Chapter 6 - Continuous Probability Distributions
|
|
- Jade Daniels
- 6 years ago
- Views:
Transcription
1 Chapter 6 - Continuous Probability s Chapter 6 Continuous Probability s Uniform Probability Normal Probability f () Uniform f () Normal Continuous Probability s A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. Slide 1 Slide 2 Continuous Probability s The probability of the random variable assuming a value within some given interval from 1 to 2 is defined to be the area under the graph of the probability density function between 1 and 2. Uniform Probability A random variable is uniformly distributed whenever the probability is proportional to the interval s length. The uniform probability density function is: f () Uniform f () Normal f () = 1/(b a) for a < < b = 0 elsewhere where: a = smallest value the variable can assume b = largest value the variable can assume Slide 3 Slide 4 Uniform Probability Epected Value of Variance of E() = (a + b)/2 Var() = (b - a) 2 /12 where: a = smallest value the variable can assume b = largest value the variable can assume Uniform Probability Eample Slater buffet customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. Uniform Probability Density Function f() = 1/10 for 5 < < 15 = 0 elsewhere where: = salad plate filling weight Slide 5 Slide 6 1
2 Chapter 6 - Continuous Probability s Uniform Probability Epected Value of E() = (a + b)/2 = (5 + 15)/2 = 10 Uniform Probability Uniform Probability for Salad Plate Filling Weight f() Variance of Var() = (b - a) 2 /12 = (15 5) 2 /12 = / Salad Weight (oz.) Slide 7 Slide 8 Uniform Probability What is the probability that a customer will take between 12 and 15 ounces of salad? 1/10 f() P(12 < < 15) = 1/10(3) =.3 Uniform Probability (Another Eample) Eample: Flight time of an airplane traveling from Chicago to New York Suppose the flight time can be any value in the interval from 120 minutes to 140 minutes. Uniform Probability Density Function Salad Weight (oz.) f() = 1/20 for 120 < < 140 = 0 elsewhere where: = Flight time of an airplane traveling from Chicago to New York Slide 9 Slide 10 Uniform Probability Another Eample Uniform Probability Epected Value of E() = (a + b)/2 = ( )/2 = 130 Variance of Var() = (b - a) 2 /12 = ( ) 2 /12 = Slide 11 Eample: Flight time of an airplane traveling from Chicago to New York Slide 12 2
3 Chapter 6 - Continuous Probability s Uniform Probability Eample: Flight time of an airplane traveling from Chicago to New York Probability of a flight time between 120 and 130 minutes P(120 < < 130) = 1/20(10) =.5 Area as a Measure of Probability The area under the graph of f() and probability are identical. This is valid for all continuous random variables. The probability that takes on a value between some lower value 1 and some higher value 2 can be found by computing the area under the graph of f() over the interval from 1 to 2. Slide 13 Slide 14 Normal Probability The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference. It has been used in a wide variety of applications including: Heights of people Rainfall amounts Test scores Scientific measurements Abraham de Moivre, a French mathematician, published The Doctrine of Chances in He derived the normal distribution. Normal Probability Normal Probability Density Function f ( ) where: 1 e 2 = mean = standard deviation = e = ( ) / 2 Slide 15 Slide 16 Normal Probability The distribution is symmetric; its skewness measure is zero. Normal Probability The entire family of normal probability distributions is defined by its mean and its standard deviation. Standard Deviation Mean Slide 17 Slide 18 3
4 Chapter 6 - Continuous Probability s Normal Probability Normal Probability The highest point on the normal curve is at the mean, which is also the median and mode. The mean can be any numerical value: negative, zero, or positive Slide 19 Slide 20 Normal Probability The standard deviation determines the width of the curve: larger values result in wider, flatter curves. = 15 Normal Probability Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and.5 to the right). = Slide 21 Slide 22 Normal Probability Normal Probability (basis for the empirical rule) 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean. (basis for the empirical rule) 99.72% 95.44% 68.26% 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean % of values of a normal random variable are within +/- 3 standard deviations of its mean Slide 23 Slide 24 4
5 Chapter 6 - Continuous Probability s A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution. The letter z is used to designate the standard normal random variable. 1 Converting to the Standard Normal z 0 z We can think of z as a measure of the number of standard deviations is from. Slide 25 Slide 26 Using ecel to compute standard normal probabilities Ecel has two functions for computing probabilities and z values for a standard normal probability distribution. Using Ecel to compute Standard Normal Probabilities Ecel Formula Worksheet NORM.S.DIST function computes the cumulative probability given a z value. NORM.S.INV function computes the z value given a cumulative probability. S in the function names reminds us that these functions relate to the standard normal probability distribution. Slide 27 Slide 28 Using Ecel to compute Standard Normal Probabilities Ecel Formula Worksheet Eample: Grear Tire Company Problem Grear Tire company has developed a new steel-belted radial tire to be sold through a chain of discount stores. But before finalizing the tire mileage guarantee policy, Grear s managers want probability information about the number of miles of tires will last. It was estimated that the mean tire mileage is 36,500 miles with a standard deviation of The manager now wants to know the probability that the tire mileage will eceed 40,000. Slide 29 P( > 40,000) =? Slide 30 5
6 Chapter 6 - Continuous Probability s Eample: Grear Tire Company Problem Solving for the Probability Step 1: Convert to standard normal distribution. z = ( - )/ = (40,000 36,500)/5,000 =.7 Step 2: Find the area under the standard normal curve to the left of z =.7. Eample: Grear Tire Company Problem Cumulative Probability Table for the Standard Normal z P(z <.7) =.7580 Slide 31 Slide 32 Eample: Grear Tire Company Problem Solving for the Probability Step 3: Compute the area under the standard normal curve to the right of z =.7 P(z >.7) = 1 P(z <.7) = =.2420 Eample: Grear Tire Company Problem Slide 33 Slide 34 Eample: Grear Tire Company Problem Eample: Grear Tire Company Problem Area =.7580 Area = =.2420 What should be the guaranteed mileage if Grear wants no more than 10% of tires to be eligible for the discount guarantee? 0.7 z (Hint: Given a probability, we can use the standard normal table in an inverse fashion to find the corresponding z value.) Slide 35 Slide 36 6
7 Chapter 6 - Continuous Probability s Eample: Grear Tire Company Problem Solving for the guaranteed mileage Eample: Grear Tire Company Problem - Solving for the guaranteed mileage Step 1: Find the z value that cuts off an area of.1 in the left tail of the standard normal distribution. z Slide Slide 38 From the table we see that z = cuts off an area of 0.1 in the lower tail. Step 2: Convert z.1 to the corresponding value of. = + z.1 = (5000) = 30,100 Thus a guarantee of 30,100 miles will meet the requirement that approimately 10% of the tires will be eligible for the guarantee. Slide 39 Using Ecel to Compute Normal Probabilities Ecel has two functions for computing cumulative probabilities and values for any normal distribution: NORM.DIST is used to compute the cumulative probability given an value. NORM.INV is used to compute the value given a cumulative probability. Slide 40 Eample: Pep Zone Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to stockouts while waiting for a replenishment order. It has been determined that demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout during replenishment lead-time. In other words, what is the probability that demand during lead-time will eceed 20 gallons? P( > 20) =? Solving for the Stockout Probability Step 1: Convert to the standard normal distribution. z = ( - )/ = (20-15)/6 =.83 Step 2: Find the area under the standard normal curve to the left of z =.83. Slide 41 Slide 42 7
8 Chapter 6 - Continuous Probability s Solving for the Stockout Probability Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z =.83. P(z >.83) = 1 P(z <.83) = =.2033 Area =.7967 Area = =.2033 Probability of a stockout P( > 20) 0.83 z Slide 43 Slide 44 If the manager of Pep Zone wants the probability of a stockout during replenishment lead-time to be no more than.05, what should the reorder point be? Solving for the Reorder Point Step 2: Convert z.05 to the corresponding value of. z = ( - )/ z ( - ) = + z = + z.05 = (6) = or 25 A reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than).05. Slide 45 Slide 46 Normal Probability Solving for the Reorder Point Solving for the Reorder Point Probability of no stockout during replenishment lead-time =.95 Probability of a stockout during replenishment lead-time =.05 By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockout decreases from about.20 to This is a significant decrease in the chance that Pep Zone will be out of stock and unable to meet a customer s desire to make a purchase. Slide 47 Slide 48 8
Chapter 6 Continuous Probability Distributions. Learning objectives
Chapter 6 Continuous s Slide 1 Learning objectives 1. Understand continuous probability distributions 2. Understand Uniform distribution 3. Understand Normal distribution 3.1. Understand Standard normal
More informationChapter ! Bell Shaped
Chapter 6 6-1 Business Statistics: A First Course 5 th Edition Chapter 7 Continuous Probability Distributions Learning Objectives In this chapter, you learn:! To compute probabilities from the normal distribution!
More informationContinuous probability distribution
Microarray Center BIOSTATISTICS Lecture 6 Continuous Probability Distributions 16-4-1 Lecture 6. Continuous probability distributions Dr. Petr Nazarov petr.nazarov@crp-sante.lu OUTLINE Lecture 1 Continuous
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
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 informationIn a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
More informationNormal Probability Distributions
Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous
More informationExercise Set 1 The normal distribution and sampling distributions
Eercise Set 1 The normal distribution and sampling distributions 1). An orange juice producer buys all his oranges from a large orange grove. The amount of juice squeezed from each of these oranges is
More informationx is a random variable which is a numerical description of the outcome of an experiment.
Chapter 5 Discrete Probability Distributions Random Variables is a random variable which is a numerical description of the outcome of an eperiment. Discrete: If the possible values change by steps or jumps.
More informationContinuous Probability Distributions
Continuous Probability Distributions Chapter 07 McGraw-Hill/Irwin Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. LEARNING OBJECTIVES LO 7-1 List the characteristics of the uniform
More informationStandard Normal Calculations
Standard Normal Calculations Section 4.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 10-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 6 The Normal Distribution 2 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find
More informationd) Find the standard deviation of the random variable X.
Q 1: The number of students using Math lab per day is found in the distribution below. x 6 8 10 12 14 P(x) 0.15 0.3 0.35 0.1 0.1 a) Find the mean for this probability distribution. b) Find the variance
More informationBiostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras
Biostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras Lecture - 05 Normal Distribution So far we have looked at discrete distributions
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 informationContinuous Probability Distributions
Continuous Probability Distributions Chapter 7 McGraw-Hill/Irwin Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. GOALS 1. Understand the difference between discrete and continuous
More informationContinuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
More informationLAB 2 INSTRUCTIONS PROBABILITY DISTRIBUTIONS IN EXCEL
LAB 2 INSTRUCTIONS PROBABILITY DISTRIBUTIONS IN EXCEL There is a wide range of probability distributions (both discrete and continuous) available in Excel. They can be accessed through the Insert Function
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationChapter 7. Sampling Distributions
Chapter 7 Sampling Distributions Section 7.1 Sampling Distributions and the Central Limit Theorem Sampling Distributions Sampling distribution The probability distribution of a sample statistic. Formed
More information. (i) What is the probability that X is at most 8.75? =.875
Worksheet 1 Prep-Work (Distributions) 1)Let X be the random variable whose c.d.f. is given below. F X 0 0.3 ( x) 0.5 0.8 1.0 if if if if if x 5 5 x 10 10 x 15 15 x 0 0 x Compute the mean, X. (Hint: First
More informationClass 16. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 16 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 7. - 7.3 Lecture Chapter 8.1-8. Review Chapter 6. Problem Solving
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More informationUniform Probability Distribution. Continuous Random Variables &
Continuous Random Variables & What is a Random Variable? It is a quantity whose values are real numbers and are determined by the number of desired outcomes of an experiment. Is there any special Random
More information22.2 Shape, Center, and Spread
Name Class Date 22.2 Shape, Center, and Spread Essential Question: Which measures of center and spread are appropriate for a normal distribution, and which are appropriate for a skewed distribution? Eplore
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More information8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions 8-1
8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions For Example: On August 8, 2011, the Dow dropped 634.8 points, sending shock waves through the financial community.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
6.1-6.2 Quiz Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the 1) X is a normally distributed random variable with a mean of 11.00. If the probability that
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationA continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
More informationIntroduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
More informationECOSOC MS EXCEL LECTURE SERIES DISTRIBUTIONS
ECOSOC MS EXCEL LECTURE SERIES DISTRIBUTIONS Module Excel provides probabilities for the following functions: (Note- There are many other functions also but here we discuss only those which will help in
More informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
More informationNormal Probability Distributions
CHAPTER 5 Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 5.2 Normal Distributions: Finding Probabilities 5.3 Normal Distributions: Finding
More informationWhat s Normal? Chapter 8. Hitting the Curve. In This Chapter
Chapter 8 What s Normal? In This Chapter Meet the normal distribution Standard deviations and the normal distribution Excel s normal distribution-related functions A main job of statisticians is to estimate
More informationContinuous random variables
Continuous random variables probability density function (f(x)) the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable),
More informationMATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION
MATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION We have examined discrete random variables, those random variables for which we can list the possible values. We will now look at continuous random variables.
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
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 informationIOP 201-Q (Industrial Psychological Research) Tutorial 5
IOP 201-Q (Industrial Psychological Research) Tutorial 5 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. To establish a cause-and-effect relation between two variables,
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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name The bar graph shows the number of tickets sold each week by the garden club for their annual flower show. ) During which week was the most number of tickets sold? ) A) Week B) Week C) Week 5
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationNormal Sampling and Modelling
8.3 Normal Sampling and Modelling Many statistical studies take sample data from an underlying normal population. As you saw in the investigation on page 422, the distribution of the sample data reflects
More informationStandard Normal, Inverse Normal and Sampling Distributions
Standard Normal, Inverse Normal and Sampling Distributions Section 5.5 & 6.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-3339 Cathy
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 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 informationChapter 6: Normal Probability Distributions
Chapter 6: Normal Probability Distributions Section Title Notes Pages 1 Review & Preview 1 2 The Standard Normal Distribution 5 9 3 Applications of Normal Distributions 10 15 4 Sampling Distributions &
More informationAMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
More information3.5 Applying the Normal Distribution (Z-Scores)
3.5 Applying the Normal Distribution (Z-Scores) The Graph: Review of the Normal Distribution Properties: - it is symmetrical; the mean, median and mode are equal and fall at the line of symmetry - it is
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 informationContinuous Random Variables and Probability Distributions
CHAPTER 5 CHAPTER OUTLINE Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables The Uniform Distribution 5.2 Expectations for Continuous Random Variables 5.3 The Normal
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More informationContinuous Probability Distributions
8.1 Continuous Probability Distributions Distributions like the binomial probability distribution and the hypergeometric distribution deal with discrete data. The possible values of the random variable
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationCHAPTERS 5 & 6: CONTINUOUS RANDOM VARIABLES
CHAPTERS 5 & 6: CONTINUOUS RANDOM VARIABLES DISCRETE RANDOM VARIABLE: Variable can take on only certain specified values. There are gaps between possible data values. Values may be counting numbers or
More informationHomework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
More information8.1 Estimation of the Mean and Proportion
8.1 Estimation of the Mean and Proportion Statistical inference enables us to make judgments about a population on the basis of sample information. The mean, standard deviation, and proportions of a population
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More informationDot Plot: A graph for displaying a set of data. Each numerical value is represented by a dot placed above a horizontal number line.
Introduction We continue our study of descriptive statistics with measures of dispersion, such as dot plots, stem and leaf displays, quartiles, percentiles, and box plots. Dot plots, a stem-and-leaf display,
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION
In Inferential Statistic, ESTIMATION (i) (ii) is called the True Population Mean and is called the True Population Proportion. You must also remember that are not the only population parameters. There
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 27 Continuous
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationInverse Normal Distribution and Approximation to Binomial
Inverse Normal Distribution and Approximation to Binomial Section 5.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 16-3339 Cathy Poliak,
More informationChapter 8. Variables. Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 Random Variables Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc. 8.1 What is a Random Variable? Random Variable: assigns a number to each outcome of a random circumstance, or,
More informationLESSON 7 INTERVAL ESTIMATION SAMIE L.S. LY
LESSON 7 INTERVAL ESTIMATION SAMIE L.S. LY 1 THIS WEEK S PLAN Part I: Theory + Practice ( Interval Estimation ) Part II: Theory + Practice ( Interval Estimation ) z-based Confidence Intervals for a Population
More informationThe Uniform Distribution
The Uniform Distribution EXAMPLE 1 The previous problem is an example of the uniform probability distribution. Illustrate the uniform distribution. The data that follows are 55 smiling times, in seconds,
More informationA LEVEL MATHEMATICS ANSWERS AND MARKSCHEMES SUMMARY STATISTICS AND DIAGRAMS. 1. a) 45 B1 [1] b) 7 th value 37 M1 A1 [2]
1. a) 45 [1] b) 7 th value 37 [] n c) LQ : 4 = 3.5 4 th value so LQ = 5 3 n UQ : 4 = 9.75 10 th value so UQ = 45 IQR = 0 f.t. d) Median is closer to upper quartile Hence negative skew [] Page 1 . a) Orders
More informationSTAT 201 Chapter 6. Distribution
STAT 201 Chapter 6 Distribution 1 Random Variable We know variable Random Variable: a numerical measurement of the outcome of a random phenomena Capital letter refer to the random variable Lower case letters
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 informationFall 2011 Exam Score: /75. Exam 3
Math 12 Fall 2011 Name Exam Score: /75 Total Class Percent to Date Exam 3 For problems 1-10, circle the letter next to the response that best answers the question or completes the sentence. You do not
More informationChapter Six Probability Distributions
6.1 Probability Distributions Discrete Random Variable Chapter Six Probability Distributions x P(x) 2 0.08 4 0.13 6 0.25 8 0.31 10 0.16 12 0.01 Practice. Construct a probability distribution for the number
More informationKey Objectives. Module 2: The Logic of Statistical Inference. Z-scores. SGSB Workshop: Using Statistical Data to Make Decisions
SGSB Workshop: Using Statistical Data to Make Decisions Module 2: The Logic of Statistical Inference Dr. Tom Ilvento January 2006 Dr. Mugdim Pašić Key Objectives Understand the logic of statistical inference
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 informationNORMAL PROBABILITY DISTRIBUTIONS
5 CHAPTER NORMAL PROBABILITY DISTRIBUTIONS 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 5.2 Normal Distributions: Finding Probabilities 5.3 Normal Distributions: Finding
More informationSTT 315 Practice Problems Chapter 3.7 and 4
STT 315 Practice Problems Chapter 3.7 and 4 Answer the question True or False. 1) The number of children in a family can be modelled using a continuous random variable. 2) For any continuous probability
More informationDescriptive Statistics
Chapter 3 Descriptive Statistics Chapter 2 presented graphical techniques for organizing and displaying data. Even though such graphical techniques allow the researcher to make some general observations
More informationNormal Curves & Sampling Distributions
Chapter 7 Name Normal Curves & Sampling Distributions Section 7.1 Graphs of Normal Probability Distributions Objective: In this lesson you learned how to graph a normal curve and apply the empirical rule
More informationLean Six Sigma: Training/Certification Books and Resources
Lean Si Sigma Training/Certification Books and Resources Samples from MINITAB BOOK Quality and Si Sigma Tools using MINITAB Statistical Software A complete Guide to Si Sigma DMAIC Tools using MINITAB Prof.
More informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationExercise Questions. Q7. The random variable X is known to be uniformly distributed between 10 and
Exercise Questions This exercise set only covers some topics discussed after the midterm. It does not mean that the problems in the final will be similar to these. Neither solutions nor answers will be
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 6. ' From the table the z value corresponding to this value Z = 1.96 or Z = 1.96 (d) P(Z >?) =
Solutions to End-of-Section and Chapter Review Problems 225 CHAPTER 6 6.1 (a) P(Z < 1.20) = 0.88493 P(Z > 1.25) = 1 0.89435 = 0.10565 P(1.25 < Z < 1.70) = 0.95543 0.89435 = 0.06108 (d) P(Z < 1.25) or Z
More informationChapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
More informationChapter 4 Random Variables & Probability. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random variable =
More informationHonors Statistics. 3. Review OTL C6#3. 4. Normal Curve Quiz. Chapter 6 Section 2 Day s Notes.notebook. May 02, 2016.
Honors Statistics Aug 23-8:26 PM 3. Review OTL C6#3 4. Normal Curve Quiz Aug 23-8:31 PM 1 May 1-9:09 PM Apr 28-10:29 AM 2 27, 28, 29, 30 Nov 21-8:16 PM Working out Choose a person aged 19 to 25 years at
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Use the Central Limit Theorem to find the indicated probability. The sample size is n,
More informationQuantitative Methods for Economics, Finance and Management (A86050 F86050)
Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationSTATS DOESN T SUCK! ~ CHAPTER 4
CHAPTER 4 QUESTION 1 The Geometric Mean Suppose you make a 2-year investment of $5,000 and it grows by 100% to $10,000 during the first year. During the second year, however, the investment suffers a 50%
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 information8. From FRED, search for Canada unemployment and download the unemployment rate for all persons 15 and over, monthly,
Economics 250 Introductory Statistics Exercise 1 Due Tuesday 29 January 2019 in class and on paper Instructions: There is no drop box and this exercise can be submitted only in class. No late submissions
More informationMath 2311 Bekki George Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment
Math 2311 Bekki George bekki@math.uh.edu Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment Class webpage: http://www.math.uh.edu/~bekki/math2311.html Math 2311 Class
More informationWeek 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics.
Week 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics. Convergent validity: the degree to which results/evidence from different tests/sources, converge on the same conclusion.
More informationChapter 6 Analyzing Accumulated Change: Integrals in Action
Chapter 6 Analyzing Accumulated Change: Integrals in Action 6. Streams in Business and Biology You will find Excel very helpful when dealing with streams that are accumulated over finite intervals. Finding
More informationHonors Statistics. Daily Agenda
Honors Statistics Daily Agenda 1. Review OTL C6#5 2. Quiz Section 6.1 A-Skip 35, 39, 40 Crickets The length in inches of a cricket chosen at random from a field is a random variable X with mean 1.2 inches
More informationCopyright 2005 Pearson Education, Inc. Slide 6-1
Copyright 2005 Pearson Education, Inc. Slide 6-1 Chapter 6 Copyright 2005 Pearson Education, Inc. Measures of Center in a Distribution 6-A The mean is what we most commonly call the average value. It is
More informationSection Random Variables and Histograms
Section 3.1 - Random Variables and Histograms 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 information