# Binomial Random Variables

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Binomial Random Variables Dr Tom Ilvento Department of Food and Resource Economics Overview A special case of a Discrete Random Variable is the Binomial This happens when the result of the eperiment is a dichotomy Success or Failure Yes or No Cured or not Cured If the discrete random variable is a binomial, we have some easier ways to solve for probabilities Formula Probability Table And the solution for the Mean and Variance is much easier to solve Binomial Random Variable In many cases the responses to an eperiment are dichotomous Sucess/Failure Yes/No Alive/Dead Support/Don t Support When our focus is conducting an eperiment n times independently and observing the number of times that one of the two outcomes occurs (Success) And the probability of success, p, remains the same from trial to trial This X is a Binomial Random Variable We can eploit this by using known formulas for a Binomial Probability Distribution Conduct an eperiment n times and observe the number of times that Success occurs 3 Characteristics of a Binomial Distribution The eperiment consists of n identical trials There are only two outcomes on each trial. Outcomes can be denoted as S for Success F for Failure The probability of S (success) remains the same from trial to trail Denoted as p the proportion The probability of F (failure) Denoted as q q=(1-p) The trials are independent of each other The binomial random variable is the number of Successes in n trials 4

2 Eample of a Binomial Random Variable: Marketing Survey Marketing survey of 100 randomly chosen consumers Record their preferences for a new and an old diet soda ask them to choose their preference Let be number of 100 who choose the new brand This is a binomial random variable Eample of a Binomial Random Variable: Fitness Eample Heart Association says only 10% of adults over 30 can pass the fitness test Suppose 4 people over 30 are selected at random Let X be the number who pass the minimum requirements Find the probability distribution for X Conduct an eperiment 100 times and observe the number of times that the subject chooses the new brand Conduct an eperiment 4 times and observe the number of times that pass occurs 5 6 How to solve this using the strategy of a Discrete Random Variable Can you solve it the probability that eactly 1 person passes the test? 1. List the events. List the sample points that refer to that event 3. Calculate the probabilities p =.1 and q = ( ) =.9 Event X Sample Points Probability All Fail FFFF (.9)(.9)(.9)(.9) =.6561 I multiply through on the probabilities because each trial is independent of the others 7 Count the ways we could have only one pass, and three failures Assign probabilities to this event SFFF FSFF FFSF FFFS For each combination, the probabilities are:.1*.9*.9*.9 And there are four ways to get one pass 4[.1*.9*.9*.9] =.916 Another way to write it is 4[.1*.9 3 ] =.916 8

3 Let s finish solving for the whole table The number of times that an adult passes in a sample of four Probability Distribution Event X Sample Points Probability 0 All Fail 1 One passes Two Pass FFFF (.9)(.9)(.9)(.9) =.6561 SFFF FSFF FFSF FFFS 4[(.1)(.9) 3 ] =.916 SSFF SFSF SFFS FSSF FSFS FFSS 6[(.1)(.9) ] =.0486 When = 0 All Fail P =.6561 When = 1 One Pass P =.916 When = Two pass P =.0486 When =3 Three pass P =.0036 When =4 Four pass P =.0001 X) Three Pass SSSF FSSS SFSS SSFS 4[(.1)3(.9)] = Four Pass SSSS (.1)(.1)(.1)(.1) = X X) Fitness Eample X Binomial Probability Distribution Formula X) Find the probability that none of the adults pass the test =0) =.6561 Find the probability that 3 of 4 adults pass the test =3) =.0036 What is the probability that or more adults pass the test? =) + =3) + =4) = = Sometimes the number of trials gets large We can also use the binomial probability distribution formula to generate the probabilities It uses factorial notation = n(n-1)(n-) (n-(n-1)) 5! = 5431 = 10 0! = 1, 1!=1,!=1=, The formula for any in n trials is:!( n!! 1

4 What defines the Binomial Distribution? p = Probability of a success on a single trial q = (1-p) probability of failure n = number of trials = number of successes in n trials Note: it uses the Combinatorial Rule as the first part of the formula!( n!! This part reflects the probabilities with each combination 13 For =3 in the fitness eample, n=4, p=.1 4! 3) = (.1) 3!(4! 3)! 3) = 4! 3!!1 3!!1 3 (.9) 4! 3 ( )( 1) (.1)3 (.9) 4"3 3) = 4 6 (.1)3 (.9) 4!3 3) = 4(.1) 3 (.9) 4!3!( n!! 3) = 4(.001)(.9) 3) = 4(.0009) =.0036 The four is how many combinations of 3 success in 4 The last part of the formula is the probability associated with each of these combinations The probability,.0036, is the eact same one we calculated earlier 14! Your Try it: For = in the fitness eample, n=4, p=.1 I will get you started ) = 4!!(4 " )! (.1) (.9) 4" ) = 6(.0081) =.0486!( n!! 15 Mean and Variance for a Binomial Random Variable Since a binomial is only a dichotomy, the formulas for the mean and the standard deviation will simplify From µ =!(!) To µ = p Our fitness eample: µ = 4*.1 =.4 The Variance changes from From To " =![(-µ)!)]! = n*p*q Our fitness eample:! = 4*.1*.9 =.36 and! =.60 16

5 I could have solved for the mean using the formula for discrete random variables To solve for the mean I would use this formula from the discrete random variable lecture: E( = (0)(.6561) + (1)(.916) + () (.0486) + (3)(.0036) + (4)(.0001) E( =.4 Binomial approach E( = n p = 4 (.1) =.4 The Binomial approach is much easier n E( = " i= 1! i i ) = µ If I know my Discrete Random Variable is distributed as a binomial random variable, it will make things much easier 17 I could have solved for the variance using the formula for discrete random variables To solve for the variance I would have: E(-µ) = (0 -.4) (.6561) + (1-.4) (.916) + (-.4) (.0486) + (3-.4) (.0036) + (4-.4) (.0001)! =.36 Binomial approach E( = n p q = 4 (.1)(.9) =.36 The Binomial approach is much easier E n [( " µ ) ] = #( i " µ ) i ) =! i= 1 If I know my Discrete Random Variable is distributed as a binomial random variable, it will make things much easier 18 Return to the Nitrous Oide Eample Nitrous Oide Eample Suppose we were recording the number of dentists that use nitrous oide (laughing gas) in their practice We know that 60% of dentists use the gas. p =.6 and q =.4 Let X = number of dentists in a random sample of five dentists use use laughing gas. n = 5 This is a Binomial Random Variable! Conduct an eperiment 5 times and observe the number of times that use Nitrous Oide 19 How to solve for these probabilities? X X) !( n!! 0

6 Probability Distribution for the Nitrous Oide Eample X X) µ = 3.00! = 1.0! = 1.01 µ = 5*.6 = 3.00! = 5*.6*.4 = 1.0! = SQRT(1.0) = 1.01 X) Probability Distribution of X Nitrous Oide Eample using Ecel Open up the file, BINOM.ls Click on the Worksheet Problem This worksheet is designed to solve problems up to n=50, for any value of p You enter in: p =.6 n = 5 The spreadsheet will do the rest! Reverse p = X p(x) Cum p(x) Cum p(>=x) q = n = Mean Variance Std Dev Binomial Formula using Ecel In Ecel, the formula for the Binomial Distribution function is: BINOMDIST(X,N,P,cumulative) X is the number of successes N is the number of independent trials P is the probability of success on each trial Cumulative is an argument - you enter TRUE or FALSE Entering TRUE gives a cumulative probability up to and including X successes (or 1) Entering FALSE gives the eact probability of X successes in N trials (or 0) Binomial Formula using Ecel For our eample of dentists BINOMDIST(,5,.6,TRUE) cumulative probability up to and including successes =.3174 BINOMDIST(,5,.6,FALSE) the eact probability of X successes in N trials =.304 BINOMDIST(3,5,.6,TRUE) 3 4

7 Binomial Table Binomial Table for n = 5 The table is arranged cumulatively For each probability, the value in the cell is the cumulative probability up to and including X The last row (in this case for = 5), the cumulative probability is Another way to get probabilities form Binomial Random Variables is via a table In eams, I will give you a table which contains cumulative probabilities for n= 5, 6, 7, 8, 9, 10, 15, 0, and 5 Each table lists values of P across the top P =.01,.05,.1,.,.3,,.95,.99 = # of successes as the rows It is a Cumulative Table The values shown are cumulative probabilities for the probability of (denoted as k in the table) PROBABILITIES (p) The probability associated with p=.3 and = 4 is.998 This means that the cumulative probability, or " 4) =.998 The actual probability of = 4) = =.09 You must subtract two values to get the actual probability of 5 6 Nitrous Oide Eample The values shown are cumulative probabilities for the probability of (denoted as k in the table) PROBABILITIES (p) Nitrous Oide Eample The values shown are cumulative probabilities for the probability of (denoted as k in the table) PROBABILITIES (p) Use the n = 5 Table for p =.6 Solve the probability for = 3 " 3) =.663 " ) =.317 =3) = =.346 This is the same value (with some rounding error) that we calculated using the formula (.3456) 7 Use the n = 5 Table for p =.6 Solve the probability for > 3 " 3) =.663 >3) = =.337 Solve the probability for " " ) =.317 8

8 The Rare Event Approach Summary What if we had 5 dentists selected randomly and none of them used nitrous oide? Given p=.6, this would be a very rare event =0) =.010 This is possible, but not probable Was this just by chance???? Or was the assumption wrong that p =.6? The Binomial is a special form of the discrete random variable There are other discrete random variables - poisson If you know it is a Binomial Random Variable it makes it easy to solve for probabilities, the mean and the variance For probabilities you can use: The Binomial Formula The Binomial Tables Ecel also has functions to solve for binomials 9 30

### Binomial Random Variables. Binomial Distribution. Examples of Binomial Random Variables. Binomial Random Variables

Binomial Random Variables Binomial Distribution Dr. Tom Ilvento FREC 8 In many cases the resonses to an exeriment are dichotomous Yes/No Alive/Dead Suort/Don t Suort Binomial Random Variables When our

### 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

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

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### 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

### Discrete Random Variables and their Probability Distributions

CHAPTER 5 Discrete Random Variables and their Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Discrete Random Variable

- Binomial Probability Distributions Definition Distribution. The procedure must have a fied number of trials.. The trials must be independent. (The outcome of any individual trial doesn t affect the probabilities

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### Chapter 9 Monté Carlo Simulation

MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### 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

### Chapter 6: Random Variables

Chapter : Random Variables Section.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter Random Variables.1 Discrete and Continuous Random Variables.2 Transforming and Combining

### 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

### Chapter 5: Normal Probability Distributions - Solutions

Chapter 5: Normal Probability Distributions - Solutions Note: All areas and z-scores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that

### Normal Distribution as an Approximation to the Binomial Distribution

Chapter 1 Student Lecture Notes 1-1 Normal Distribution as an Approximation to the Binomial Distribution : Goals ONE TWO THREE 2 Review Binomial Probability Distribution applies to a discrete random variable

### 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

### 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

### Sampling Central Limit Theorem Proportions. Outline. 1 Sampling. 2 Central Limit Theorem. 3 Proportions

Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Populations and samples When we use statistics, we are trying to find out information about

### Chapter 5 Discrete Probability Distribution. Learning objectives

Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able

### AP STATISTICS 2010 SCORING GUIDELINES

2010 SCORING GUIDELINES Question 4 Intent of Question The primary goals of this question were to (1) assess students ability to calculate an expected value and a standard deviation; (2) recognize the applicability

### CHAPTER 5 THE BINOMIAL DISTRIBUTION AND RELATED TOPICS

CHAPTER 5 THE BINOMIAL DISTRIBUTION AND RELATED TOPICS THE BINOMIAL PROBABILITY DISTRIBUTION (SECTIONS 5.1, 5.2 OF UNDERSTANDABLE STATISTICS) The binomial probability distribution is discussed in Chapter

### DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

### 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

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### 6.2. Discrete Probability Distributions

6.2. Discrete Probability Distributions Discrete Uniform distribution (diskreetti tasajakauma) A random variable X follows the dicrete uniform distribution on the interval [a, a+1,..., b], if it may attain

### Basic Elements of a Hypothesis Test. Hypothesis Testing of Proportions and Small Sample Means. Proportions. Proportions

Hypothesis Testing of Proportions and Small Sample Means Dr. Tom Ilvento FREC 408 Basic Elements of a Hypothesis Test H 0 : H a : : : Proportions The Pepsi Challenge asked soda drinkers to compare Diet

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

### Probability Distributions

Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.

### SAMPLING DISTRIBUTIONS

0009T_c07_308-352.qd 06/03/03 20:44 Page 308 7Chapter SAMPLING DISTRIBUTIONS 7.1 Population and Sampling Distributions 7.2 Sampling and Nonsampling Errors 7.3 Mean and Standard Deviation of 7.4 Shape of

### 2 Binomial, Poisson, Normal Distribution

2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability

### 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

### Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### Practice Questions Chapter 4 & 5

Practice Questions Chapter 4 & 5 Use the following to answer questions 1-3: Ignoring twins and other multiple births, assume babies born at a hospital are independent events with the probability that a

GNH7/GEOLGG9/GEOL2 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD TUTORIAL (6): EARTHQUAKE STATISTICS Question. Questions and Answers How many distinct 5-card hands can be dealt from a standard 52-card deck?

### 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

Using Your TI-NSpire Calculator: Binomial Probability Distributions Dr. Laura Schultz Statistics I This handout describes how to use the binompdf and binomcdf commands to work with binomial probability

### AP Statistics Solutions to Packet 8

AP Statistics Solutions to Packet 8 The Binomial and Geometric Distributions The Binomial Distributions The Geometric Distributions 54p HW #1 1 5, 7, 8 8.1 BINOMIAL SETTING? In each situation below, is

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

### Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

### SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2

### 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

### indicate how accurate the is and how we are that the result is correct. We use intervals for this purpose.

Sect. 6.1: Confidence Intervals Statistical Confidence When we calculate an, say, of the population parameter, we want to indicate how accurate the is and how we are that the result is correct. We use

### Review the following from Chapter 5

Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

### Lecture 10: Depicting Sampling Distributions of a Sample Proportion

Lecture 10: Depicting Sampling Distributions of a Sample Proportion Chapter 5: Probability and Sampling Distributions 2/10/12 Lecture 10 1 Sample Proportion 1 is assigned to population members having a

### Math 461 Fall 2006 Test 2 Solutions

Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two

### Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

### MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities

Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and

### 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

### Summary of Probability

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

### MAT 155. Key Concept. September 22, 2010. 155S5.3_3 Binomial 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

### 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

### 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.

### Chapter 5: Discrete Probability Distributions

Chapter 5: Discrete Probability Distributions Section 5.1: Basics of Probability Distributions As a reminder, a variable or what will be called the random variable from now on, is represented by the letter

### Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

### 11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which

### of course the mean is p. That is just saying the average sample would have 82% answering

Sampling Distribution for a Proportion Start with a population, adult Americans and a binary variable, whether they believe in God. The key parameter is the population proportion p. In this case let us

### Chapter 5 - Practice Problems 1

Chapter 5 - Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level

### 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

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### Lecture 5 : The Poisson Distribution

Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

### 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

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

### 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

### PROBABILITIES AND PROBABILITY DISTRIBUTIONS

Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL

### Sampling Distribution of a Sample Proportion

Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given

### Power and Sample Size Determination

Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### Normal Approximation. Contents. 1 Normal Approximation. 1.1 Introduction. Anthony Tanbakuchi Department of Mathematics Pima Community College

Introductory Statistics Lectures Normal Approimation To the binomial distribution Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission

### Applied Reliability ------------------------------------------------------------------------------------------------------------ Applied Reliability

Applied Reliability Techniques for Reliability Analysis with Applied Reliability Tools (ART) (an EXCEL Add-In) and JMP Software AM216 Class 6 Notes Santa Clara University Copyright David C. Trindade, Ph.

### Key Concept. February 25, 2011. 155S6.5_3 The Central Limit Theorem. Chapter 6 Normal Probability Distributions. Central Limit Theorem

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

### ECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003

ECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003 1. (4 points) The number of claims for missing baggage for a well-known airline

### 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

### P(X = x k ) = 1 = k=1

74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k

### 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

### 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

### 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

### Computing Binomial Probabilities

The Binomial Model The binomial probability distribution is a discrete probability distribution function Useful in many situations where you have numerical variables that are counts or whole numbers Classic

### Continuous Random Variables and the Normal Distribution

CHAPTER 6 Continuous Random Variables and the Normal Distribution CHAPTER OUTLINE 6.1 The Standard Normal Distribution 6.2 Standardizing a Normal Distribution 6.3 Applications of the Normal Distribution

### MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p

### 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

### Chapter 6 ATE: Random Variables Alternate Examples and Activities

Probability Chapter 6 ATE: Random Variables Alternate Examples and Activities [Page 343] Alternate Example: NHL Goals In 2010, there were 1319 games played in the National Hockey League s regular season.

### Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

### Part I Learning about SPSS

STATS 1000 / STATS 1004 / STATS 1504 Statistical Practice 1 Practical Week 5 2015 Practical Outline In this practical, we will look at how to do binomial calculations in Excel. look at how to do normal

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

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

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

### 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

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### 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

### MATH 3090 Spring 2014 Test 2 Version A

Multiple Choice: (Questions 1 24) Answer the following questions on the scantron provided. Give the response that best answers the question. Each multiple choice correct response is worth 2.5 points. Please

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

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

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

### MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample

MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of

### 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