NPTEL STRUCTURAL RELIABILITY

Size: px
Start display at page:

Download "NPTEL STRUCTURAL RELIABILITY"

Transcription

1 NPTEL Course On STRUCTURAL RELIABILITY Module # 02 Lecture 6 Course Format: Web Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati

2 6. Lecture 06: Hypothesis Testing Tests of Distributions Generally, the data available to us is either through experimental observations or recorded signals. They are not bound to exactly follow any mathematical probability distribution model. Behaviour of the data may be of approximately matching any defined probability distribution. Hence, data analysis is required in concern to see whether it is following any distribution or not, if yes than with how much significance. Following sections will discuss two very popular distribution tests, they are Chi Square Test and Kolmogorov Smirnov Test. Chi Square Test Chi square test is used to analyze whether a given random sample follows any theoretically defined probability distribution or not. The basic idea is based on evaluating the cumulative error between the probability density of the random sample and theoretical probability distribution. Stepwise detailed discussion for conducting Chi square test in view of checking probability distribution with its statistical parameters are stated below. Step 1. Firstly, null and alternate hypotheses are formulated based on the probability distribution and statistical parameters of the random data. Both hypotheses can be presented as H 0 : X ~ f a, b H A : X f a, b where H 0 and H A denotes null and alternative hypotheses, respectively. X is the random variable and f is the assumed probability distribution. Rejection of null hypothesis states that the given information either doesn't follows the distribution or assumed parameters or both. Step 2. Null hypothesis is defined by assuming an appropriate model for the observed data. The model must be defined with probability distribution and corresponding statistical parameters. This will stand as basis for estimating the expected frequencies. According to null hypothesis, chi square test is defined on the basis of comparison of observed 1

3 frequencies and expected frequencies as per the assumed model. The test statistics is expressed as χ 2 = k i=1 O i E i 2 E i where, χ 2 represents the chi square distribution, O i and E i are observed frequencies of the given sample and expected frequencies of the assumed model of i th interval, respectively and k is total number of intervals of the histogram. Step 3. Level of significance α is selected based on the importance or priority of the data. Usually, a level of significance of 5% is selected for the general data. This value can be reduced for higher priority or critical data. Step 4. Histogram of the observed data is formed to evaluate observed frequencies. Similarly, expected frequencies are also evaluated from assumed probability distribution and its parameters. Step 5. Accepting or rejecting of null hypothesis is dependent on the degrees of freedom of chi square distribution, level of significance and chi square value. This gives a region of rejection that above a certain χ 2 value the hypothesis is rejected. This value is based on the degrees of freedom and level of significance. The degree of freedom is defined by (k j), where j is number of quantities estimated from the given sample for use in calculating the expected frequencies. Generally, these quantities are number of observations, mean and standard deviation of sample data, hence, a total of 3 degrees of freedom are subtracted (i.e., k 3). Note, if only the number of observations is considered than the degrees of freedom is increased (k 1). Now, for a specific degrees of freedom and level of significance one can find the χ 2 value from a χ 2 value table with respect to level of significance. This is clearly explained in the following example. Step 6. If null hypothesis is rejected than an alternative hypothesis is selected and the assumed model is again chosen to conduct the chi square test from Step 1. Example Ex # 01. A series of random data of sample size 40 is mentioned below from an experimental outcome. Assumed probability distribution model is exponential distribution with level of significance 5% and 1%. Check whether the null hypothesis is rejected or accepted by conducting Chi square test

4 Solu. Initially, null hypothesis and alternate hypothesis is defined as H 0 : X ~ λ e λx H A : X λ e λx where, λ e λx is probability density function [f X x ] for exponential distribution. Before forming a histogram, one must find out mean, standard deviation, the number of intervals and interval class. Mean of the observed data is evaluated as X = x i n = For exponential distribution standard deviation is equal to mean. Parameter λ is evaluated from mean as λ = 1 X = Number of intervals k can be evaluated from as shown below k = log 10 n = log = Interval class is selected based on the difference between the minimum value and maximum value of the above mentioned sample per interval. Thus, the interval class comes out to be Class = = Now, the histogram for the observed data is formed as shown in table below Class Interval O i < E i = nf X (x i 1 < x < x i ) nf X (x i 1 < x < x i ) exp ) exp ) exp ) exp ) exp ) = E i O i E i 2 E i

5 exp ) exp ) Lecture 06: Hypothesis Testing O i = 40 E i = 40 χ 2 = Now, degrees of freedom, for this example, is evaluated as (7 3 = 4). Based on this and level of significance one can obtain χ 2 value as (for α = 5%) and (for α = 1%). According to Chi square test, null hypothesis is rejected for 5% level of significance whereas for 1% level of significance it is accepted. Kolmogorov Smirnov Test Chi square test considers the probability density whereas Kolmogorov Smirnov (KS) test considers cumulative distribution function. The philosophy KS behind is determining the maximum absolute difference between the values of cumulative distribution of given random data and assumed model as per null hypothesis. Steps for conducting KS test on a given random sample and with assumed model and its parameters are explained below. Step 1. Similar to Chi square test, null and alternate hypotheses are formulated in terms of probability distribution and statistical parameters of the random data. Also, level of significance α is also selected (generally, α = 5% is selected). Step 2. As defined above, cumulative mass density of the observed sample F O x is calculated as shown in equation below F O x = 0 for x < x 1 i for x n i < x < x i+1 1 for x x n where, x is random data placed in ascending order, n is sample size and i ranges from 1,2,, n. Step 3. Cumulative distribution of the random sample as per the assumed probability distribution and its parameters, i.e. F X x is calculated. Step 4. Finally, maximum absolute difference between the cumulative function of the observed and expected is evaluated as shown below 4

6 KS = max F X x 1 F O x 0, F X x 1 F O x 1, F X x 2 F O x 1, F X x 2 F O x 2,, F X x i F O x i 1, F X x i F O x i, F X x n F O x n 1, F X x n F O x n Step 5. Critical KS value with respect to α and n is calculated from a KS value table for comparing the observed KS value evaluated as per Eq Step 6. Like Chi square test, null hypothesis is rejected if the computed KS value is more than critical value from Step 5. Example Ex # 02. Considering Ex # 01 check whether the null hypothesis is rejected or accepted by conducting Kolmogorov Smirnov test. For ease the random data is arranged in ascending order Solu. Again, null hypothesis and alternate hypothesis, mean of the observed data and parameter λ is taken from Ex # 01. Now, for performing KS test one have to evaluate cumulative mass distribution as per Eq ,i.e. F O x i and F X x i are given in table below Rank i x i F O x i F X x i F X x i F O x i 1 F X x i F O x i

7

8 max( F X x i F O x i 1, F X x i F O x i ) = KS value observed from the random data is Critical KS values based on sample size 40 and level of significance α (5% and 1%) are evaluated as KS 5% = KS 1% = 1.36, n for n > 35 = , n for n > 35 = Thus, according to KS test, for both the cases null hypothesis is accepted as the observed value is less than the critical values. 7

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course On STRUCTURAL RELIABILITY Module # 0 Lecture Course Format: eb Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati . Lecture 0: System

More information

12.5: CHI-SQUARE GOODNESS OF FIT TESTS

12.5: CHI-SQUARE GOODNESS OF FIT TESTS 125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability

More information

3.6: General Hypothesis Tests

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.

More information

Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

More information

2. DATA AND EXERCISES (Geos2911 students please read page 8)

2. DATA AND EXERCISES (Geos2911 students please read page 8) 2. DATA AND EXERCISES (Geos2911 students please read page 8) 2.1 Data set The data set available to you is an Excel spreadsheet file called cyclones.xls. The file consists of 3 sheets. Only the third is

More information

Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

More information

Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics

Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Jan Krhovják Basic Idea Behind the Statistical Tests Generated random sequences properties as sample drawn from uniform/rectangular

More information

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

More information

Exact Confidence Intervals

Exact Confidence Intervals Math 541: Statistical Theory II Instructor: Songfeng Zheng Exact Confidence Intervals Confidence intervals provide an alternative to using an estimator ˆθ when we wish to estimate an unknown parameter

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

More information

Unit 29 Chi-Square Goodness-of-Fit Test

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

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

More information

START Selected Topics in Assurance

START Selected Topics in Assurance START Selected Topics in Assurance Related Technologies Table of Contents Introduction Some Statistical Background Fitting a Normal Using the Anderson Darling GoF Test Fitting a Weibull Using the Anderson

More information

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

More information

. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.

. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i. Chapter 3 Kolmogorov-Smirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric

More information

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 7 Multiple Linear Regression (Contd.) This is my second lecture on Multiple Linear Regression

More information

Comparing Multiple Proportions, Test of Independence and Goodness of Fit

Comparing Multiple Proportions, Test of Independence and Goodness of Fit Comparing Multiple Proportions, Test of Independence and Goodness of Fit Content Testing the Equality of Population Proportions for Three or More Populations Test of Independence Goodness of Fit Test 2

More information

Testing Random- Number Generators

Testing Random- Number Generators Testing Random- Number Generators Raj Jain Washington University Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse574-08/

More information

DOEACC Society Ministry of C&IT, Govt. of India. PSIT, Kanpur, U.P., India

DOEACC Society Ministry of C&IT, Govt. of India. PSIT, Kanpur, U.P., India Necessity of Goodness of Fit Tests in Research and Development 1 Sanjeev Kumar Jha, 2 Dr. A.K.D.Dwivedi, 3 Dr. Amod Tiwari 1,2 DOEACC Society Ministry of C&IT, Govt. of India 3 PSIT, Kanpur, U.P., India

More information

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

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

More information

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures. Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

More information

Chapter 8 Introduction to Hypothesis Testing

Chapter 8 Introduction to Hypothesis Testing Chapter 8 Student Lecture Notes 8-1 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate

More information

9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test. Neyman-Pearson lemma 9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

More information

VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA

VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA Csilla Csendes University of Miskolc, Hungary Department of Applied Mathematics ICAM 2010 Probability density functions A random variable X has density

More information

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015. Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

More information

Chapter 9, Part A Hypothesis Tests. Learning objectives

Chapter 9, Part A Hypothesis Tests. Learning objectives Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population

More information

Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test

Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test Nonparametric Two-Sample Tests Sign test Mann-Whitney U-test (a.k.a. Wilcoxon two-sample test) Kolmogorov-Smirnov Test Wilcoxon Signed-Rank Test Tukey-Duckworth Test 1 Nonparametric Tests Recall, nonparametric

More information

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 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

More information

Hypothesis testing - Steps

Hypothesis testing - Steps Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

More information

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test... Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

More information

Lecture Outline. Hypothesis Testing. Simple vs. Composite Testing. Stat 111. Hypothesis Testing Framework

Lecture Outline. Hypothesis Testing. Simple vs. Composite Testing. Stat 111. Hypothesis Testing Framework Stat 111 Lecture Outline Lecture 14: Intro to Hypothesis Testing Sections 9.1-9.3 in DeGroot 1 Hypothesis Testing Consider a statistical problem involving a parameter θ whose value is unknown but must

More information

CHI-SQUARE: TESTING FOR GOODNESS OF FIT

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

More information

Non Parametric Inference

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

More information

How to Conduct a Hypothesis Test

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

More information

4. Sum the results of the calculation described in step 3 for all classes of progeny

4. Sum the results of the calculation described in step 3 for all classes of progeny F09 Biol 322 chi square notes 1. Before proceeding with the chi square calculation, clearly state the genetic hypothesis concerning the data. This hypothesis is an interpretation of the data that gives

More information

Reliability and Risk Analysis. Analysis of Model Tasks from Selected Application Areas Using a Computer (software R, Excel)

Reliability and Risk Analysis. Analysis of Model Tasks from Selected Application Areas Using a Computer (software R, Excel) Reliability and Risk Analysis Analysis of Model Tasks from Selected Application Areas Using a Computer (software R, Excel) Task The aim of this study is to assess the wastewater treatment plant (WWTP)

More information

Having a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails.

Having a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails. Chi-square Goodness of Fit Test The chi-square test is designed to test differences whether one frequency is different from another frequency. The chi-square test is designed for use with data on a nominal

More information

The Chi Square Test. Diana Mindrila, Ph.D. Phoebe Balentyne, M.Ed. Based on Chapter 23 of The Basic Practice of Statistics (6 th ed.

The Chi Square Test. Diana Mindrila, Ph.D. Phoebe Balentyne, M.Ed. Based on Chapter 23 of The Basic Practice of Statistics (6 th ed. The Chi Square Test Diana Mindrila, Ph.D. Phoebe Balentyne, M.Ed. Based on Chapter 23 of The Basic Practice of Statistics (6 th ed.) Concepts: Two-Way Tables The Problem of Multiple Comparisons Expected

More information

Stats Review Chapters 9-10

Stats Review Chapters 9-10 Stats Review Chapters 9-10 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

START Selected Topics in Assurance

START Selected Topics in Assurance START Selected Topics in Assurance Related Technologies Table of Contents Introduction Some Statistical Background Fitting a Normal Using the Anderson Darling GoF Test Fitting a Weibull Using the Anderson

More information

Lecture 13 More on hypothesis testing

Lecture 13 More on hypothesis testing Lecture 13 More on hypothesis testing Thais Paiva STA 111 - Summer 2013 Term II July 22, 2013 1 / 27 Thais Paiva STA 111 - Summer 2013 Term II Lecture 13, 07/22/2013 Lecture Plan 1 Type I and type II error

More information

MATH 10: Elementary Statistics and Probability Chapter 11: The Chi-Square Distribution

MATH 10: Elementary Statistics and Probability Chapter 11: The Chi-Square Distribution MATH 10: Elementary Statistics and Probability Chapter 11: The Chi-Square Distribution Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

More information

Lecture Notes Module 1

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

More information

Traffic Behavior Analysis with Poisson Sampling on High-speed Network 1

Traffic Behavior Analysis with Poisson Sampling on High-speed Network 1 Traffic Behavior Analysis with Poisson Sampling on High-speed etwork Guang Cheng Jian Gong (Computer Department of Southeast University anjing 0096, P.R.China) Abstract: With the subsequent increasing

More information

Projects Involving Statistics (& SPSS)

Projects Involving Statistics (& SPSS) Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,

More information

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST UNDERSTANDING THE DEPENDENT-SAMPLES t TEST A dependent-samples t test (a.k.a. matched or paired-samples, matched-pairs, samples, or subjects, simple repeated-measures or within-groups, or correlated groups)

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

Chi Square Tests. Chapter 10. 10.1 Introduction

Chi Square Tests. Chapter 10. 10.1 Introduction Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

More information

Module 2 Probability and Statistics

Module 2 Probability and Statistics Module 2 Probability and Statistics BASIC CONCEPTS Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The standard deviation of a standard normal distribution

More information

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

More information

11-2 Goodness of Fit Test

11-2 Goodness of Fit Test 11-2 Goodness of Fit Test In This section we consider sample data consisting of observed frequency counts arranged in a single row or column (called a one-way frequency table). We will use a hypothesis

More information

Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling

Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling Instructor: Sargur N. University at Buffalo The State University of New York srihari@cedar.buffalo.edu Topics 1. Hypothesis Testing 1.

More information

Basic concepts and introduction to statistical inference

Basic concepts and introduction to statistical inference Basic concepts and introduction to statistical inference Anna Helga Jonsdottir Gunnar Stefansson Sigrun Helga Lund University of Iceland (UI) Basic concepts 1 / 19 A review of concepts Basic concepts Confidence

More information

Non-Inferiority Tests for One Mean

Non-Inferiority Tests for One Mean Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random

More information

Chapter 9: Hypothesis Testing Sections

Chapter 9: Hypothesis Testing Sections Chapter 9: Hypothesis Testing Sections - we are still here Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.5 The t Test 9.6 Comparing the

More information

Pearson's Correlation Tests

Pearson's Correlation Tests Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation

More information

For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random

For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random Lecture.9 Test of significance Basic concepts null hypothesis alternative hypothesis level of significance Standard error and its importance steps in testing Test of Significance Objective To familiarize

More information

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 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

More information

Inferential Statistics

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

More information

ORF 245 Fundamentals of Statistics Chapter 8 Hypothesis Testing

ORF 245 Fundamentals of Statistics Chapter 8 Hypothesis Testing ORF 245 Fundamentals of Statistics Chapter 8 Hypothesis Testing Robert Vanderbei Fall 2015 Slides last edited on December 11, 2015 http://www.princeton.edu/ rvdb Coin Tossing Example Consider two coins.

More information

Dongfeng Li. Autumn 2010

Dongfeng Li. Autumn 2010 Autumn 2010 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis

More information

Chapter 10 Monte Carlo Methods

Chapter 10 Monte Carlo Methods 411 There is no result in nature without a cause; understand the cause and you will have no need for the experiment. Leonardo da Vinci (1452-1519) Chapter 10 Monte Carlo Methods In very broad terms one

More information

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96 1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

More information

How Does My TI-84 Do That

How Does My TI-84 Do That How Does My TI-84 Do That A guide to using the TI-84 for statistics Austin Peay State University Clarksville, Tennessee How Does My TI-84 Do That A guide to using the TI-84 for statistics Table of Contents

More information

CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES

CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES CHAPTER 11. GOODNESS OF FIT AND CONTINGENCY TABLES The chi-square distribution was discussed in Chapter 4. We now turn to some applications of this distribution. As previously discussed, chi-square is

More information

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

More information

First-year Statistics for Psychology Students Through Worked Examples

First-year Statistics for Psychology Students Through Worked Examples First-year Statistics for Psychology Students Through Worked Examples 1. THE CHI-SQUARE TEST A test of association between categorical variables by Charles McCreery, D.Phil Formerly Lecturer in Experimental

More information

Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

1 The Pareto Distribution

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

More information

Descriptive Statistics

Descriptive Statistics Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

More information

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

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

More information

NEW TABLE AND NUMERICAL APPROXIMATIONS FOR KOLMOGOROV-SMIRNOV/LILLIEFORS/VAN SOEST NORMALITY TEST

NEW TABLE AND NUMERICAL APPROXIMATIONS FOR KOLMOGOROV-SMIRNOV/LILLIEFORS/VAN SOEST NORMALITY TEST NEW TABLE AND NUMERICAL APPROXIMATIONS FOR KOLMOGOROV-SMIRNOV/LILLIEFORS/VAN SOEST NORMALITY TEST PAUL MOLIN AND HERVÉ ABDI Abstract. We give new critical values for the Kolmogorov-Smirnov/Lilliefors/Van

More information

Chapter 3 RANDOM VARIATE GENERATION

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.

More information

Testing on proportions

Testing on proportions Testing on proportions Textbook Section 5.4 April 7, 2011 Example 1. X 1,, X n Bernolli(p). Wish to test H 0 : p p 0 H 1 : p > p 0 (1) Consider a related problem The likelihood ratio test is where c is

More information

Chapter 1 Hypothesis Testing

Chapter 1 Hypothesis Testing Chapter 1 Hypothesis Testing Principles of Hypothesis Testing tests for one sample case 1 Statistical Hypotheses They are defined as assertion or conjecture about the parameter or parameters of a population,

More information

Chapter 11 Chi square Tests

Chapter 11 Chi square Tests 11.1 Introduction 259 Chapter 11 Chi square Tests 11.1 Introduction In this chapter we will consider the use of chi square tests (χ 2 tests) to determine whether hypothesized models are consistent with

More information

Non-Parametric Tests (I)

Non-Parametric Tests (I) Lecture 5: Non-Parametric Tests (I) KimHuat LIM lim@stats.ox.ac.uk http://www.stats.ox.ac.uk/~lim/teaching.html Slide 1 5.1 Outline (i) Overview of Distribution-Free Tests (ii) Median Test for Two Independent

More information

PASS Sample Size Software. Linear Regression

PASS Sample Size Software. Linear Regression Chapter 855 Introduction Linear regression is a commonly used procedure in statistical analysis. One of the main objectives in linear regression analysis is to test hypotheses about the slope (sometimes

More information

Chi Square for Contingency Tables

Chi Square for Contingency Tables 2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure

More information

Choosing the correct statistical test made easy

Choosing the correct statistical test made easy Classroom Choosing the correct statistical test made easy N Gunawardana Senior Lecturer in Community Medicine, Faculty of Medicine, University of Colombo Gone are the days where researchers had to perform

More information

3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples 3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

More information

Gamma Distribution Fitting

Gamma Distribution Fitting Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics

More information

Lesson19: Comparing Predictive Accuracy of two Forecasts: Th. Diebold-Mariano Test

Lesson19: Comparing Predictive Accuracy of two Forecasts: Th. Diebold-Mariano Test Lesson19: Comparing Predictive Accuracy of two Forecasts: The Diebold-Mariano Test Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it

More information

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE IN STATISTICS, 2014 MODULE 3 : Basic statistical methods Time allowed: One and a half hours Candidates should answer THREE questions. Each

More information

Study Guide for the Final Exam

Study Guide for the Final Exam Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

More information

Develop hypothesis and then research to find out if it is true. Derived from theory or primary question/research questions

Develop hypothesis and then research to find out if it is true. Derived from theory or primary question/research questions Chapter 12 Hypothesis Testing Learning Objectives Examine the process of hypothesis testing Evaluate research and null hypothesis Determine one- or two-tailed tests Understand obtained values, significance,

More information

Chapter 9: Hypothesis Testing Sections

Chapter 9: Hypothesis Testing Sections Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.5 The t Test 9.6

More information

1. Comparing Two Means: Dependent Samples

1. Comparing Two Means: Dependent Samples 1. Comparing Two Means: ependent Samples In the preceding lectures we've considered how to test a difference of two means for independent samples. Now we look at how to do the same thing with dependent

More information

Pr(X = x) = f(x) = λe λx

Pr(X = x) = f(x) = λe λx Old Business - variance/std. dev. of binomial distribution - mid-term (day, policies) - class strategies (problems, etc.) - exponential distributions New Business - Central Limit Theorem, standard error

More information

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

More information

1. The Classical Linear Regression Model: The Bivariate Case

1. The Classical Linear Regression Model: The Bivariate Case Business School, Brunel University MSc. EC5501/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 018956584) Lecture Notes 3 1.

More information

MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators... MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

More information

Randomized Block Analysis of Variance

Randomized Block Analysis of Variance Chapter 565 Randomized Block Analysis of Variance Introduction This module analyzes a randomized block analysis of variance with up to two treatment factors and their interaction. It provides tables of

More information

Statistics Review PSY379

Statistics Review PSY379 Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

More information

Elementary Statistics Sample Exam #3

Elementary Statistics Sample Exam #3 Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to

More information

HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

More information

Mobile Systems Security. Randomness tests

Mobile Systems Security. Randomness tests Mobile Systems Security Randomness tests Prof RG Crespo Mobile Systems Security Randomness tests: 1/6 Introduction (1) [Definition] Random, adj: lacking a pattern (Longman concise English dictionary) Prof

More information