The Central Limit Theorem. Sec. 8.2: The Random Variable. it s Distribution. it s Distribution
|
|
|
- Nigel Hancock
- 5 years ago
- Views:
Transcription
1 The Central Limit Theorem Sec. 8.1: The Random Variable it s Distribution Sec. 8.2: The Random Variable it s Distribution X p and and
2 How Should You Think of a Random Variable? Imagine a bag with numbers in it. You draw one number from the bag and the random variable X is the number that you drew from the bag. How did the bag with numbers get there? From some experiment in the background whose outcomes we relabeled with numbers. But the end product is the bag with numbers and that s what we care about. If you know the probability distribution of X then you can answer questions about what numbers can be drawn from the bag and what the chances are (probability) that various numbers are drawn from the bag.
3 Bag with numbers How Should You Think of a Random Variable?
4 Bag with numbers How Should You Think of a Random Variable?
5 Bag with numbers How Should You Think of a Random Variable?
6 Bag with numbers How Should You Think of a Random Variable?
7 How Should You Think of a Random Variable? Bag with numbers The experiment in the background could be Betting on the draw of a card from a deck Counting the total number of heads when you flip a coin 4 times or something else
8 How Should You Think of a Random Variable? Bag with numbers If you know the probability distribution, say x P(X=x) Then I know The possible numbers that can be drawn from the bag The probabilities for the different numbers that can come out of the bag The average (EV), standard deviation and variance of all numbers in the bag
9 How Should You Think of a Note about random variables: Random Variable? A random variable is not a fixed quantity, it varies (variable). Every time you perform the experiment (or draw a number from the bag), you get a new value for the random variable.
10 Recall: Population Population Data (unknown) Population Parameters (unknown) p 2 Sample Sample Data (known) Sample Statistics (known) pˆ x s 2 s These are Random Variables!
11 Sec. 8.1: How is X a Random Variable? Original Bag Start off with a bag of numbers (population data). The random variable X is the number you get if you draw 1 number from the bag. The population parameter μ = the average of all of the numbers in the entire original bag, which is usually unknown, is known in this section. σ will also be known
12 Sec. 8.1: How is X a Random Variable? New Bag Form a new bag of numbers as follows: Take a sample of numbers of size n from the original bag (population data) and calculate the average of the numbers in your sample. This is X. Do this for all possible samples of size n to form your new bag. The random variable X is the number you get when you draw 1 number from this new bag. You can think of getting a value of X in 2 different ways: 1. Draw a sample of size n from the original bag and calculate the average, OR 2. Draw a single number from the new X bag
13 Sec. 8.1: How is X a Random Variable? Picture of the situation X Take a sample of size n Calculate X (the average of the numbers in sample) Put answer in bag on right X Do this for all possible samples of size n Original bag of numbers (population data) μ = the average of all numbers in the entire original bag is known in this section (usually unknown). is also known σ New bag has all sample averages in it Taking a sample of size n and calculating X is equivalent to drawing 1 number from the new bag
14 Sec. 8.1: How is X a Random Variable? Notes: There are 3 different ways to draw a sample of size n from a bag: 1. Draw one by one with replacement 2. Draw one by one without replacement 3. Draw all n at once When we draw samples, we will always think of them as drawn the first way (with replacement), but as long as the condition n 0.05N is met, all probabilities regarding X will be essentially the same.
15 Sec. 8.1: The Probability Distribution of X What does the probability distribution of look like? Ex: Suppose a bag contains the numbers 1, 2, 3, 4, 5, 6 in it, each appearing only once in the bag. Let X = The number you get when you draw 1 number from this original bag = The average of a sample of size 2 drawn from the original bag (or the number you get when you draw 1 number from the X 2 bag = The average of a sample of size 3 drawn from the original bag (or the number you get when you draw 1 number from the bag X 2 X 3 X 3 X
16 Sec. 8.1: The Probability Distribution of X What does the probability distribution of X look like? X
17 Sec. 8.1: The Probability Distribution of X What does the probability distribution of X look like? X 2
18 Sec. 8.1: The Probability Distribution of X What does the probability distribution of X look like? X 3
19 Sec. 8.1: The Probability Distribution of X Central Limit Theorem (for X ) Suppose n 0.05N If X is any random variable whatsoever and n 30, then X has a normal distribution. If X has a normal distribution to begin with, then X has a normal distribution no matter what n is. μ X = μ X and σ X = σ X n
20 Ex 1: A bag of numbers has a normal distribution with a mean of 10 and a standard deviation of 4. Find a) the probability that a single number drawn from the bag is less than 21 b) the probability that a single number drawn from the bag is larger than 5 c) the probability that a single number drawn from the bag is between 4.5 and 11.3 d) the probability that in a sample of size 20, the average is less than 9.5 e) the probability that in a sample of size 20, the average is at least 11.3 f) the probability that in a sample of size 20, the average is between 10.4 and 11.7
21 Ex 2: The amount of credit card debt among all Americans ages 18 to 24 has a mean of $2,982 and a standard deviation of $315. What is the probability that in a randomly selected group of 36 Americans between the ages of 18 and 24 that a) their average credit card debt is at most $3050? b) their average credit card debt is more than $2850? c) their average credit card debt is between $2900 and $3000?
22 Sec. 8.2: How is p a Random Variable? Original Bag Start off with a bag of numbers consisting of 0 s and 1 s only (population data). Think of 0 s as standing for no s and 1 s as standing for yes s. The random variable X is the number you get if you draw 1 number from the bag. The population parameter p = the percentage of 1 s in the entire original bag, which is usually unknown, is known in this section.
23 Sec. 8.2: How is p a Random Variable? New Bag Form a new bag of numbers as follows: Take a sample of 0 s and 1 s of size n from the original bag (population data) and calculate the percentage of 1 s in your sample. This is p. Do this for all possible samples of size n to form your new bag. The random variable p is the number you get when you draw 1 number from this new bag. This new bag does not contain 0 s and 1 s in it, but instead has many different percentages in it. You can think of getting a value of p in 2 different ways: 1. Draw a sample of size n from the original bag and calculate the percentage of 1 s in the sample, OR 2. Draw a single number from the new bag p
24 Sec. 8.2: How is p a Random Variable? Picture of the situation X Take a sample of size n Calculate p (the percentage of 1 s in sample) Put answer in bag on right p Original bag of 0 s and 1 s (population data, yes=1, no=0) p = the percentage of 1 s in the entire original bag is known in this section (usually unknown) Do this for all possible samples of size n New bag has all sample percentages in it instead of 0 s and 1 s Taking a sample of size n and calculating p is equivalent to drawing 1 number from the new bag
25 Sec. 8.2: How is p a Random Variable? Notes: There are 3 different ways to draw a sample of size n from a bag: 1. Draw one by one with replacement 2. Draw one by one without replacement 3. Draw all n at once When we draw samples, we will always think of them as drawn the first way (with replacement), but as long as the condition n 0.05N is met, all probabilities regarding p will be essentially the same. p is a kind of x (explained on board). Therefore central limit applies to p. But instead of the condition n 30, we have the condition npq 10. (p + q = 1)
26 Sec. 8.2: The Probability Distribution of p Central Limit Theorem for p As long as the conditions n 0.05N npq 10 are satisfied, p (p + q = 1) has a NORMAL DISTRIBUTION with μ p = p and σ p = pq n
27 Ex 3: A bag of numbers contains only 0 s and 1 s in it (think of 0 s as no s and 1 s as yes s). The percentage of 1 s in the bag is 72%. If a sample of 60 numbers is drawn from the bag, a) what is the probability that the percentage of 1 s in the sample is less than 65%? b) what is the probability that the percentage of 1 s in the sample is more than 68%? c) what is the probability that the percentage of 1 s in the sample is between 73% and 78%
28 Ex 4: 56% of the students who enter America s colleges and universities graduate within six years. If 250 college freshman are randomly selected from American colleges and universities, a) what is the probability that less than 64% of them will graduate within 6 years? b) what is the probability that at least 60% of them will graduate within 6 years? c) what is the probability that between 48% and 55% of them will graduate within 6 years?
29 How is s a Random Variable? Picture of the situation X Take a sample of size n Calculate s (the standard deviation of the numbers in sample) Put answer in bag on right s Do this for all possible samples of size n Original bag of numbers (population data) New bag has all sample standard deviations in it Taking a sample of size n and calculating s is equivalent to drawing 1 number from the new bag
30 How is s 2 Picture of the situation X a Random Variable? Take a sample of size n Calculate s 2 (the variance of the numbers in sample) Put answer in bag on right s 2 Do this for all possible samples of size n Original bag of numbers (population data) New bag has all sample variances in it Taking a sample of size n and calculating s 2 is equivalent to drawing 1 number from the new bag
Version A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
Chapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 9. Introduction This chapter connects the material in Chapters 4 through 8 (numerical descriptive statistics, sampling, and probability distributions, in particular) with
Central Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
Law of Large Numbers, Central Limit Theorem
November 14, 2017 November 15 18 Ribet in Providence on AMS business. No SLC office hour tomorrow. Thursday s class conducted by Teddy Zhu. November 21 Class on hypothesis testing and p-values December
The Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
Statistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
Part 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
Chapter 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
Binomial Distributions
Binomial Distributions Binomial Experiment The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials There are only two possible outcomes of interest
Statistics vs. statistics
Statistics vs. statistics Question: What is Statistics (with a capital S)? Definition: Statistics is the science of collecting, organizing, summarizing and interpreting data. Note: There are 2 main ways
The Central Limit Theorem
Section 6-5 The Central Limit Theorem I. Sampling Distribution of Sample Mean ( ) Eample 1: Population Distribution Table 2 4 6 8 P() 1/4 1/4 1/4 1/4 μ (a) Find the population mean and population standard
Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5
Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5 Steve Dunbar Due Fri, October 9, 7. Calculate the m.g.f. of the random variable with uniform distribution on [, ] and then
Elementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
Probability is the tool used for anticipating what the distribution of data should look like under a given model.
AP Statistics NAME: Exam Review: Strand 3: Anticipating Patterns Date: Block: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%-30%) Probability is the tool used
Discrete Random Variables
Discrete Random Variables In this chapter, we introduce a new concept that of a random variable or RV. A random variable is a model to help us describe the state of the world around us. Roughly, a RV can
Stat511 Additional Materials
Binomial Random Variable Stat511 Additional Materials The first discrete RV that we will discuss is the binomial random variable. The binomial random variable is a result of observing the outcomes from
Sampling Distributions
AP Statistics Ch. 7 Notes Sampling Distributions A major field of statistics is statistical inference, which is using information from a sample to draw conclusions about a wider population. Parameter:
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data
Statistics 511 Additional Materials
Discrete Random Variables In this section, we introduce the concept of a random variable or RV. A random variable is a model to help us describe the state of the world around us. Roughly, a RV can be thought
Stat 213: Intro to Statistics 9 Central Limit Theorem
1 Stat 213: Intro to Statistics 9 Central Limit Theorem H. Kim Fall 2007 2 unknown parameters Example: A pollster is sure that the responses to his agree/disagree questions will follow a binomial distribution,
As you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
Statistics and Their Distributions
Statistics and Their Distributions Deriving Sampling Distributions Example A certain system consists of two identical components. The life time of each component is supposed to have an expentional distribution
Previously, when making inferences about the population mean, μ, we were assuming the following simple conditions:
Chapter 17 Inference about a Population Mean Conditions for inference Previously, when making inferences about the population mean, μ, we were assuming the following simple conditions: (1) Our data (observations)
work to get full credit.
Chapter 18 Review Name Date Period Write complete answers, using complete sentences where necessary.show your work to get full credit. MULTIPLE CHOICE. Choose the one alternative that best completes the
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
Chapter 7 Study Guide: The Central Limit Theorem
Chapter 7 Study Guide: The Central Limit Theorem Introduction Why are we so concerned with means? Two reasons are that they give us a middle ground for comparison and they are easy to calculate. In this
STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions 5/31/11 Lecture 14 1 Statistic & Its Sampling Distribution
Sampling and sampling distribution
Sampling and sampling distribution September 12, 2017 STAT 101 Class 5 Slide 1 Outline of Topics 1 Sampling 2 Sampling distribution of a mean 3 Sampling distribution of a proportion STAT 101 Class 5 Slide
Statistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
6. THE BINOMIAL DISTRIBUTION
6. THE BINOMIAL DISTRIBUTION Eg: For 1000 borrowers in the lowest risk category (FICO score between 800 and 850), what is the probability that at least 250 of them will default on their loan (thereby rendering
The Normal Approximation to the Binomial
Lecture 16 The Normal Approximation to the Binomial We can calculate l binomial i probabilities bbilii using The binomial formula The cumulative binomial tables When n is large, and p is not too close
MATH 264 Problem Homework I
MATH Problem Homework I Due to December 9, 00@:0 PROBLEMS & SOLUTIONS. A student answers a multiple-choice examination question that offers four possible answers. Suppose that the probability that the
Module 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
Chapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
Midterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
Bernoulli and Binomial Distributions
Bernoulli and Binomial Distributions Bernoulli Distribution a flipped coin turns up either heads or tails an item on an assembly line is either defective or not defective a piece of fruit is either damaged
L04: Homework Answer Key
L04: Homework Answer Key Instructions: You are encouraged to collaborate with other students on the homework, but it is important that you do your own work. Before working with someone else on the assignment,
4.2 Probability Distributions
4.2 Probability Distributions Definition. A random variable is a variable whose value is a numerical outcome of a random phenomenon. The probability distribution of a random variable tells us what the
AMS7: 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:
Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances
Physical Principles in Biology Biology 3550 Fall 2018 Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances Monday, 10 September 2018 c David P. Goldenberg University
Chapter 7 - Lecture 1 General concepts and criteria
Chapter 7 - Lecture 1 General concepts and criteria January 29th, 2010 Best estimator Mean Square error Unbiased estimators Example Unbiased estimators not unique Special case MVUE Bootstrap General Question
Statistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
MidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
Distribution of the Sample Mean
Distribution of the Sample Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Experiment (1 of 3) Suppose we have the following population : 4 8 1 2 3 4 9 1
Section 7-2 Estimating a Population Proportion
Section 7- Estimating a Population Proportion 1 Key Concept In this section we present methods for using a sample proportion to estimate the value of a population proportion. The sample proportion is the
STA258H5. Al Nosedal and Alison Weir. Winter Al Nosedal and Alison Weir STA258H5 Winter / 41
STA258H5 Al Nosedal and Alison Weir Winter 2017 Al Nosedal and Alison Weir STA258H5 Winter 2017 1 / 41 NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION. Al Nosedal and Alison Weir STA258H5 Winter 2017
S = 1,2,3, 4,5,6 occurs
Chapter 5 Discrete Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. Discrete random variables associated with these experiments
Tutorial 6. Sampling Distribution. ENGG2450A Tutors. 27 February The Chinese University of Hong Kong 1/6
Tutorial 6 Sampling Distribution ENGG2450A Tutors The Chinese University of Hong Kong 27 February 2017 1/6 Random Sample and Sampling Distribution 2/6 Random sample Consider a random variable X with distribution
I. Standard Error II. Standard Error III. Standard Error 2.54
1) Original Population: Match the standard error (I, II, or III) with the correct sampling distribution (A, B, or C) and the correct sample size (1, 5, or 10) I. Standard Error 1.03 II. Standard Error
Confidence Intervals and Sample Size
Confidence Intervals and Sample Size Chapter 6 shows us how we can use the Central Limit Theorem (CLT) to 1. estimate a population parameter (such as the mean or proportion) using a sample, and. determine
Theoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
8.4: The Binomial Distribution
c Dr Oksana Shatalov, Spring 2012 1 8.4: The Binomial Distribution Binomial Experiments have the following properties: 1. The number of trials in the experiment is fixed. 2. There are 2 possible outcomes
Chapter 9. Sampling Distributions. A sampling distribution is created by, as the name suggests, sampling.
Chapter 9 Sampling Distributions 9.1 Sampling Distributions A sampling distribution is created by, as the name suggests, sampling. The method we will employ on the rules of probability and the laws of
MLLunsford 1. Activity: Central Limit Theorem Theory and Computations
MLLunsford 1 Activity: Central Limit Theorem Theory and Computations Concepts: The Central Limit Theorem; computations using the Central Limit Theorem. Prerequisites: The student should be familiar with
STAT Chapter 6: Sampling Distributions
STAT 515 -- Chapter 6: Sampling Distributions Definition: Parameter = a number that characterizes a population (example: population mean ) it s typically unknown. Statistic = a number that characterizes
STAT Chapter 7: Confidence Intervals
STAT 515 -- Chapter 7: Confidence Intervals With a point estimate, we used a single number to estimate a parameter. We can also use a set of numbers to serve as reasonable estimates for the parameter.
Engineering Statistics ECIV 2305
Engineering Statistics ECIV 2305 Section 5.3 Approximating Distributions with the Normal Distribution Introduction A very useful property of the normal distribution is that it provides good approximations
6 Central Limit Theorem. (Chs 6.4, 6.5)
6 Central Limit Theorem (Chs 6.4, 6.5) Motivating Example In the next few weeks, we will be focusing on making statistical inference about the true mean of a population by using sample datasets. Examples?
MATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Tommy Khoo Your friendly neighbourhood graduate student. It is Time for Homework Again! ( ω `) Please hand in your homework. Third homework will be posted on the website,
STAT 509: Statistics for Engineers Dr. Dewei Wang. Copyright 2014 John Wiley & Sons, Inc. All rights reserved.
STAT 509: Statistics for Engineers Dr. Dewei Wang Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger 7 Point CHAPTER OUTLINE 7-1 Point Estimation 7-2
Lecture 2 INTERVAL ESTIMATION II
Lecture 2 INTERVAL ESTIMATION II Recap Population of interest - want to say something about the population mean µ perhaps Take a random sample... Recap When our random sample follows a normal distribution,
Central Limit Theorem
Central Limit Theorem Lots of Samples 1 Homework Read Sec 6-5. Discussion Question pg 329 Do Ex 6-5 8-15 2 Objective Use the Central Limit Theorem to solve problems involving sample means 3 Sample Means
8.1 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
6.3: The Binomial Model
6.3: The Binomial Model The Normal distribution is a good model for many situations involving a continuous random variable. For experiments involving a discrete random variable, where the outcome of the
CHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Discrete and Continuous Random
Determine whether the given procedure results in a binomial distribution. If not, state the reason why.
Math 5.3 Binomial Probability Distributions Name 1) Binomial Distrbution: Determine whether the given procedure results in a binomial distribution. If not, state the reason why. 2) Rolling a single die
Statistics for Business and Economics
Statistics for Business and Economics Chapter 7 Estimation: Single Population Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-1 Confidence Intervals Contents of this chapter: Confidence
AP Statistics Ch 8 The Binomial and Geometric Distributions
Ch 8.1 The Binomial Distributions The Binomial Setting A situation where these four conditions are satisfied is called a binomial setting. 1. Each observation falls into one of just two categories, which
Review: Population, sample, and sampling distributions
Review: Population, sample, and sampling distributions A population with mean µ and standard deviation σ For instance, µ = 0, σ = 1 0 1 Sample 1, N=30 Sample 2, N=30 Sample 100000000000 InterquartileRange
Confidence Intervals Introduction
Confidence Intervals Introduction A point estimate provides no information about the precision and reliability of estimation. For example, the sample mean X is a point estimate of the population mean μ
1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by. Cov(X, Y ) = E(X E(X))(Y E(Y ))
Correlation & Estimation - Class 7 January 28, 2014 Debdeep Pati Association between two variables 1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by Cov(X, Y ) = E(X E(X))(Y
(Practice Version) Midterm Exam 1
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran September 19, 2014 (Practice Version) Midterm Exam 1 Last name First name SID Rules. DO NOT open
Interval estimation. September 29, Outline Basic ideas Sampling variation and CLT Interval estimation using X More general problems
Interval estimation September 29, 2017 STAT 151 Class 7 Slide 1 Outline of Topics 1 Basic ideas 2 Sampling variation and CLT 3 Interval estimation using X 4 More general problems STAT 151 Class 7 Slide
The Binomial Distribution
MATH 382 The Binomial Distribution Dr. Neal, WKU Suppose there is a fixed probability p of having an occurrence (or success ) on any single attempt, and a sequence of n independent attempts is made. Then
Statistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 5 Discrete Probability Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 014 Pearson Education, Inc. Chap 5-1 Learning
Probability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
No, 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
Chapter 5 Student Lecture Notes 5-1. Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 5 Student Lecture Notes 5-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals
15.063: Communicating with Data Summer Recitation 3 Probability II
15.063: Communicating with Data Summer 2003 Recitation 3 Probability II Today s Goal Binomial Random Variables (RV) Covariance and Correlation Sums of RV Normal RV 15.063, Summer '03 2 Random Variables
E509A: Principle of Biostatistics. GY Zou
E509A: Principle of Biostatistics (Week 2: Probability and Distributions) GY Zou gzou@robarts.ca Reporting of continuous data If approximately symmetric, use mean (SD), e.g., Antibody titers ranged from
Section 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
2.) What is the set of outcomes that describes the event that at least one of the items selected is defective? {AD, DA, DD}
Math 361 Practice Exam 2 (Use this information for questions 1 3) At the end of a production run manufacturing rubber gaskets, items are sampled at random and inspected to determine if the item is Acceptable
The normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
AP Statistics Test 5
AP Statistics Test 5 Name: Date: Period: ffl If X is a discrete random variable, the the mean of X and the variance of X are given by μ = E(X) = X xp (X = x); Var(X) = X (x μ) 2 P (X = x): ffl If X is
Making Sense of Cents
Name: Date: Making Sense of Cents Exploring the Central Limit Theorem Many of the variables that you have studied so far in this class have had a normal distribution. You have used a table of the normal
Week 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
Lesson 97 - Binomial Distributions IBHL2 - SANTOWSKI
Lesson 97 - Binomial Distributions IBHL2 - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin 3 times where P(H) = / (b) THUS, find the probability
Opening Exercise: Lesson 91 - Binomial Distributions IBHL2 - SANTOWSKI
08-0- Lesson 9 - Binomial Distributions IBHL - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin times where P(H) = / (b) THUS, find the probability
Sampling Distribution
MAT 2379 (Spring 2012) Sampling Distribution Definition : Let X 1,..., X n be a collection of random variables. We say that they are identically distributed if they have a common distribution. Definition
Discrete Random Variables
Discrete Random Variables ST 370 A random variable is a numerical value associated with the outcome of an experiment. Discrete random variable When we can enumerate the possible values of the variable
Part V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
Lecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
CH 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
Binomial and Geometric Distributions
Binomial and Geometric Distributions Section 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office hours: T Th 2:30 pm - 5:15 pm 620 PGH Department of Mathematics University of Houston February 11, 2016
MA131 Lecture 9.1. = µ = 25 and σ X P ( 90 < X < 100 ) = = /// σ X
The Central Limit Theorem (CLT): As the sample size n increases, the shape of the distribution of the sample means taken with replacement from the population with mean µ and standard deviation σ will approach
HOMEWORK: Due Mon 11/8, Chapter 9: #15, 25, 37, 44
This week: Chapter 9 (will do 9.6 to 9.8 later, with Chap. 11) Understanding Sampling Distributions: Statistics as Random Variables ANNOUNCEMENTS: Shandong Min will give the lecture on Friday. See website
Business 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:
Shifting our focus. We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why?
Probability Introduction Shifting our focus We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why? What is Probability? Probability is used