5.4 Normal Approximation of the Binomial Distribution
|
|
- Rodger Snow
- 5 years ago
- Views:
Transcription
1 5.4 Normal Approximation of the Binomial Distribution Bernoulli Trials have 3 properties: 1. Only two outcomes - PASS or FAIL 2. n identical trials Review from yesterday. 3. Trials are independent - probability is constant
2 Binomial Probability Distribution In a binomial experiment with n Bernoulli trials, each with a probability of success p, the probability of k successes in the n trials is given by: Where X is the discrete random variable corresponding to the number of successes. Warm-Up a) What is the probability of getting exactly 30 heads if a coin is tossed 50 times? X = # of heads in 50 tosses n = # of Bernoulli Trials (50) p = probability of success (.5)
3 b) Determine the expected number of heads in 50 trials For a binomial distribution, expected value can simply be calculated by: E(X) = np Note: n = # of trials p = probability of success c) Find the probability of tossing between 20 and 30 heads, inclusive, in 50 tosses of a coin. P(20 X 30) This is where the real problem occurs with more complex situations. You could calculate it with 11 such calculations: P(20 X 30) = P(X = 20) + P(X = 21) + P (X = 22) + P(X = 30) This calculation would be very time consuming! To help simplify the calculation, look at the graphical representation of the binomial distribution. A binomial distribution can be approximated by a normal distribution as long as the number of trials is large!
4 - It is mound shaped and the normal curve is a close fit. - To use the normal curve, you must know values for the mean and standard deviation because you must calculate the z-score in order to translate the data from the normal curve to the standard normal curve. This will allow you to determine the percentage of data that has an equal or lower z-score. The percentage is equivalent to the area under the curve. What is the Standard Normal Curve? The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of
5 Mean and Standard Deviation of the Normal Approximation to the Binomial Distribution Mean of a Binomial Distribution: (** same as expected value) Standard Deviation: Practice: If you were to toss a coin 50 times, calculate. a) the mean ( x ) number of heads in 50 tosses b) the standard deviation ( σ ) of the number of heads in 50 tosses
6 From Discrete to Continuous There is one major issue.. - A binomial distribution represents a discrete random variable - A normal distribution is continuous - In order to use the normal distribution to approximate the binomial distribution, you must consider a range of values rather than specific discrete values. Example: To find a range of values to represent the discrete value of 5, you must include all numbers that round to 5. Therefore, the range of values between 4.5 and 5.5 can be used to represent the discrete value of 5. P(X = 5) = P(4.5 < X < 5.5)
7 Practice finding the range of continuous values for a discrete value. a) What range of continuous values can be used to represent P(X = 2)? b) What range of continuous values can be used to represent P(2 X 4)?
8 c) What range of continuous values can be used to represent P(X > 3)? d) What range of continuous values can be used to represent P(X 3)?
9 e) What range of continuous values can be used to represent P(2 < X < 5)? f) What range of continuous values can be used to represent P(X < 11)?
10 g) What range of continuous values can be used to represent P(X 7)? Example 1 Fran tosses a fair coin 50 times. Estimate the probability that she will get tails less than 20 times. This means you must find P(X < 20) where X is the number of tails in 50 tosses) Step 1: Find the mean and standard deviation Note: Let a success be a toss of tails
11 Step 2: Determine continuous values for P(X < 20) Step 3: Find the z-score Note: finding the z-score standardizes the distribution to the standard normal curve Step 4: Use your z-score to determine the probability using the chart [Find P(z < -1.55)] P(X < 19.5) = P(z < -1.55) = Therefore there is a % chance that Fran will toss less than 20 tails in 50 attempts.
12 Example 2 Calculate the probability that, in 100 rolls of a fair die, a 6 appears between 10 and 20 times, inclusive. [Find P(10 X 20)] Step 1: Find the mean and standard deviation Step 2: Find the continuous range for the discrete values of P(10 X 20) Step 3: Find the z-scores
13 Step 4: Use your z-scores to determine the probability using the chart P(9.5 < X < 20.5) = P(-1.92 < z <1.03) = P(z < 1.03) - P(z < -1.92) Therefore the probability of rolling between 10 and 20 (inclusive) sixes on a fair die that is rolled 100 times is %. Using TI-83 Example 2 using your calculator A TI-83 calculator can be used to calculate the area under the normal distribution. Use the command: normalcdf(lower X value, upper X value, mean, stand. dev.) access the command by pressing: 2nd --> distr --> 2
14 Example 3 A hamburger patty producer claims that its burgers contain 400 grams of beef. It has been determined that 85% of burgers contain 400 grams or more. An inspector will only accept a shipment if at least 90% of a sample of 250 burgers contain more than 400 grams. What is the probability that a shipment is accepted? Step 1: Find the mean and standard deviation Step 2: Find the continuous range for the discrete values of P(X 225) Note: we are using 225 because 90% of 250 burgers is 225. Step 3: Find the z-score
15 Step 4: Use your z-score to determine the probability using the chart P(X > 224.5) = P(z > 2.12) = 1 - P(z < 2.12) = There is a % chance that the shipment is accepted. WARNING Not all binomial distributions can be approximated using normal distributions. If X is a binomial random variable with n independent trials with a probability of success = p, AND if np > 5 and n(1-p) > 5 then you can approximate it using normal distribution with mean=np and S.D.= np(1-p)
16 Was a normal approximation of the binomial distribution appropriate for example 3? np = n(1 - p) = Since np > 5 and 1(n - p) > 5, the binomial distribution can be approximated by the normal curve.
5.4 Normal Approximation of the Binomial Distribution Lesson MDM4U Jensen
5.4 Normal Approximation of the Binomial Distribution Lesson MDM4U Jensen Review From Yesterday Bernoulli Trials have 3 properties: 1. 2. 3. Binomial Probability Distribution In a binomial experiment with
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationChapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationSection 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
More informationProbability Distributions. Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution
Probability Distributions Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution Definitions Random Variable: a variable that has a single numerical value
More informationLecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationThe binomial distribution p314
The binomial distribution p314 Example: A biased coin (P(H) = p = 0.6) ) is tossed 5 times. Let X be the number of H s. Fine P(X = 2). This X is a binomial r. v. The binomial setting p314 1. There are
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationguessing Bluman, Chapter 5 2
Bluman, Chapter 5 1 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question.
More informationConsider the following examples: ex: let X = tossing a coin three times and counting the number of heads
Overview Both chapters and 6 deal with a similar concept probability distributions. The difference is that chapter concerns itself with discrete probability distribution while chapter 6 covers continuous
More informationchapter 13: Binomial Distribution Exercises (binomial)13.6, 13.12, 13.22, 13.43
chapter 13: Binomial Distribution ch13-links binom-tossing-4-coins binom-coin-example ch13 image Exercises (binomial)13.6, 13.12, 13.22, 13.43 CHAPTER 13: Binomial Distributions The Basic Practice of Statistics
More informationCHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
CHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS 8.1 Distribution of Random Variables Random Variable Probability Distribution of Random Variables 8.2 Expected Value Mean Mean is the average value of
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationSection Random Variables
Section 6.2 - Random Variables According to the Bureau of the Census, the latest family data pertaining to family size for a small midwestern town, Nomore, is shown in Table 6.. If a family from this town
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationChapter 6: Random Variables. Ch. 6-3: Binomial and Geometric Random Variables
Chapter : Random Variables Ch. -3: Binomial and Geometric Random Variables X 0 2 3 4 5 7 8 9 0 0 P(X) 3???????? 4 4 When the same chance process is repeated several times, we are often interested in whether
More informationEcon 6900: Statistical Problems. Instructor: Yogesh Uppal
Econ 6900: Statistical Problems Instructor: Yogesh Uppal Email: yuppal@ysu.edu Lecture Slides 4 Random Variables Probability Distributions Discrete Distributions Discrete Uniform Probability Distribution
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More informationMath Week in Review #10. Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
Math 141 Spring 2006 c Heather Ramsey Page 1 Section 8.4 - Binomial Distribution Math 141 - Week in Review #10 Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
More informationChapter 5. Discrete Probability Distributions. McGraw-Hill, Bluman, 7 th ed, Chapter 5 1
Chapter 5 Discrete Probability Distributions McGraw-Hill, Bluman, 7 th ed, Chapter 5 1 Chapter 5 Overview Introduction 5-1 Probability Distributions 5-2 Mean, Variance, Standard Deviation, and Expectation
More informationVIDEO 1. A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.
Part 1: Probability Distributions VIDEO 1 Name: 11-10 Probability and Binomial Distributions A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.
More informationMA 1125 Lecture 18 - Normal Approximations to Binomial Distributions. Objectives: Compute probabilities for a binomial as a normal distribution.
MA 25 Lecture 8 - Normal Approximations to Binomial Distributions Friday, October 3, 207 Objectives: Compute probabilities for a binomial as a normal distribution.. Normal Approximations to the Binomial
More informationRandom Variable: Definition
Random Variables Random Variable: Definition A Random Variable is a numerical description of the outcome of an experiment Experiment Roll a die 10 times Inspect a shipment of 100 parts Open a gas station
More informationBinomial Distribution. Normal Approximation to the Binomial
Binomial Distribution Normal Approximation to the Binomial /29 Homework Read Sec 6-6. Discussion Question pg 337 Do Ex 6-6 -4 2 /29 Objectives Objective: Use the normal approximation to calculate 3 /29
More informationTheoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
More informationSTA258H5. Al Nosedal and Alison Weir. Winter Al Nosedal and Alison Weir STA258H5 Winter / 41
STA258H5 Al Nosedal and Alison Weir Winter 2017 Al Nosedal and Alison Weir STA258H5 Winter 2017 1 / 41 NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION. Al Nosedal and Alison Weir STA258H5 Winter 2017
More informationLecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances
Physical Principles in Biology Biology 3550 Fall 2018 Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances Monday, 10 September 2018 c David P. Goldenberg University
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationElementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
More informationSTOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions 5/31/11 Lecture 14 1 Statistic & Its Sampling Distribution
More informationSection 5 3 The Mean and Standard Deviation of a Binomial Distribution!
Section 5 3 The Mean and Standard Deviation of a Binomial Distribution! Previous sections required that you to find the Mean and Standard Deviation of a Binomial Distribution by using the values from a
More informationDiscrete Probability Distributions
Page 1 of 6 Discrete Probability Distributions In order to study inferential statistics, we need to combine the concepts from descriptive statistics and probability. This combination makes up the basics
More informationExamples: Random Variables. Discrete and Continuous Random Variables. Probability Distributions
Random Variables Examples: Random variable a variable (typically represented by x) that takes a numerical value by chance. Number of boys in a randomly selected family with three children. Possible values:
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More information5.2 Random Variables, Probability Histograms and Probability Distributions
Chapter 5 5.2 Random Variables, Probability Histograms and Probability Distributions A random variable (r.v.) can be either continuous or discrete. It takes on the possible values of an experiment. It
More informationPROBABILITY DISTRIBUTIONS. Chapter 6
PROBABILITY DISTRIBUTIONS Chapter 6 6.1 Summarize Possible Outcomes and their Probabilities Random Variable Random variable is numerical outcome of random phenomenon www.physics.umd.edu 3 Random Variable
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 Fall 216 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / Fall 216 1
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationChapter 5 Probability Distributions. Section 5-2 Random Variables. Random Variable Probability Distribution. Discrete and Continuous Random Variables
Chapter 5 Probability Distributions Section 5-2 Random Variables 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation for the Binomial Distribution Random
More informationThe binomial distribution
The binomial distribution The coin toss - three coins The coin toss - four coins The binomial probability distribution Rolling dice Using the TI nspire Graph of binomial distribution Mean & standard deviation
More informationBinomal and Geometric Distributions
Binomal and Geometric Distributions Sections 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 7-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationThe Binomial and Geometric Distributions. Chapter 8
The Binomial and Geometric Distributions Chapter 8 8.1 The Binomial Distribution A binomial experiment is statistical experiment that has the following properties: The experiment consists of n repeated
More informationRandom Variables and Probability Functions
University of Central Arkansas Random Variables and Probability Functions Directory Table of Contents. Begin Article. Stephen R. Addison Copyright c 001 saddison@mailaps.org Last Revision Date: February
More informationBinomial and Normal Distributions. Example: Determine whether the following experiments are binomial experiments. Explain.
Binomial and Normal Distributions Objective 1: Determining if an Experiment is a Binomial Experiment For an experiment to be considered a binomial experiment, four things must hold: 1. The experiment is
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationSection 8.4 The Binomial Distribution
Section 8.4 The Binomial Distribution Binomial Experiment A binomial experiment has the following properties: 1. The number of trials in the experiment is fixed. 2. There are two outcomes of each trial:
More informationThe Binomial Distribution
MATH 382 The Binomial Distribution Dr. Neal, WKU Suppose there is a fixed probability p of having an occurrence (or success ) on any single attempt, and a sequence of n independent attempts is made. Then
More informationProbability Distribution Unit Review
Probability Distribution Unit Review Topics: Pascal's Triangle and Binomial Theorem Probability Distributions and Histograms Expected Values, Fair Games of chance Binomial Distributions Hypergeometric
More informationPart 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
More information4.3 Normal distribution
43 Normal distribution Prof Tesler Math 186 Winter 216 Prof Tesler 43 Normal distribution Math 186 / Winter 216 1 / 4 Normal distribution aka Bell curve and Gaussian distribution The normal distribution
More informationLECTURE 6 DISTRIBUTIONS
LECTURE 6 DISTRIBUTIONS OVERVIEW Uniform Distribution Normal Distribution Random Variables Continuous Distributions MOST OF THE SLIDES ADOPTED FROM OPENINTRO STATS BOOK. NORMAL DISTRIBUTION Unimodal and
More informationBinomial and Normal Distributions
Binomial and Normal Distributions Bernoulli Trials A Bernoulli trial is a random experiment with 2 special properties: The result of a Bernoulli trial is binary. Examples: Heads vs. Tails, Healthy vs.
More information4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
More informationMATH 118 Class Notes For Chapter 5 By: Maan Omran
MATH 118 Class Notes For Chapter 5 By: Maan Omran Section 5.1 Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Ex1: The test scores
More informationProbability Distributions for Discrete RV
Probability Distributions for Discrete RV Probability Distributions for Discrete RV Definition The probability distribution or probability mass function (pmf) of a discrete rv is defined for every number
More informationMAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
More informationContinuous Random Variables and the Normal Distribution
Chapter 6 Continuous Random Variables and the Normal Distribution Continuous random variables are used to approximate probabilities where there are many possible outcomes or an infinite number of possible
More information7 THE CENTRAL LIMIT THEOREM
CHAPTER 7 THE CENTRAL LIMIT THEOREM 373 7 THE CENTRAL LIMIT THEOREM Figure 7.1 If you want to figure out the distribution of the change people carry in their pockets, using the central limit theorem and
More information***SECTION 8.1*** The Binomial Distributions
***SECTION 8.1*** The Binomial Distributions CHAPTER 8 ~ The Binomial and Geometric Distributions In practice, we frequently encounter random phenomenon where there are two outcomes of interest. For example,
More informationThe Normal Probability Distribution
102 The Normal Probability Distribution C H A P T E R 7 Section 7.2 4Example 1 (pg. 71) Finding Area Under a Normal Curve In this exercise, we will calculate the area to the left of 5 inches using a normal
More informationChapter 11. Data Descriptions and Probability Distributions. Section 4 Bernoulli Trials and Binomial Distribution
Chapter 11 Data Descriptions and Probability Distributions Section 4 Bernoulli Trials and Binomial Distribution 1 Learning Objectives for Section 11.4 Bernoulli Trials and Binomial Distributions The student
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationBinomial and Geometric Distributions
Binomial and Geometric Distributions Section 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH Department of Mathematics University of Houston February 11, 2016
More informationMATH 446/546 Homework 1:
MATH 446/546 Homework 1: Due September 28th, 216 Please answer the following questions. Students should type there work. 1. At time t, a company has I units of inventory in stock. Customers demand the
More informationMidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
More informationMA 1125 Lecture 12 - Mean and Standard Deviation for the Binomial Distribution. Objectives: Mean and standard deviation for the binomial distribution.
MA 5 Lecture - Mean and Standard Deviation for the Binomial Distribution Friday, September 9, 07 Objectives: Mean and standard deviation for the binomial distribution.. Mean and Standard Deviation of the
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More information. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
More informationCHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
More informationMean of a Discrete Random variable. Suppose that X is a discrete random variable whose distribution is : :
Dr. Kim s Note (December 17 th ) The values taken on by the random variable X are random, but the values follow the pattern given in the random variable table. What is a typical value of a random variable
More informationCentral Limit Theorem 11/08/2005
Central Limit Theorem 11/08/2005 A More General Central Limit Theorem Theorem. Let X 1, X 2,..., X n,... be a sequence of independent discrete random variables, and let S n = X 1 + X 2 + + X n. For each
More informationBinomial formulas: The binomial coefficient is the number of ways of arranging k successes among n observations.
Chapter 8 Notes Binomial and Geometric Distribution Often times we are interested in an event that has only two outcomes. For example, we may wish to know the outcome of a free throw shot (good or missed),
More informationSTA 6166 Fall 2007 Web-based Course. Notes 10: Probability Models
STA 6166 Fall 2007 Web-based Course 1 Notes 10: Probability Models We first saw the normal model as a useful model for the distribution of some quantitative variables. We ve also seen that if we make a
More informationChapter 3. Discrete Probability Distributions
Chapter 3 Discrete Probability Distributions 1 Chapter 3 Overview Introduction 3-1 The Binomial Distribution 3-2 Other Types of Distributions 2 Chapter 3 Objectives Find the exact probability for X successes
More informationLesson 97 - Binomial Distributions IBHL2 - SANTOWSKI
Lesson 97 - Binomial Distributions IBHL2 - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin 3 times where P(H) = / (b) THUS, find the probability
More informationOpening Exercise: Lesson 91 - Binomial Distributions IBHL2 - SANTOWSKI
08-0- Lesson 9 - Binomial Distributions IBHL - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin times where P(H) = / (b) THUS, find the probability
More informationRandom Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES
Random Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES Essential Question How can I determine whether the conditions for using binomial random variables are met? Binomial Settings When the
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationChapter 8: The Binomial and Geometric Distributions
Chapter 8: The Binomial and Geometric Distributions 8.1 Binomial Distributions 8.2 Geometric Distributions 1 Let me begin with an example My best friends from Kent School had three daughters. What is the
More informationStatistics for Business and Economics: Random Variables:Continuous
Statistics for Business and Economics: Random Variables:Continuous STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Murray Bourne (interactive
More informationSTOR Lecture 7. Random Variables - I
STOR 435.001 Lecture 7 Random Variables - I Shankar Bhamidi UNC Chapel Hill 1 / 31 Example 1a: Suppose that our experiment consists of tossing 3 fair coins. Let Y denote the number of heads that appear.
More informationHomework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
More information8.1 Binomial Distributions
8.1 Binomial Distributions The Binomial Setting The 4 Conditions of a Binomial Setting: 1.Each observation falls into 1 of 2 categories ( success or fail ) 2 2.There is a fixed # n of observations. 3.All
More informationBusiness Statistics. Chapter 5 Discrete Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability
More informationCentral Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
More informationPlease have out... - notebook - calculator
Please have out... - notebook - calculator May 6 8:36 PM 6.3 How can we find probabilities when each observation has two possible outcomes? 1 What are we learning today? John Doe claims to possess ESP.
More informationProbability Models. Grab a copy of the notes on the table by the door
Grab a copy of the notes on the table by the door Bernoulli Trials Suppose a cereal manufacturer puts pictures of famous athletes in boxes of cereal, in the hope of increasing sales. The manufacturer announces
More informationChapter 5 Student Lecture Notes 5-1. Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 5 Student Lecture Notes 5-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals
More informationECO220Y Sampling Distributions of Sample Statistics: Sample Proportion Readings: Chapter 10, section
ECO220Y Sampling Distributions of Sample Statistics: Sample Proportion Readings: Chapter 10, section 10.1-10.3 Fall 2011 Lecture 9 (Fall 2011) Sampling Distributions Lecture 9 1 / 15 Sampling Distributions
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
More information6. THE BINOMIAL DISTRIBUTION
6. THE BINOMIAL DISTRIBUTION Eg: For 1000 borrowers in the lowest risk category (FICO score between 800 and 850), what is the probability that at least 250 of them will default on their loan (thereby rendering
More information