# The normal approximation to the binomial

Size: px
Start display at page:

## Transcription

1 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 large number ( ) n k with a potentially very small one p k ( p) n k. Fortunately, the normal distribution provides a very good approximation to the Binomial when n is large enough (say np > 5 and n( p) > 5). The approximation involves two steps, one obvious and one not so obvious: () Instead of using the B(n, p) distribution, using the N(µ, σ 2 ) distribution, where µ = np and σ 2 = np( p). (2) Use a continuity correction to correct for the use of a continuous distribution to approximate a discrete one. The need for a continuity correction can be seen in the following situation. Consider an experiment where we toss a fair coin 2 times and observe the number of heads. Suppose we want to compute the probability of obtaining exactly 4 heads. Whereas a discrete random variable can have only a specified value (such as 4), a continuous random variable used to approximate it could take on any values within an interval around that specified value, as demonstrated in this figure: Probability The goal is to approximate the area of the bar above 4 with an area under the normal c 2, Jeffrey S. Simonoff

2 curve. Using the probability corresponding to P(X = 4) is clearly wrong, since that is ; by using the area between 3.5 and 4.5 (the shaded area above), the area between the bar and curve missed on the left is roughly made up for on the right. This is the essence of the calculation: whenever the Binomial probability for an integer value k is needed, use the area under the corresponding normal curve for (k.5, k +.5). Example. Let H be the number of heads in 2 flips of a fair coin. What is the probability of observing between 3 and 5 heads, inclusive; that is, P(3 H 5)? In this case we can calculate the value exactly: since ( ) 2 P(H = k) = (.5) k (.5) 2 k, k =,, 2,..., 2, k we have P(3 H 5) = P(H = 3) + P(H = 4) + P(H = 5) = = To approximate this we first need the mean and standard deviation µ = np = (2)(.5) = 6 σ = np( p) = (2)(.5)(.5) =.732. Since H is a binomial random variable, the following statement (based on the continuity correction) is exactly correct: P(3 H 5) = P(2.5 < H < 5.5). Note that this statement is not an approximation it is exactly correct! The reason for this is that we are adding the events 2.5 < H < 3 and 5 < H < 5.5 to get from the left side of the equation to the right side of the equation, but for the binomial random variable, these events have probability zero. The continuity correction is not where the approximation comes in; that comes when we approximate H using a normal distribution with mean µ = 6 and standard deviation σ =.732: P(3 H 5) = P(2.5 < H < 5.5) ( P.732 < Z < ).732 = P( 2.2 < Z <.29) = P(Z <.29) P(Z < 2.2) = =.3642 c 2, Jeffrey S. Simonoff 2

3 Note that the approximation is only off by about %, which is pretty good for such a small sample size. Example. Suppose that a sample of n =, 6 tires of the same type are obtained at random from an ongoing production process in which 8% of all such tires produced are defective. What is the probability that in such a sample not more than 5 tires will be defective? Answer. We approximate the B(6,.8) random variable T with a normal, with mean (6)(.8) = 28 and standard deviation (6)(.8)(.92) =.85. The probability calculation is thus P(T 5) = P(T < 5.5) ( ) P Z <.85 = P(Z < 2.7) =.988. Note that if the question had been about the probability of there being fewer than 5 defective tires, the calculation would have been P(T < 5) = P(T < 49.5) ( ) P Z <.85 = P(Z <.98) =.976, the difference in these two values (.47) being the estimated probability of exactly 5 tires with defects. The normal approximation to the binomial is the underlying principle to an important tool in statistical quality control, the Np chart. Say we have an assembly line that turns out thousands of units per day. Periodically (daily, say), we sample n items from the assembly line, and count up the number of defective items, D. What distribution does D have? B(n, p), of course, with p being the probability that a particular item is defective. Thus, examination of D allows us to see if the probability of a defective item is changing over time; that is, the process is getting out of control. Consider the following situation. Our assembly line has been running for a while, and based on this historical data, we ve seen that the probability of an individual item being defective is.2 (this would come from sampling and getting the empirical frequency of defectives). Our online quality control system samples items per day, and counts up c 2, Jeffrey S. Simonoff 3

4 the number of defectives D. We know that D B(,.2), so E(D) = ()(.2) = 2 and S(D) = ()(.2)(.98) = Thus, D is approximately normally distributed with mean µ = 2 and standard deviation σ = This allows us to assess whether future values of D are unusual, by seeing whether they get too far from 2. Here is an example of an Np chart. NP Chart for In contr 4 Sample Count SL=33.28 NP=2. -3.SL=6.78 Sample Number 2 The chart consists of the values for the process plotted with three lines: the expected number of defects in the center, and two control limits at µ±3σ (the number 3 is arbitrary, but standard). Since D is roughly normally distributed, we know that the probability of D being outside the control limits, assuming that p is staying at.2, is P( Z > 3) =.26. So in this case, where there are 2 days worth of data, we re not surprised to see a day or two outside the limits. There are actually three, but this is just random fluctuation; we know that because the process settles down to its correct value immediately. In fact, these data do come from a stable process. Now consider this chart: c 2, Jeffrey S. Simonoff 4

5 Sample Count NP Chart for Number o 3.SL=33.28 NP=2. Sample Number 2-3.SL=6.78 This chart is very different. About days into the sample, the process starts to go out of control. Eventually the D values move outside the control limits, and the process should be stopped and corrected. In fact, for these data, starting with the st observation, I increased p steadily by.3 per day. Control charts also can be constructed based on other statistics, such as means and standard deviations. They are an integral part of the idea of kaizen, or continuous improvement, that has revolutionized manufacturing around the world in recent years. Minitab commands To obtain an Np chart, click on Stat Control Charts NP. Enter the variable with the binomial counts in it next to Variable:, and insert the number of binomial trials each observation refers to next to Subgroup size:. If there is a value of p that is known from historical data to correspond to the process being in control, enter that value next to Historical p:. c 2, Jeffrey S. Simonoff 5

### The normal approximation to the binomial

The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### Stats on the TI 83 and TI 84 Calculator

Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and

### 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

### 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.

### Sample Questions for Mastery #5

Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could

### 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.

### IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem

IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have

### 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 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

### Binomial random variables

Binomial and Poisson Random Variables Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance

### 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

### 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

### 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

### 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

### TEACHER NOTES MATH NSPIRED

Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when

### 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

### Binomial Distribution Problems. Binomial Distribution SOLUTIONS. Poisson Distribution Problems

1 Binomial Distribution Problems (1) A company owns 400 laptops. Each laptop has an 8% probability of not working. You randomly select 20 laptops for your salespeople. (a) What is the likelihood that 5

### 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,

### 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 ()

### 5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### Example 1. so the Binomial Distrubtion can be considered normal

Chapter 6 8B: Examples of Using a Normal Distribution to Approximate a Binomial Probability Distribution Example 1 The probability of having a boy in any single birth is 50%. Use a normal distribution

### 39.2. The Normal Approximation to the Binomial Distribution. Introduction. Prerequisites. Learning Outcomes

The Normal Approximation to the Binomial Distribution 39.2 Introduction We have already seen that the Poisson distribution can be used to approximate the binomial distribution for large values of n and

### 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

### 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.

### 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

### 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

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22

Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability

### 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

### 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

### Binomial random variables (Review)

Poisson / Empirical Rule Approximations / Hypergeometric Solutions STAT-UB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die

### 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

### STAT 200 QUIZ 2 Solutions Section 6380 Fall 2013

STAT 200 QUIZ 2 Solutions Section 6380 Fall 2013 The quiz covers Chapters 4, 5 and 6. 1. (8 points) If the IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. (a) (3 pts)

### 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

### Key Concept. Density Curve

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

### The Normal Distribution

The Normal Distribution Continuous Distributions A continuous random variable is a variable whose possible values form some interval of numbers. Typically, a continuous variable involves a measurement

### Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers

Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers 5.18 Attitudes toward drinking and behavior studies. Some of the methods in this section are approximations

### 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

### Practice Problems for Homework #6. Normal distribution and Central Limit Theorem.

Practice Problems for Homework #6. Normal distribution and Central Limit Theorem. 1. Read Section 3.4.6 about the Normal distribution and Section 4.7 about the Central Limit Theorem. 2. Solve the practice

### 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

### 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

### 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

### 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.

### The Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University

The Normal Distribution Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu Spring 2006 R Notes 1 Copyright c 2006 Alan T. Arnholt 2 Continuous Random

### 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

### 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

### AP Statistics Solutions to Packet 2

AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

### Hypothesis testing. c 2014, Jeffrey S. Simonoff 1

Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there

### 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,

### MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem

MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

### Binomial Probability Distribution

Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are

### 4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

### Chapter 4. iclicker Question 4.4 Pre-lecture. Part 2. Binomial Distribution. J.C. Wang. iclicker Question 4.4 Pre-lecture

Chapter 4 Part 2. Binomial Distribution J.C. Wang iclicker Question 4.4 Pre-lecture iclicker Question 4.4 Pre-lecture Outline Computing Binomial Probabilities Properties of a Binomial Distribution Computing

### Statistics 100A Homework 4 Solutions

Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation

### Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part 1

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part Note: Assume missing data (if any) and mention the same. Q. Suppose X has a normal

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

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

### 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

### 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

### 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

### Statistiek I. Proportions aka Sign Tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. http://www.let.rug.nl/nerbonne/teach/statistiek-i/

Statistiek I Proportions aka Sign Tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://www.let.rug.nl/nerbonne/teach/statistiek-i/ John Nerbonne 1/34 Proportions aka Sign Test The relative frequency

### 1 SAMPLE SIGN TEST. Non-Parametric Univariate Tests: 1 Sample Sign Test 1. A non-parametric equivalent of the 1 SAMPLE T-TEST.

Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

### 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 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

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

STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.

### 39.2. The Normal Approximation to the Binomial Distribution. Introduction. Prerequisites. Learning Outcomes

The Normal Approximation to the Binomial Distribution 39.2 Introduction We have already seen that the Poisson distribution can be used to approximate the binomial distribution for large values of n and

### 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

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### 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

### Section 5-3 Binomial Probability Distributions

Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial

### 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

### Binomial Distribution n = 20, p = 0.3

This document will describe how to use R to calculate probabilities associated with common distributions as well as to graph probability distributions. R has a number of built in functions for calculations

### 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

### The Normal distribution

The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

### 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 =

### 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

### The Binomial Distribution. Summer 2003

The Binomial Distribution Summer 2003 Internet Bubble Several industry experts believe that 30% of internet companies will run out of cash in 6 months and that these companies will find it very hard to

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 0.4987 B) 0.9987 C) 0.0010 D) 0.

Ch. 5 Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 1 Find Areas Under the Standard Normal Curve 1) Find the area under the standard normal

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

### Point and Interval Estimates

Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### 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)

### 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

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### 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

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### 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