Continuous Random Variables & Probability Distributions
|
|
- Sabrina Gaines
- 7 years ago
- Views:
Transcription
1 Continuous Random Variables & Probability Distributions
2 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 Variable? We can categorize random variables in two groups: Discrete random variable Continuous random variable
3 What are Discrete Random Variables? It is a numerical value associated with the desired outcomes and has either a finite number of values or infinitely many values but countable such as whole numbers 0,,2,3,. What are Continuous Random Variables? It has infinitely numerical values associated with any interval on the number line system without any gaps or breaks.
4 What is a Probability Distribution? It is a description and often given in the form of a graph, formula, or table that provides the probability for all possible random variables of the desired outcomes. Are there any Requirements? Let x be any random variable and P(x) be the probability of the random variable x, then P(x) = 0 P(x)
5 What is Discrete Probability Distributions? It is a probability distribution for a discrete random variable x with probability P(x) such that P(x) =, and 0 P(x). Here are some examples of continuous probability distribution: Discrete Probability Distribution Binomial Probability Distribution
6 What is Continuous Probability Distributions? It is a probability distribution for a continuous random variable x with probability P(x) such that P(x) =, 0 P(x), and P(x = a) = 0. Here are some examples of continuous probability distribution: Uniform Probability Distribution Standard Normal Probability Distribution Normal Probability Distribution
7 What is Uniform Probability Distributions? It is a probability distribution for a continuous random variable x that can assumes all values on the interval [a,b] such that All values are evenly spread over the interval [a,b], The graph of the distribution has a rectangular shape, and P(c < x < d) = (d c) as displayed below. b a b a a c d b
8 Example: The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 2 minutes, inclusive. Draw and label the uniform probability distribution, Find the probability that a randomly selected person has to wait for a bus is between 7.5 to 0 minutes? Find k = P 90, and explain what this number represents. Solution: We have a uniform probability distribution with a = 0, b = 2, and a rectangular graph with the width of b a = 2
9 Continued Solution: Draw and label the uniform probability distribution, Find the probability that a randomly selected person has to wait for a bus is between 7.5 to 0 minutes? We need to redraw our uniform probability distribution, label, and shade a region that represents the probability for wait time from 7.5 to 0 minutes.
10 Continued Solution: Now to the find the probability, P(7.5 < x < 0) = (0 7.5) = =
11 Continued Solution: Find k = P 90, and explain what this number represents. The gray area below is 0.9 since it represents k = P 90, k 2 P(x < k) = (k 0) = k 2 k = 0.8 So 90% of the customers have a wait time that is below 0.8 minutes.
12 Finding Mean, Variance, and Standard Deviation of a Uniform Probability Distribution: Given a continuous random variable x that can assumes all values on the interval [a, b] with a uniform probability distribution, then Mean µ = b +a 2, Variance σ 2 (b a)2 =, and 2 Standard Deviation σ = σ 2.
13 Example: A continuous random variable x that assumes all values from x = 5 and x = 30 with uniform probability distribution. Draw and label the uniform probability distribution, Find the mean of the distribution. Find the variance of the distribution. Find the standard deviation of the distribution. Solution: We have a uniform probability distribution with a = 5, b = 30, and a rectangular graph with the width of b a = 25.
14 Continued Solution: A continuous random variable x that assumes all values from x = 5 and x = 30 has a uniform probability distribution. Draw and label the uniform probability distribution, Find the mean of the distribution. µ = b +a 2 = 30+5 µ = 7.5 2
15 Continued Solution: Find the variance of the distribution. σ 2 = (b a)2 2 = (30 5)2 2 σ 2 = Find the standard deviation of the distribution. σ = σ = 2 σ 7.27
16 Example: The amount of coffee dispensed by a certain machine into a cup is a continuous random variable x that assumes all values from x = and x = 25 ounces with uniform probability distribution. Draw and label the uniform probability distribution, Find the probability that a cup filled by this machine will contain at least 4.5 ounces. Find the probability that a cup filled by this machine will contain at most 20 ounces. Find the probability that a cup filled by this machine will contain more than 6.5 ounces but less than 8 ounces. Find k such that P(x > k) = 0.3. Explain what this number represents.
17 Solution: In this example, we have a uniform probability distribution with a =, b = 25, and a rectangular graph with the width of b a = 4 Draw and label the uniform probability distribution, 4 25 Find the probability that a cup filled by this machine will contain at least 4.5 ounces. P(x 4.5) = P(x > 4.5) = (25 4.5) 4 = 0.75
18 Continued Solution: Find the probability that a cup filled by this machine will contain at most 20 ounces. P(x 20) = P(x < 20) = (20 ) Find the probability that a cup filled by this machine will contain more than 6.5 ounces but less than 8 ounces. P(6.5 < x < 8) = (8 6.5) Find k such that P(x > k) = 0.3. Explain what this number represents. (25 k) = 0.3 k = The probability that the machine will fill a cup more than 20.8 ounces in 0.3.
19 What is Standard Normal Distributions? It is a probability distribution for a continuous random variable z that can assumes any values such that The graph of the distribution is symmetric, and bell-shaped, The total are under the curve is equal to, Mean, mode, and median are all equal, µ = 0 and σ =, and P(a < z < b) is the area under the curve on the interval (a,b). P(a < z < b) a µ = 0 σ = b
20
21 Example: Find P(.5 < z < ). Solution: We start by drawing the normal curve, then shade and label accordingly. P(.5 < z < ).5 µ = 0 σ = Now we can use the normal distribution chart or different technology to compute the area which is the probability that we are looking for. In this case, P(.5 < z < ) =
22 Example: Find P(z >.5). Solution: We start by drawing the normal curve, then shade and label accordingly. P(z >.5) µ = 0 σ =.5 Now we can use the normal distribution chart or different technology to compute the area which is the probability that we are looking for. In this case, P(z >.5) =
23 Example: Use any method that you prefer to find the following and very your results with the drawing given below. P(0 < z < ) 2 P( 2 < z < 2) 3 P(z > ) 4 P(z < 2)
24 How do we find Z Score from Known Area? It is a probability distribution for a continuous random variable z that can assumes any values such that Draw a bell-shaped curve, Clearly identify and shade the region that represents the known area, Working with the cumulative area from the left, use the normal distribution chart or different technology to find the corresponding Z score. Needed Area Z µ = 0 σ =
25 Example: Use any method that you prefer to find k such that P(z > k) = Solution: Needed Area 0.5 Known Area 0.85 k µ = 0 σ = Using any method, we get k =.036, that is P(z >.036) = 0.85 and P(z <.036) = 0.5. We can also conclude that k = P 5 =.036.
26 What is Z α? It is a notation that describes a z score with an area α to its right. How do we find Z α? We use the fact that Z α = Z α. Draw a bell-shaped curve, clearly identify, and shade the regions that represent the area for α, and α, Working with the cumulative area from the left, use the normal distribution chart or different technology to find the corresponding Z score. α α µ = 0 σ = Z α
27 Example: Use any method that you prefer to find Z Solution: Given Z 0.025, we have α = 0.025, and α = 0.975, so we draw and label our normal curve µ = 0 σ = Z Using any method, we get Z =.960.
28 What is Normal Distributions? It is a probability distribution for a continuous random variable x that can assumes any values such that The graph of the distribution is symmetric, and bell-shaped, The total are under the curve is equal to, Mean, mode, and median are all equal, µ and σ are given, and P(a < x < b) is the area under the curve on the interval (a,b). P(a < x < b) a µ σ b
29 Example: Consider a normal distribution with the mean of 5 and standard deviation of 0. Find P(00 < x < 30). Solution: We start by drawing the normal curve, then shade and label accordingly. P(00 < x < 30) 00 µ = 5 30 σ = 0 Now we can use different technology to compute the area which is the probability that we are looking for. In this case, P(00 < x < 30) =
30 Example: Consider a normal distribution with the mean of 75 and standard deviation of 7.5. Find P(x > 90). Solution: We start by drawing the normal curve, then shade and label accordingly. P(x > 90) µ = 75 σ = 7.5 Now we can use different technology to compute the area which is the probability that we are looking for. In this case, P(x > 90) =
31 Example: Consider a normal distribution with the mean of 6375 and standard deviation of 200. Find P(x < 6000). Solution: We start by drawing the normal curve, then shade and label accordingly. P(x < 6000) 6000 µ = 6375 σ = 200 Now we can use different technology to compute the area which is the probability that we are looking for. In this case, P(x < 6000) =
32 Example: Consider a normal distribution with the mean of 25 and standard deviation of 2. Find x = P 80. Solution: µ = 25 σ = 2 P 80 Using any method, we get x = P 80 = 35., that is P(x < 35.) = 0.85 and P(x > 35.) = 0.2.
33 Normal Probability Distributions & TI Normal Distribution P(a < x < b) P(x < a) P(x > a) P k Z α TI Command normalcdf(a,b,µ,σ) normalcdf( E99,a,µ,σ) normalcdf(a,e99,µ,σ) invnorm(k%,µ,σ) invnorm( α,0,)
34
The Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More informationAP 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
More informationKey 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
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationAP 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
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationContinuous Random Variables
Chapter 5 Continuous Random Variables 5.1 Continuous Random Variables 1 5.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand continuous
More information3.4 The Normal Distribution
3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous
More informationNormal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1.
Normal Distribution Definition A continuous random variable has a normal distribution if its probability density e -(y -µ Y ) 2 2 / 2 σ function can be written as for < y < as Y f ( y ) = 1 σ Y 2 π Notation:
More informationAn 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
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationProbability. Distribution. Outline
7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationProbability Distributions
CHAPTER 6 Probability Distributions Calculator Note 6A: Computing Expected Value, Variance, and Standard Deviation from a Probability Distribution Table Using Lists to Compute Expected Value, Variance,
More informationTEACHER 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
More informationNormal 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
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationNotes 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
More informationStats 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
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.
More informationSummarizing and Displaying Categorical Data
Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency
More informationMATH 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
More informationDef: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.
Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.
More information4. 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
More informationStatistical Data analysis With Excel For HSMG.632 students
1 Statistical Data analysis With Excel For HSMG.632 students Dialog Boxes Descriptive Statistics with Excel To find a single descriptive value of a data set such as mean, median, mode or the standard deviation,
More informationWEEK #22: PDFs and CDFs, Measures of Center and Spread
WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook
More informationRandom 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 ()
More informationChapter 9 Monté Carlo Simulation
MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?
More information39.2. 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
More informationSection 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
More informationChapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs
Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)
More informationEXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!
STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.
More informationProbability 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.
More informationMATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables
MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,
More informationChapter 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
More informationNormal Probability Distribution
Normal Probability Distribution The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use
More informationChapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions
Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the
More informationEngineering Problem Solving and Excel. EGN 1006 Introduction to Engineering
Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques
More informationThe 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
More informationThe normal approximation to the binomial
The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationStatistics Revision Sheet Question 6 of Paper 2
Statistics Revision Sheet Question 6 of Paper The Statistics question is concerned mainly with the following terms. The Mean and the Median and are two ways of measuring the average. sumof values no. of
More information4.1 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.
More informationChapter 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
More informationCHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS
CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM (SECTION 7.2 OF UNDERSTANDABLE STATISTICS) The Central Limit Theorem says that if x is a random variable with any distribution having
More informationMULTIPLE 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.
More informationCHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution,
More informationThe right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median
CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box
More informationProbability, Mean and Median
Proaility, Mean and Median In the last section, we considered (proaility) density functions. We went on to discuss their relationship with cumulative distriution functions. The goal of this section is
More informationChapter 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
More informationMind on Statistics. Chapter 8
Mind on Statistics Chapter 8 Sections 8.1-8.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable
More informationStatistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013
Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives
More informationExploratory data analysis (Chapter 2) Fall 2011
Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More information5. 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
More informationCHAPTER 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,
More informationTHE BINOMIAL DISTRIBUTION & PROBABILITY
REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationUNIT I: RANDOM VARIABLES PART- A -TWO MARKS
UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0
More informationA Correlation of. to the. South Carolina Data Analysis and Probability Standards
A Correlation of to the South Carolina Data Analysis and Probability Standards INTRODUCTION This document demonstrates how Stats in Your World 2012 meets the indicators of the South Carolina Academic Standards
More informationExploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More informationHISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
More informationAP STATISTICS REVIEW (YMS Chapters 1-8)
AP STATISTICS REVIEW (YMS Chapters 1-8) Exploring Data (Chapter 1) Categorical Data nominal scale, names e.g. male/female or eye color or breeds of dogs Quantitative Data rational scale (can +,,, with
More informationChapter 5. Discrete Probability Distributions
Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A researcher for an airline interviews all of the passengers on five randomly
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More informationz-scores AND THE NORMAL CURVE MODEL
z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationSection 1.3 Exercises (Solutions)
Section 1.3 Exercises (s) 1.109, 1.110, 1.111, 1.114*, 1.115, 1.119*, 1.122, 1.125, 1.127*, 1.128*, 1.131*, 1.133*, 1.135*, 1.137*, 1.139*, 1.145*, 1.146-148. 1.109 Sketch some normal curves. (a) Sketch
More informationDiagrams and Graphs of Statistical Data
Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationSimulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes
Simulation Exercises to Reinforce the Foundations of Statistical Thinking in Online Classes Simcha Pollack, Ph.D. St. John s University Tobin College of Business Queens, NY, 11439 pollacks@stjohns.edu
More informationSTAT 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)
More informationExercise 1.12 (Pg. 22-23)
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
More informationData Analysis Tools. Tools for Summarizing Data
Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool
More informationMULTIPLE 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
More informationUniversity of Arkansas Libraries ArcGIS Desktop Tutorial. Section 2: Manipulating Display Parameters in ArcMap. Symbolizing Features and Rasters:
: Manipulating Display Parameters in ArcMap Symbolizing Features and Rasters: Data sets that are added to ArcMap a default symbology. The user can change the default symbology for their features (point,
More informationLesson 20. Probability and Cumulative Distribution Functions
Lesson 20 Probability and Cumulative Distribution Functions Recall If p(x) is a density function for some characteristic of a population, then Recall If p(x) is a density function for some characteristic
More informationDescriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion
Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research
More informationLecture 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,
More informationCharacteristics of Binomial Distributions
Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation
More informationRandom Variables and Probability
CHAPTER 9 Random Variables and Probability IN THIS CHAPTER Summary: We ve completed the basics of data analysis and we now begin the transition to inference. In order to do inference, we need to use the
More informationBowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition
Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology Step-by-Step - Excel Microsoft Excel is a spreadsheet software application
More information7 CONTINUOUS PROBABILITY DISTRIBUTIONS
7 CONTINUOUS PROBABILITY DISTRIBUTIONS Chapter 7 Continuous Probability Distributions Objectives After studying this chapter you should understand the use of continuous probability distributions and the
More informationThe 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
More informationVariables. Exploratory Data Analysis
Exploratory Data Analysis Exploratory Data Analysis involves both graphical displays of data and numerical summaries of data. A common situation is for a data set to be represented as a matrix. There is
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover
More informationAP * Statistics Review. Descriptive Statistics
AP * Statistics Review Descriptive Statistics Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationLecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000
Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event
More information7. Normal Distributions
7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped
More informationThursday, November 13: 6.1 Discrete Random Variables
Thursday, November 13: 6.1 Discrete Random Variables Read 347 350 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) ±1.88 B) ±1.645 C) ±1.96 D) ±2.
Ch. 6 Confidence Intervals 6.1 Confidence Intervals for the Mean (Large Samples) 1 Find a Critical Value 1) Find the critical value zc that corresponds to a 94% confidence level. A) ±1.88 B) ±1.645 C)
More informationUSING A TI-83 OR TI-84 SERIES GRAPHING CALCULATOR IN AN INTRODUCTORY STATISTICS CLASS
USING A TI-83 OR TI-84 SERIES GRAPHING CALCULATOR IN AN INTRODUCTORY STATISTICS CLASS W. SCOTT STREET, IV DEPARTMENT OF STATISTICAL SCIENCES & OPERATIONS RESEARCH VIRGINIA COMMONWEALTH UNIVERSITY Table
More information