Lecture 10: Depicting Sampling Distributions of a Sample Proportion
|
|
- Cornelius Day
- 7 years ago
- Views:
Transcription
1 Lecture 10: Depicting Sampling Distributions of a Sample Proportion Chapter 5: Probability and Sampling Distributions 2/10/12 Lecture 10 1
2 Sample Proportion 1 is assigned to population members having a specified characteristic and 0 is assigned to those who don t. The parameter of interest in this situation is p (or called π), the proportion of the population that has the characteristic of interest. ˆp Denote as the proportion of members having such characteristic (or say, successes), also called the sample proportion, in a random sample of size n. 2/10/12 Lecture 10 2
3 Sampling Distribution of Sample Proportion If X ~ B(n, p), the sample proportion is defined as X p ˆ = = n count of successes in sample size of sample. Mean & variance of a sample proportion: µ pˆ = p σ, pˆ = p(1 p) / n. 2/10/12 Lecture 10 3
4 2/10/12 Lecture 10 4
5 Example: Clinton's vote 43% of the population voted for Clinton in Suppose we survey a sample of size 2300 and see if they voted for Clinton or not in We are interested in the sampling distribution of the sample proportion samples of size What's the mean and variance of pˆ? pˆ, for 2/10/12 Lecture 10 5
6 Count & Proportion of Success A BoilerMaker basketball player is a 95% freethrow shooter. Suppose he will shoot 5 free-throws during each practice. X: number of free-throws he makes during practice. pˆ : proportion of made-free-throws during practice. pˆ P( =0.6) =? P(X=3) =? 2/10/12 Lecture 10 6
7 Normal Approximation for Counts and Proportions Let X ~ B(n, p) and pˆ = X / n is the sample proportion. If n is large*, then X is approx. N( np, np(1-p)) pˆ is approx. N( p, p(1-p) / n). *Rule of Thumb: np 5, n(1 - p) 5. 2/10/12 Lecture 10 7
8 Switches Inspection A quality engineer selects an SRS of 100 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the specifications. What is the probability that at most 9 switches fail the standard test in the sample? Use Normal Approximation here. Try to use continuity correction as well. (Similar to ex. 43 in Hw4) 2/10/12 Lecture 10 8
9 Switches Inspection Express the probability in terms of X: P(X 9) The normal approximation to the probability of no more than 9 bad switches is the area to the left of X = 9 under the normal curve, which has µ X = np = ( 100)(.1) = 10, σ = np(1 p) = 100(.1)(.9) = In this case np = 10, and n(1-p) = 90, both satisfying the condition for rule of thumb. X 3. Also, don t forget the CONTINUITY CORRECTION when applying normal approximation to the Binomial distribution. 2/10/12 Lecture 10 9
10 Continuity Correction The normal approximation is more accurate if we consider X = 9 to extend from 8.5 to 9.5 Example (Cont.): X PX ( 9) = PX ( 9.5) = P( ) 3 3 PZ (.17) = /10/12 Lecture 10 10
11 Continuity Correction P(X 8) replaced by P(X < 8.5) P(X 14) replaced by P(X > 13.5) P(X < 8) = P(X<=7), then replaced by P(X < 7.5) For large n the effects of the continuity correction factor is very small and will be omitted. 2/10/12 Lecture 10 11
12 Coin Tossing Example Toss a fair coin 200 times, what is the probability that the total number of heads is between 90 and 110? Use the normal approximation and continuity correction for Binomial distribution to solve the problem. X= the total number of heads X ~ Bin(200, 0.5). Want: P(90 X 110). µ X =200.5 = 100, σ X = ( ) 1/2 = With continuity correction: P(90 X 110) = P(89.5 X 110.5) P( Z ( )/7.07) - P (Z ( )/7.07) = P (Z 1.48) - P (Z -1.48)= /10/12 Lecture 10 12
13 Example Note: If not required to use Binomial knowledge, we can use the sampling distribution for sample proportion as well. The Laurier company s brand has a market share of 30%. In a survey 1000 consumers were asked which brand they prefer. What is the probability that more than 32% of the respondents say they prefer the Laurier brand? Solution: The number of respondents who prefer Laurier is binomial with n = 1000 and p =.30. Also, np = 1000(.3) >5, n(1-p) = 1000(1-.3) > 5. Therefore we can use normal approximation to sample proportion here: ˆ ( ˆ.32) p p P p > = P > = P( Z > 1.38) = (1 ) p p n 2/10/12 Lecture 10 13
14 Summary Sampling Distribution Sampling distribution for a sample mean Mean and Variance/S.D. for a sample mean Central Limit Theorem Normal calculation Sampling distribution for a sample proportion Continuity correction for Binomial Distribution Rule of thumb, when n is large 2/10/12 Lecture 10 14
15 Review Sec 5.4 through 5.6. After Class Start Hw 4 due by next Monday, Feb 14th Review all the notes up to today, go over the practice test. Come to Office Hours for Hw#1-#4 grading issues. About Exam 1. Start editing your own cheat-sheet Practice Test is posted. Try to do it in an hour 2/10/12 Lecture 10 15
4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
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 informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)
More informationIntroduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers
Introduction to the Practice of Statistics Sixth Edition Moore, McCabe Section 5.1 Homework Answers 5.18 Attitudes toward drinking and behavior studies. Some of the methods in this section are approximations
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 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 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 informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationThe normal approximation to the binomial
The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationStatistics 104: Section 6!
Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC
More informationThe Normal distribution
The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ
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 information6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.
Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.
More informationof course the mean is p. That is just saying the average sample would have 82% answering
Sampling Distribution for a Proportion Start with a population, adult Americans and a binary variable, whether they believe in God. The key parameter is the population proportion p. In this case let us
More informationBinomial Probability Distribution
Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are
More informationIntroduction to Hypothesis Testing
I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true
More informationLecture 2: Discrete Distributions, Normal Distributions. Chapter 1
Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables
More informationBayes Theorem. Bayes Theorem- Example. Evaluation of Medical Screening Procedure. Evaluation of Medical Screening Procedure
Bayes Theorem P(C A) P(A) P(A C) = P(C A) P(A) + P(C B) P(B) P(E B) P(B) P(B E) = P(E B) P(B) + P(E A) P(A) P(D A) P(A) P(A D) = P(D A) P(A) + P(D B) P(B) Cost of procedure is $1,000,000 Data regarding
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 information2. Discrete random variables
2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationDepartment of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part 1
Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part Note: Assume missing data (if any) and mention the same. Q. Suppose X has a normal
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 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 informationPractice problems for Homework 11 - Point Estimation
Practice problems for Homework 11 - Point Estimation 1. (10 marks) Suppose we want to select a random sample of size 5 from the current CS 3341 students. Which of the following strategies is the best:
More informationMath 251, Review Questions for Test 3 Rough Answers
Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationPractice Problems for Homework #6. Normal distribution and Central Limit Theorem.
Practice Problems for Homework #6. Normal distribution and Central Limit Theorem. 1. Read Section 3.4.6 about the Normal distribution and Section 4.7 about the Central Limit Theorem. 2. Solve the practice
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationMath 151. Rumbos Spring 2014 1. Solutions to Assignment #22
Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability
More informationWeek 3&4: Z tables and the Sampling Distribution of X
Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal
More information東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文
東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文 指 導 教 授 楊 朝 棟 博 士 以 網 路 功 能 虛 擬 化 實 作 網 路 即 時 流 量 監 控 服 務 研 究 生 楊 曜 佑 中 華 民 國 一 零 四 年 五 月 摘 要 與 的 概 念 一 同 發 展 的, 是 指 利 用 虛 擬 化 的 技 術, 將 現 有 的 網 路 硬 體 設 備, 利 用 軟 體 來 取 代 其
More informationChapter 5: Normal Probability Distributions - Solutions
Chapter 5: Normal Probability Distributions - Solutions Note: All areas and z-scores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that
More informationJoint Exam 1/P Sample Exam 1
Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question
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 informationExample 1. so the Binomial Distrubtion can be considered normal
Chapter 6 8B: Examples of Using a Normal Distribution to Approximate a Binomial Probability Distribution Example 1 The probability of having a boy in any single birth is 50%. Use a normal distribution
More informationBinomial random variables (Review)
Poisson / Empirical Rule Approximations / Hypergeometric Solutions STAT-UB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die
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 informationSample Questions for Mastery #5
Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could
More informationHYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationNormal Approximation. Contents. 1 Normal Approximation. 1.1 Introduction. Anthony Tanbakuchi Department of Mathematics Pima Community College
Introductory Statistics Lectures Normal Approimation To the binomial distribution Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission
More informationUniversity of California, Los Angeles Department of Statistics. Random variables
University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.
More informationDETERMINE whether the conditions for a binomial setting are met. COMPUTE and INTERPRET probabilities involving binomial random variables
1 Section 7.B Learning Objectives After this section, you should be able to DETERMINE whether the conditions for a binomial setting are met COMPUTE and INTERPRET probabilities involving binomial random
More informationPr(X = x) = f(x) = λe λx
Old Business - variance/std. dev. of binomial distribution - mid-term (day, policies) - class strategies (problems, etc.) - exponential distributions New Business - Central Limit Theorem, standard error
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 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 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 informationSampling Distributions
Sampling Distributions You have seen probability distributions of various types. The normal distribution is an example of a continuous distribution that is often used for quantitative measures such as
More informationBinomial random variables
Binomial and Poisson Random Variables Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance
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 information2WB05 Simulation Lecture 8: Generating random variables
2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating
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 information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
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 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 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 informationMath 425 (Fall 08) Solutions Midterm 2 November 6, 2008
Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationAP Statistics 7!3! 6!
Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!
More informationSome special discrete probability distributions
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationYou flip a fair coin four times, what is the probability that you obtain three heads.
Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.
More informationOpgaven Onderzoeksmethoden, Onderdeel Statistiek
Opgaven Onderzoeksmethoden, Onderdeel Statistiek 1. What is the measurement scale of the following variables? a Shoe size b Religion c Car brand d Score in a tennis game e Number of work hours per week
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 information16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION
6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of
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 informationSAMPLING DISTRIBUTIONS
0009T_c07_308-352.qd 06/03/03 20:44 Page 308 7Chapter SAMPLING DISTRIBUTIONS 7.1 Population and Sampling Distributions 7.2 Sampling and Nonsampling Errors 7.3 Mean and Standard Deviation of 7.4 Shape of
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 informationSTAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE
STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
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 informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
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 information2 Binomial, Poisson, Normal Distribution
2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability
More informationStat 104: Quantitative Methods for Economists. Study Guide Solutions, part 2
Stat 104: Quantitative Methods for Economists Study Guide Solutions, part 2 1) The table below shows, for credit card holders with one to three cards, the joint probabilities for number of cards owned
More informationLecture 4: BK inequality 27th August and 6th September, 2007
CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound
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 informationLecture 6: Discrete & Continuous Probability and Random Variables
Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September
More informationReview #2. Statistics
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers 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 informationECE302 Spring 2006 HW4 Solutions February 6, 2006 1
ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationPractice problems for Homework 12 - confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.
Practice problems for Homework 1 - confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the
More informationPractice Midterm Exam #2
The Islamic University of Gaza Faculty of Engineering Department of Civil Engineering 12/12/2009 Statistics and Probability for Engineering Applications 9.2 X is a binomial random variable, show that (
More informationChapter 4 Statistical Inference in Quality Control and Improvement. Statistical Quality Control (D. C. Montgomery)
Chapter 4 Statistical Inference in Quality Control and Improvement 許 湘 伶 Statistical Quality Control (D. C. Montgomery) Sampling distribution I a random sample of size n: if it is selected so that the
More informationLecture 3: Continuous distributions, expected value & mean, variance, the normal distribution
Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing
More informationIn the general population of 0 to 4-year-olds, the annual incidence of asthma is 1.4%
Hypothesis Testing for a Proportion Example: We are interested in the probability of developing asthma over a given one-year period for children 0 to 4 years of age whose mothers smoke in the home In the
More informationFinal Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin
Final Mathematics 51, Section 1, Fall 24 Instructor: D.A. Levin Name YOU MUST SHOW YOUR WORK TO RECEIVE CREDIT. A CORRECT ANSWER WITHOUT SHOWING YOUR REASONING WILL NOT RECEIVE CREDIT. Problem Points Possible
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 informationBinomial Random Variables
Binomial Random Variables Dr Tom Ilvento Department of Food and Resource Economics Overview A special case of a Discrete Random Variable is the Binomial This happens when the result of the eperiment is
More informationPoint and Interval Estimates
Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number
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 informationStatistics 100A Homework 4 Solutions
Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation
More informationHYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...
HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men
More information5.1 Identifying the Target Parameter
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying
More information