Probability and Probability Distributions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Probability and Probability Distributions"

Transcription

1 Chapter 1-3 Probability and Probability Distributions 1-3-1

2 Probability The relative frequency of occurrence of an event is known as its probability. It is the ratio of the number of favorable events to the total number of possible events

3 Example An inspector checks a sample of 1000 parts for conformance to standard. 40 do not meet standard. What is the probability of a defect? P(d) = 40/1000 =

4 Probability Definitions Two events are mutually exclusive if they have no outcomes in common or if when one occurs the other cannot. Two events are non-mutually exclusive when they share a common area of occurrence or can both happen at the same time

5 Addition Rule for Mutually Exclusive Events P(A or B) = P(A) + P(B) 1-3-5

6 Example A sample of 100 employees is selected at random from the company. Within this sample are 61 males and 39 females. Find the probability of Finding a male Finding a female Finding a male or a female 1-3-6

7 Solution Male: 61/100 =.61 Female: 39/100 =.39 Male or Female = = 1.00 (Addition Rule) 1-3-7

8 Replacement Sampling A sample or group of samples is returned to the population after sampling so that the probabilities associated with selecting additional items are not changed

9 Multiplication Rule for Independent Events The probability that one event will occur and that a second independent event will also occur requires the use of the multiplication rule. P(A and B) = P(A)P(B) 1-3-9

10 Example What is the probability of selecting two clubs in two draws from a deck of cards if each card is replaced in the deck after it is drawn? P(club) = 13/52 P(club and club) = (13/52)(13/52) = (169)/(2704) =

11 Probability Distributions Complete listing of all possible outcomes along with the likelihood that each will occur. Discrete distribution is one in which the observed characteristics fit into a finite number of categories. Continuous distribution is one in which the observed characteristic may take on any value within a given range

12 Practical Uses of Probability Distributions Some practical uses of probability distributions are: To calculate confidence intervals for parameters and to calculate critical regions for hypothesis tests. For univariate data, it is often useful to determine a reasonable distributional model for the data. Statistical intervals and hypothesis tests are often based on specific distributional assumptions. Before computing an interval or test based on a distributional assumption, we need to verify that the assumption is justified for the given data set. In this case, the distribution does not need to be the best-fitting distribution for the data, but an adequate enough model so that the statistical technique yields valid conclusions. Simulation studies with random numbers generated from using a specific probability distribution are often needed

13 Probability Distributions The values of a probability distribution must be numbers on the interval from 0 to 1. The sum of all the values of a probability distribution must be equal to

14 Sample Probability Distribution Dice Outcome Ways to Achieve Ways Total: Data Set Relative Frequency

15 Binomial Distribution A discrete probability distribution used in defining the probability of favorable occurrences in a sample. Often used to define the probability of finding defects in a sample. Each event must have a constant probability of occurrence and each must be independent

16 Binomial Probability Formula n! P( x) x!( n x)! p Mean np Stdev np( 1 p) x ( p) nx 1 P(x) is the probability of a favorable event x n is the sample size p is probability of single favorable event! Is the symbol for factorial x is the favorable event

17 Example A component supplied by a certain vendor has a 10 percent chance of being defective. Upon receipt of a lot a sample of 10 is selected at random. What is the probability the sample will contain 2 defects 2 or fewer defects

18 Solution Part a Because an item can be good or defective this is binomial. n = 10 p =.1 x = 2 P(x) = ((n!)/x!(n-x)!)(p) x (1-p) n-x P(2) = ((10!)/(2!(10-2)!))(.1) 2 (.90) 8 P(2) = (45)(.010)(.4305) =

19 Solution Part b The probability of 2 or fewer is the probability of 2 or 1 or 0. P(2) =.1937 P(1) = ((10!)/(1!9!))(.1) 1 (.9) 9 =.3874 P(0) = ((10!)/(0!10!))(.1) 0 (.9) 10 =.3487 P(2 or less) = P(2 or less) =

20 Probability Table Values of the binomial probability distribution are included in the appendix to this training book and are found in Table A

21 1-3-21

22 1-3-22

23 SPCXL

24 1-3-24

25 Poisson Distribution The Poisson distribution is a discrete probability distribution used to determine the probability of x occurrences in a sample of n where the probability of a favorable event is constant, but relatively small. Often used when the probability is expressed as a rate

26 Poisson Probabilities P( x) Mean Stdev ( np) x x! np np e np P(x) is the probability of a favorable event x n is the sample size p is probability of single favorable event! Is the symbol for factorial x is the favorable event e =

27 Example Use the Poisson table (Table B) to answer the following: A textile manufacturer has kept records on the number of defects per yard of material. The probability has been.04. What is the probability that 20 yards of material will have 2 defects 3 or fewer defects

28 Excel

29 SPCXL

30 Normal Probability Distribution A continuous probability distribution used when there is a concentration of observations about the mean and equal likelihood that observations will occur above and below the mean

31 THE NORMAL LAW OF ERROR STANDS OUT IN THE EXPERIENCE OF MANKIND AS ONE OF THE BROADEST GENERALIZATIONS OF NATURAL PHILOSOPHY. IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES IN THE PHYSICAL AND SOCIAL SCIENCES AND IN MEDICINE, AGRICULTURE, AND ENGINEERING. IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION

32 Normal Probability Distribution The mean and standard deviation are calculated with standard procedures. Distribution is symmetrical about the mean. y e 1 x ( ) 2 s s

33 Standard Normal Curve Normal probability distribution with a mean of 0 and a standard deviation of 1. Values tabulated z x s x (Table C) Table accessed with z transform (typical format shown)

34 Example Sixteen samples of product weight were measured. They averaged 80 with a standard deviation of 6. What is the probability that a sample taken at random from future production will be a. Greater than 83? b. Less than 78?

35 Example A product has specs of and A sample of 25 has an average of and a standard deviation of.1 What is the probability that an item will be a. Greater than 16.05? b. Less than 16.00? c. Between and 16.04?

36 1-3-36

37 1-3-37

38 Distribution of Sample Means z x - s n A population is normally distributed with a mean of 62 and a standard deviation of 3.5. What is the probability that a sample mean taken from a sample of size 16 will be larger than 60?

39 Chi Square Distribution The chi-square distribution results when independent variables with standard normal distributions are squared and summed. The formula for the probability density function of the chi-square distribution is f ( x) e 2 x 2 n 2 x n 1 2 n G( ) 2 for x 0 where n is the shape parameter and G is the gamma function. The formula for the gamma function is G( a) t 0 a1 e l dt

40 1-3-40

41 F Distribution The F distribution is the ratio of two chi-square distributions with degrees of freedom n 1 and n 2, respectively, where each chi-square has first been divided by its degrees of freedom. The formula for the probability density function of the F distribution is f n1 n 2 n1 G( )( ) 2 n n1 2 2 ( x) n1n 2 n1 n 2 n1x 2 G( ) G( )(1 ) 2 2 n where n 1 and n 2 are the shape parameters and G is the gamma function. 2 x n

42 1-3-42

43 Student t Distribution The t distribution is used instead of the normal distribution whenever the standard deviation is estimated. The t distribution has relatively more scores in its tails than does the normal distribution. As the degrees of freedom increases, the t distribution approaches the normal distribution

44 Student t Distribution The formula for the probability density function of the t distribution is f ( x) 2 ( n 1) x 2 (1 ) n b (0.5,0.5n ) where b is the beta function and n is a positive integer shape parameter. The formula for the beta function is n b 1 b 1 (, b) t (1 t) dt

45 1-3-45

46 Hypergeometric Distribution A distribution closely related to the binomial that has a wide applicability in acceptance sampling. x is the number of favorable events N is the lot size n is the sample size D is the number of favorable events in the population P(x) D x N n N n D x [ ] Notation means combinations

47 Which Distribution to Use? The question continues to arise regarding which probability distribution should be used when. The following conditions, originally proposed by Burr in Elementary Statistical Quality Control address this issue. Conditions for a Binomial Distribution Constant probability of a defective on each draw of a piece is established. Results on drawings are independent of preceding results. n draws of a piece are taken. Number of defectives in the n draws are counted. Either/Or

48 Conditions for a Poisson Distribution Samples provide equal areas of opportunity for defects (the same size of unit, subassembly, length, area, or quantity.) Defects occur randomly and independently of each other. The average number remains constant. The possible number of defects is far greater than the average number of defects. Rate

49 Properties of a Standard Normal Distribution The total area lying between the curve and the horizontal z axis is 1, representing a total probability of 1. The height of the curve represents relative frequency. The relative frequency is greatest at z = 0. Relative frequencies steadily decrease as z moves away from 0 in either direction. The relative frequency rapidly approaches 0 in both directions. The curve is symmetrical around z = 0. Desired probabilities for ranges of z may be found from published tables of normal curve areas

50 Some Common Applications of Distributions Distribution Uses Hypergeometric Sampling Binomial Go-No Go Sampling Confidence Intervals - Proportions Control Charts-Proportion Defective Poisson Sampling Queueing - Arrival Rates Control Charts-Defects per Unit Normal Process Capability Tests of Hypothesis-Means, Correlation Coefficients Confidence Intervals - Means Student t Tests of Hypothesis Means Confidence Intervals - Means Chi Square Goodness of Fit Testing Confidence Intervals Standard Deviations F Tests of Hypothesis Standard Deviations Analysis of Variance Exponential Reliability Queueing - Service Rates Weibull Reliability

51 Descriptive Statistics Distribution Mean Standard Deviation Binomial np np(1 - p) Poisson np np Normal Student t x n x n (x n (x n x ) x ) 2 2 Exponential x n x n

52 Practice Problems In problems 1-3, use a binomially distributed sample with p =.2 and n = 14 to determine the probability of each event. 1. x is equal to x is less than x is greater than x is between 2 and Calculate the mean and the standard deviation for a binomially distributed sample with p =.2 and n =

53 Practice Problems In problems 6-10 use a binomially distributed sample with p =.6 and n = 10 to determine the probability of each event. 6. x is equal to x is greater than x is less than x is greater than x is between 0 and

54 Practice Problems In problems use a binomially distributed sample with p =.45 and n = 20 to determine the probability of each event. 11. x is greater than x is less than x is equal to x is between 2 and 7. In problems given that a sample follows the Poisson distribution and has a mean of.95, determine the requested value. 15. The standard deviation. 16. The probability that x is greater than The probability that x is less than The probability that x is between 1 and

55 Practice Problems 19. Given a normal probability distribution with a mean of 79 and a standard deviation of 4, at what value would we expect to find at least as large as 70? 20. Given a normal probability distribution with a mean of 20 and a standard deviation of.5. If we want to have 99% of all the process output between symmetrical specifications what should those specifications be? 21. What z value corresponds to having a probability between the point and minus infinity of.025? 22. What z value corresponds to have a probaiblity between the point and positive infinity of.100?

56 Practice Problems In problems 23-32, determine the probability of each event for a normally distributed process with a mean of 120 and a standard deviation of x is greater than x is less than x is greater than x is less than x is greater than x is between 90 and x is between 125 and x is between 128 and x is between 116 and x is equal to

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

More information

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

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

More information

Normal distribution. ) 2 /2σ. 2π σ

Normal distribution. ) 2 /2σ. 2π σ Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

More information

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4) Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Choosing Probability Distributions in Simulation

Choosing Probability Distributions in Simulation MBA elective - Models for Strategic Planning - Session 14 Choosing Probability Distributions in Simulation Probability Distributions may be selected on the basis of Data Theory Judgment a mix of the above

More information

The Procedures of Monte Carlo Simulation (and Resampling)

The Procedures of Monte Carlo Simulation (and Resampling) 154 Resampling: The New Statistics CHAPTER 10 The Procedures of Monte Carlo Simulation (and Resampling) A Definition and General Procedure for Monte Carlo Simulation Summary Until now, the steps to follow

More information

STAT 35A HW2 Solutions

STAT 35A HW2 Solutions STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

More information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

More information

Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2. Probability Distributions for Discrete Random Variables Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

More information

Characteristics of Binomial Distributions

Characteristics of Binomial Distributions Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

More information

EMPIRICAL FREQUENCY DISTRIBUTION

EMPIRICAL FREQUENCY DISTRIBUTION INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION - described by mathematical models 2 1 when some empirical distribution approximates

More information

Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2

Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2 Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable

More information

Statistical Functions in Excel

Statistical Functions in Excel Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

More information

UNIVERSITY of MASSACHUSETTS DARTMOUTH Charlton College of Business Decision and Information Sciences Fall 2010

UNIVERSITY of MASSACHUSETTS DARTMOUTH Charlton College of Business Decision and Information Sciences Fall 2010 UNIVERSITY of MASSACHUSETTS DARTMOUTH Charlton College of Business Decision and Information Sciences Fall 2010 COURSE: POM 500 Statistical Analysis, ONLINE EDITION, Fall 2010 Prerequisite: Finite Math

More information

Chapter 4. Probability Distributions

Chapter 4. Probability Distributions Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

More information

Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

More information

Point Biserial Correlation Tests

Point Biserial Correlation Tests Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the product-moment correlation calculated between a continuous random variable

More information

Association Between Variables

Association Between Variables Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

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

BINOMIAL DISTRIBUTION

BINOMIAL DISTRIBUTION MODULE IV BINOMIAL DISTRIBUTION A random variable X is said to follow binomial distribution with parameters n & p if P ( X ) = nc x p x q n x where x = 0, 1,2,3..n, p is the probability of success & q

More information

Quantitative Methods for Finance

Quantitative Methods for Finance Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain

More information

Probability Distributions

Probability Distributions CHAPTER 6 Probability Distributions Calculator Note 6A: Computing Expected Value, Variance, and Standard Deviation from a Probability Distribution Table Using Lists to Compute Expected Value, Variance,

More information

Simple Random Sampling

Simple Random Sampling Source: Frerichs, R.R. Rapid Surveys (unpublished), 2008. NOT FOR COMMERCIAL DISTRIBUTION 3 Simple Random Sampling 3.1 INTRODUCTION Everyone mentions simple random sampling, but few use this method for

More information

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology Step-by-Step - Excel Microsoft Excel is a spreadsheet software application

More information

6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS 6 PROBABILITY GENERATING FUNCTIONS Certain derivations presented in this course have been somewhat heavy on algebra. For example, determining the expectation of the Binomial distribution (page 5.1 turned

More information

Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

More information

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

More information

Exploratory Data Analysis

Exploratory Data Analysis Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k. REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

More information

3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.

3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved. 3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately

More information

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different

More information

The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math. P (x) = 5! = 1 2 3 4 5 = 120. The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

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

Normality Testing in Excel

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

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Probability and Probability Distributions 1. Introduction 2. Probability 3. Basic rules of probability 4. Complementary events 5. Addition Law for

More information

Unit 4 The Bernoulli and Binomial Distributions

Unit 4 The Bernoulli and Binomial Distributions PubHlth 540 4. Bernoulli and Binomial Page 1 of 19 Unit 4 The Bernoulli and Binomial Distributions Topic 1. Review What is a Discrete Probability Distribution... 2. Statistical Expectation.. 3. The Population

More information

Confidence Intervals in Public Health

Confidence Intervals in Public Health Confidence Intervals in Public Health When public health practitioners use health statistics, sometimes they are interested in the actual number of health events, but more often they use the statistics

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

Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000 Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

More information

Aachen Summer Simulation Seminar 2014

Aachen Summer Simulation Seminar 2014 Aachen Summer Simulation Seminar 2014 Lecture 07 Input Modelling + Experimentation + Output Analysis Peer-Olaf Siebers pos@cs.nott.ac.uk Motivation 1. Input modelling Improve the understanding about how

More information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

More information

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

More information

AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

More information

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

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

Generalized Linear Models

Generalized Linear Models Generalized Linear Models We have previously worked with regression models where the response variable is quantitative and normally distributed. Now we turn our attention to two types of models where the

More information

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010 Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

2. Discrete random variables

2. Discrete random variables 2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be

More information

16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION 6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of

More information

Chi-square test Fisher s Exact test

Chi-square test Fisher s Exact test Lesson 1 Chi-square test Fisher s Exact test McNemar s Test Lesson 1 Overview Lesson 11 covered two inference methods for categorical data from groups Confidence Intervals for the difference of two proportions

More information

Probability Distributions

Probability Distributions CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution

More information

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

More information

MAT 155. Key Concept. September 27, 2010. 155S5.5_3 Poisson Probability Distributions. Chapter 5 Probability Distributions

MAT 155. Key Concept. September 27, 2010. 155S5.5_3 Poisson Probability Distributions. Chapter 5 Probability Distributions MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance and Standard

More information

9. Sampling Distributions

9. Sampling Distributions 9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling

More information

Confidence Intervals for One Standard Deviation Using Standard Deviation

Confidence Intervals for One Standard Deviation Using Standard Deviation Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

More information

CHAPTER 2 Estimating Probabilities

CHAPTER 2 Estimating Probabilities CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a

More information

COMMON CORE STATE STANDARDS FOR

COMMON CORE STATE STANDARDS FOR COMMON CORE STATE STANDARDS FOR Mathematics (CCSSM) High School Statistics and Probability Mathematics High School Statistics and Probability Decisions or predictions are often based on data numbers in

More information

LAB : THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

LAB : THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics Period Date LAB : THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

More information

University of Chicago Graduate School of Business. Business 41000: Business Statistics Solution Key

University of Chicago Graduate School of Business. Business 41000: Business Statistics Solution Key Name: OUTLINE SOLUTIONS University of Chicago Graduate School of Business Business 41000: Business Statistics Solution Key Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper

More information

STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4

STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate

More information

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

More information

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

More information

DETERMINE whether the conditions for a binomial setting are met. COMPUTE and INTERPRET probabilities involving binomial random variables

DETERMINE whether the conditions for a binomial setting are met. COMPUTE and INTERPRET probabilities involving binomial random variables 1 Section 7.B Learning Objectives After this section, you should be able to DETERMINE whether the conditions for a binomial setting are met COMPUTE and INTERPRET probabilities involving binomial random

More information

Advanced Statistical Analysis of Mortality. Rhodes, Thomas E. and Freitas, Stephen A. MIB, Inc. 160 University Avenue. Westwood, MA 02090

Advanced Statistical Analysis of Mortality. Rhodes, Thomas E. and Freitas, Stephen A. MIB, Inc. 160 University Avenue. Westwood, MA 02090 Advanced Statistical Analysis of Mortality Rhodes, Thomas E. and Freitas, Stephen A. MIB, Inc 160 University Avenue Westwood, MA 02090 001-(781)-751-6356 fax 001-(781)-329-3379 trhodes@mib.com Abstract

More information

AP STATISTICS REVIEW (YMS Chapters 1-8)

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

More information

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

DECISION MAKING UNDER UNCERTAINTY:

DECISION MAKING UNDER UNCERTAINTY: DECISION MAKING UNDER UNCERTAINTY: Models and Choices Charles A. Holloway Stanford University TECHNISCHE HOCHSCHULE DARMSTADT Fachbereich 1 Gesamtbibliothek Betrtebswirtscrtaftslehre tnventar-nr. :...2>2&,...S'.?S7.

More information

Notes on the Negative Binomial Distribution

Notes on the Negative Binomial Distribution Notes on the Negative Binomial Distribution John D. Cook October 28, 2009 Abstract These notes give several properties of the negative binomial distribution. 1. Parameterizations 2. The connection between

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) If two events are mutually exclusive, what is the probability that one or the other occurs? A)

More information

A Tutorial on Probability Theory

A Tutorial on Probability Theory Paola Sebastiani Department of Mathematics and Statistics University of Massachusetts at Amherst Corresponding Author: Paola Sebastiani. Department of Mathematics and Statistics, University of Massachusetts,

More information

What is the purpose of this document? What is in the document? How do I send Feedback?

What is the purpose of this document? What is in the document? How do I send Feedback? This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Statistics

More information

MINITAB ASSISTANT WHITE PAPER

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

More information

Binomial Sampling and the Binomial Distribution

Binomial Sampling and the Binomial Distribution Binomial Sampling and the Binomial Distribution Characterized by two mutually exclusive events." Examples: GENERAL: {success or failure} {on or off} {head or tail} {zero or one} BIOLOGY: {dead or alive}

More information

MAS108 Probability I

MAS108 Probability I 1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

More information

Descriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics

Descriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),

More information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

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

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm

More information

List of Examples. Examples 319

List of Examples. Examples 319 Examples 319 List of Examples DiMaggio and Mantle. 6 Weed seeds. 6, 23, 37, 38 Vole reproduction. 7, 24, 37 Wooly bear caterpillar cocoons. 7 Homophone confusion and Alzheimer s disease. 8 Gear tooth strength.

More information

CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution,

More information

The normal approximation to the binomial

The normal approximation to the binomial The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

More information

Probability and statistical hypothesis testing. Holger Diessel holger.diessel@uni-jena.de

Probability and statistical hypothesis testing. Holger Diessel holger.diessel@uni-jena.de Probability and statistical hypothesis testing Holger Diessel holger.diessel@uni-jena.de Probability Two reasons why probability is important for the analysis of linguistic data: Joint and conditional

More information

Sample Term Test 2A. 1. A variable X has a distribution which is described by the density curve shown below:

Sample Term Test 2A. 1. A variable X has a distribution which is described by the density curve shown below: Sample Term Test 2A 1. A variable X has a distribution which is described by the density curve shown below: What proportion of values of X fall between 1 and 6? (A) 0.550 (B) 0.575 (C) 0.600 (D) 0.625

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

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

Introduction to Hypothesis Testing

Introduction to Hypothesis Testing I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

More information

Crosstabulation & Chi Square

Crosstabulation & Chi Square Crosstabulation & Chi Square Robert S Michael Chi-square as an Index of Association After examining the distribution of each of the variables, the researcher s next task is to look for relationships among

More information

http://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions

http://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp.

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

6 POISSON DISTRIBUTIONS

6 POISSON DISTRIBUTIONS 6 POISSON DISTRIBUTIONS Chapter 6 Poisson Distributions Objectives After studying this chapter you should be able to recognise when to use the Poisson distribution; be able to apply the Poisson distribution

More information

STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE

STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about

More information

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

More information

Practice Problems #4

Practice Problems #4 Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

The Binomial Distribution

The Binomial Distribution The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing

More information

Education & Training Plan. Accounting Math Professional Certificate Program with Externship

Education & Training Plan. Accounting Math Professional Certificate Program with Externship Office of Professional & Continuing Education 301 OD Smith Hall Auburn, AL 36849 http://www.auburn.edu/mycaa Contact: Shavon Williams 334-844-3108; szw0063@auburn.edu Auburn University is an equal opportunity

More information

STAT 360 Probability and Statistics. Fall 2012

STAT 360 Probability and Statistics. Fall 2012 STAT 360 Probability and Statistics Fall 2012 1) General information: Crosslisted course offered as STAT 360, MATH 360 Semester: Fall 2012, Aug 20--Dec 07 Course name: Probability and Statistics Number

More information