lee Petersen Introduction. A broad approach to using correlation coefficients for parameter estimation

### AP STATISTICS REVIEW (YMS Chapters 1-8)

AP STATISTICS REVIEW (YMS Chapters 1-8) Exploring Data (Chapter 1) Categorical Data nominal scale, names e.g. male/female or eye color or breeds of dogs Quantitative Data rational scale (can +,,, with

### Spearman s correlation

Spearman s correlation Introduction Before learning about Spearman s correllation it is important to understand Pearson s correlation which is a statistical measure of the strength of a linear relationship

### Problems with OLS Considering :

Problems with OLS Considering : we assume Y i X i u i E u i 0 E u i or var u i E u i u j 0orcov u i,u j 0 We have seen that we have to make very specific assumptions about u i in order to get OLS estimates

### 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

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### Combining information from different survey samples - a case study with data collected by world wide web and telephone

Combining information from different survey samples - a case study with data collected by world wide web and telephone Magne Aldrin Norwegian Computing Center P.O. Box 114 Blindern N-0314 Oslo Norway E-mail:

### BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

### CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

### Hypothesis Testing - Relationships

- Relationships Session 3 AHX43 (28) 1 Lecture Outline Correlational Research. The Correlation Coefficient. An example. Considerations. One and Two-tailed Tests. Errors. Power. for Relationships AHX43

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Example 1: Calculate and compare RiskMetrics TM and Historical Standard Deviation Compare the weights of the volatility parameter using,, and.

3.6 Compare and contrast different parametric and non-parametric approaches for estimating conditional volatility. 3.7 Calculate conditional volatility using parametric and non-parametric approaches. Parametric

### 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: DESCRIPTIVE STATISTICS - FREQUENCY DISTRIBUTIONS AND AVERAGES: Inferential and Descriptive Statistics: There are four

### PERFORMANCE EVALUATION OF EQUITY MUTUAL FUNDS IN INDIA: AN EMPIRICAL EXPLORATION. Soumya Guha Deb (THESIS ADVISORS)

PERFORMANCE EVALUATION OF EQUITY MUTUAL FUNDS IN INDIA: AN EMPIRICAL EXPLORATION by Soumya Guha Deb (THESIS ADVISORS) PROF. B. B.CHAKRABARTI PROF. ASHOK BANERJEE ABSTRACT The mutual fund industry in India

### 1291/2 Physics Lab Report Format

1291/2 Physics Lab Report Format General Remarks: Writing a lab report is the only way your TA will know what you have done during the lab and how well you have understood the process and the results.

### FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits

Technical Paper Series Congressional Budget Office Washington, DC FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Albert D. Metz Microeconomic and Financial Studies

### SPSS: Descriptive and Inferential Statistics. For Windows

For Windows August 2012 Table of Contents Section 1: Summarizing Data...3 1.1 Descriptive Statistics...3 Section 2: Inferential Statistics... 10 2.1 Chi-Square Test... 10 2.2 T tests... 11 2.3 Correlation...

### Draft 1, Attempted 2014 FR Solutions, AP Statistics Exam

Free response questions, 2014, first draft! Note: Some notes: Please make critiques, suggest improvements, and ask questions. This is just one AP stats teacher s initial attempts at solving these. I, as

### The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker

HYPOTHESIS TESTING PHILOSOPHY 1 The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker Question: So I'm hypothesis testing. What's the hypothesis I'm testing? Answer: When you're

### 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

### The aspect of the data that we want to describe/measure is the degree of linear relationship between and The statistic r describes/measures the degree

PS 511: Advanced Statistics for Psychological and Behavioral Research 1 Both examine linear (straight line) relationships Correlation works with a pair of scores One score on each of two variables ( and

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### Efficient Curve Fitting Techniques

15/11/11 Life Conference and Exhibition 11 Stuart Carroll, Christopher Hursey Efficient Curve Fitting Techniques - November 1 The Actuarial Profession www.actuaries.org.uk Agenda Background Outline of

### Evaluation of Value-at-Risk Models Using Historical Data

Evaluation of Value-at-Risk Models Using Historical Data Darryll Hendricks The views expressed in this article are those of the authors and do not necessarily reflect the position of the Federal Reserve

### Appendix A: Sampling Methods

Appendix A: Sampling Methods What is Sampling? Sampling is used in an @RISK simulation to generate possible values from probability distribution functions. These sets of possible values are then used to

### 2013 IAA Education Syllabus

This version was approved at the Council meeting on 26 May 2012 and replaces the 2007 document. 1. Financial Mathematics To provide a grounding in the techniques of financial mathematics and their applications.

### Unit 3 Sample Test. Name: Class: Date: True/False Indicate whether the statement is true or false.

Name: Class: Date: Unit 3 Sample Test True/False Indicate whether the statement is true or false. 1. An example of qualitative data is the colour of a person s eyes. 2. When a researcher conducts an experiment,

### 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

### Survey Process White Paper Series The Six Steps in Conducting Quantitative Marketing Research

Survey Process White Paper Series The Six Steps in Conducting Quantitative Marketing Research POLARIS MARKETING RESEARCH, INC. 1455 LINCOLN PARKWAY, SUITE 320 ATLANTA, GEORGIA 30346 404.816.0353 www.polarismr.com

### Introduction to Credibility 1

Introduction to Credibility 1 RPM Workshop 4: Basic Ratemaking Introduction to Credibility Ken Doss, FCAS, MAAA State Farm Insurance March 2011 Antitrust Notice The Casualty Actuarial Society is committed

### Regression. In this class we will:

AMS 5 REGRESSION Regression The idea behind the calculation of the coefficient of correlation is that the scatter plot of the data corresponds to a cloud that follows a straight line. This idea can be

### ANOVA MULTIPLE CHOICE QUESTIONS. In the following multiple-choice questions, select the best answer.

ANOVA MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, select the best answer. 1. Analysis of variance is a statistical method of comparing the of several populations. a. standard

### Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

### Means, standard deviations and. and standard errors

CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

### 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

### Hints for Success on the AP Statistics Exam. (Compiled by Zack Bigner)

Hints for Success on the AP Statistics Exam. (Compiled by Zack Bigner) The Exam The AP Stat exam has 2 sections that take 90 minutes each. The first section is 40 multiple choice questions, and the second

### MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance