Class 13. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
|
|
- Richard Cross
- 5 years ago
- Views:
Transcription
1 Class 13 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 017 by D.B. Rowe 1
2 Agenda: Recap Chapter Lecture Chapter Review Chapter 5 for Eam 3. Problem Solving Session.
3 Recap Chapter
4 6: Normal Probability Distributions 6.3 Applications of Normal Distributions Eample: Assume that IQ scores are normally distributed with a mean μ of 100 and a standard deviation σ of 16. If a person is picked at random, what is the probability that his or her IQ is between 100 and 115? i.e. P( )? μ μ Figures from Johnson & Kuby, 01. 4
5 6: Normal Probability Distributions 6.3 Applications of Normal Distributions IQ scores normally distributed μ=100 and σ=16. P( ) z z z Figures from Johnson & Kuby, 01. 5
6 6: Normal Probability Distributions 6.3 Applications of Normal Distributions Now we can use the table. = - P(0 z 0.94) P( z 0.94) P( z 0) Figures from Johnson & Kuby, 01. 6
7 6: Normal Probability Distributions 6.4 Notation Eample: Let α=0.05. Let s find z(0.05). P(z>z(0.05))=0.05. P(z>z(0.05)) Same as finding P(z<z(0.05))= z =P(z>z(0.05)) Figures from Johnson & Kuby, 01. 7
8 6: Normal Probability Distributions 6.4 Notation Eample: Same as finding P(z<z(0.05))= Figures from Johnson & Kuby, 01. 8
9 6: Normal Probability Distributions 6.5 Normal Approimation of the Binomial Distribution Approimate binomial probabilities with normal areas. Use a normal with np, np(1 p) n=14 p=1/ (14)(.5) 7 (14)(.5)(1.5) 3.5 Figures from Johnson & Kuby, 01. 9
10 6: Normal Probability Distributions 6.5 Normal Approimation of the Binomial Distribution n=14, p=1/ We then approimate binomial probabilities with normal areas. P ( 4) from the binomial formula is approimately P( ) from the normal with 7, 3.5 the ±.5 is called a continuity correction Figures from Johnson & Kuby,
11 6: Normal Probability Distributions 6.5 Normal Approimation of the Binomial Distribution From the binomial formula 14! P(4) (.5) (1.5) 4!(14 4)! P ( 4) P( 1.87 z 1.34) P( 1.87 z 1.34) Pz ( 1.34) From the Normal Distribution n=14, p=1/ P( ) 7, z z Pz ( 1.87)
12 6: Normal Probability Distributions Questions? Homework: Chapter 6 # 7, 9, 13, 17, 19, 9, 31, 33, 41, 45, 47, 53, 61, 75, 95, 99 Read Chapter 7. 1
13 Lecture Chapter
14 Chapter 7: Sample Variability Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science 14
15 7: Sample Variability 7. The Sampling Distribution of Sample Means When we take a random sample 1,, n from a population, one of the things that we do is compute the sample mean. The value of is not μ. Each time we take a random sample of size n, we get a different set of values 1,, n and a different value for. 15
16 7: Sample Variability 7. The Sampling Distribution of Sample Means Recall: When we take a sample of data 1,, n from a population, then compute an estimate of a parameter it is called a sample statistic. i.e. for μ Sampling Distribution of a sample statistic: The distribution of values for a sample statistic obtained from repeated samples, all of the same size and all drawn from the same population. 16
17 7: Sample Variability 7. The Sampling Distribution of Sample Means Let s discuss the relationship between the sample mean and the population mean. Assume that we have a population of items with population mean μ and population standard deviation σ. If we take a random sample of size n and compute sample mean,. The collection of all possible means is called the sampling distribution of the sample mean. 17
18 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. S={ } Prob. of each value = 18
19 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. Population data values: 0,, 4, 6, 8. 5 possible values S={0,, 4, 6, 8} 0, occurs one time, occurs one time 4, occurs one time 6, occurs one time 8, occurs one time Prob. of each value = 1/5 = 0. 19
20 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. Population data values: 0,, 4, 6, P( ) 0 1/ 5 1/ 5 4 1/ 5 6 1/ 5 8 1/ 5 5 possible values P()
21 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. Population data values: 0,, 4, 6, possible values P( ) 0 1/ 5 1/ 5 4 1/ 5 6 1/ 5 8 1/ 5 [ P( )] 1
22 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n=1 with replacement. Population data values: 0,, 4, 6, possible values P( ) [( ) P( )] 0 1/ 5 1/ 5 4 1/ 5 6 1/ 5 8 1/ 5 3
23 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5 balls in bucket, select n= with replacement. Population data values: ,, 4, 6, 8. (0,0) (,0) (4,0) (6,0) (8,0) (0,) (,) (4,) (6,) (8,) (0,4) (,4) (4,4) (6,4) (8,4) (0,6) (,6) (4,6) (6,6) (8,6) (0,8) (,8) (4,8) (6,8) (8,8) 5 possible samples 5
24 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: There are N=5 items in the population. Population data values: 0,, 4, 6, 8. Take samples of size n= (with replacement). There are 5 possible samples. Each sample has mean. (0,0) (,0) (4,0) (6,0) (8,0)????? (0,) (,) (4,) (6,) (8,) (0,4) (,4) (4,4) (6,4) (8,4) (0,6) (,6) (4,6) (6,6) (8,6) (0,8) (,8) (4,8) (6,8) (8,8)???????????????????? 6
25 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). 5 possible samples. Each possible sample is equally likely. Prob. of each sample = 1/5 = 0.04 P[( i, j)] 1/ 5 i 0,,4,6,8 j 0,,4,6,8 There are 5 possible samples. (0,0) (,0) (4,0) (6,0) (8,0) (0,) (,) (4,) (6,) (8,) (0,4) (,4) (4,4) (6,4) (8,4) (0,6) (,6) (4,6) (6,6) (8,6) (0,8) (,8) (4,8) (6,8) (8,8) 8
26 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). 5 possible samples. Each possible sample is equally likely. Prob. of each samples mean = 1/5 = 0.04????????????????????????? There are 5 possible samples. (0,0) (,0) (4,0) (6,0) (8,0) (0,) (,) (4,) (6,) (8,) (0,4) (,4) (4,4) (6,4) (8,4) (0,6) (,6) (4,6) (6,6) (8,6) (0,8) (,8) (4,8) (6,8) (8,8) 9
27 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). 5 possible samples. Prob. of each samples mean = 1/5 = 0.04??????????????????????????, occurs times?, occurs times?, occurs times?, occurs times?, occurs times?, occurs times?, occurs times?, occurs times?, occurs times 31
28 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). 5 possible samples. Prob. of each samples mean = 1/5 = 0.04????????????????????????? P (?) P (?) P (?) P (?) P (?) P (?) P (?) P (?) P (?) 33
29 7: Sample Variability 7. The Sampling Distribution of Sample Means Don t forget that the two values that we draw are random. That is, we may know the sample space of possible outcomes but we do not know eactly which ones we will get! Random Sample: A sample obtained in such a way that each possible sample of fied size n has an equal probability of being selected. 36
30 7: Sample Variability 7. The Sampling Distribution of Sample Means if samples increases the empirical dist. turns into theoretical dist. empirical distribution true distribution with population parameters, Figure from Johnson & Kuby, 01. As the number of samples increases the empirical dist. turns into theoretical dist. 37
31 7: Sample Variability 7. The Sampling Distribution of Sample Means Sample distribution of sample means (SDSM): If all possible random samples, each of size n, are taken from any population with mean μ and standard deviation σ, then the sampling distribution of sample means will have the following: 1. A mean equal to μ. A standard deviation equal to n Furthermore, if the sampled population has a normal distribution, then the sampling distribution of will also be normal for all samples of all sizes. Discuss Later: What if the sampled population does not have a normal distribution? 38
32 7: Sample Variability 7. The Sampling Distribution of Sample Means umber if samples increases the empirical dist. turns into theoretical dist. empirical distribution Sample Sample 1 parameter of interest, μ true distribution with population parameters, 8 Sample All Other Samples many more values portion of Figure from Johnson & Kuby, 01. As the number of samples increases the empirical dist. turns into theoretical dist. 3 39
33 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). Instead of drawing two values with replacement and computing the sample mean, we can think of this as drawing one of the sample means with replacement. The probability for each sample mean is 40
34 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). P( ) P (?) P (?) P (?) P (?) P (?) P (?) P (?) P (?) P (?) 4
35 7: Sample Variability 7. The Sampling Distribution of Sample Means Eample: N=5, values: 0,, 4, 6, 8, n= (with replacement). ( ) P( ) P (?) P (?) P (?) P (?) P (?) P (?) P (?) P (?) P (?) 44
36 7: Sample Variability Questions? Homework: Chapter 7 # 6, 1, 3, 9, 33, 35 46
37 Review Chapters 5 (Eam 3 Chapter) Just the highlights! 47
38 5: Probability Distributions (Discrete Variables) 5. Probability Distributions of a Discrete Random Variable Random Variables: assumes a unique value for each of the outcomes in the sample space. Probability Function: A rule P() that assigns probabilities to the values of the random variable. Eample: Let = # of heads when we flip a coin twice. ={0,1,}! 1 1 P ( )!( )! 0 1 P( )
39 5: Probability Distributions (Discrete Variables) 5. Mean and Variance of a Discrete Random Variable Mean of a discrete random variable (epected value): The mean, μ, of a discrete random variable is found by multiplying each possible value of by its own probability, P(), and then adding all of the products together: mean of : mu = sum of (each multiplied by its own probability) n [ ip( i)] i1 (5.1) 49
40 5: Probability Distributions (Discrete Variables) 5. Mean and Variance of a Discrete Random Variable n [ i P( i )] 1P( 1 ) P( )... np( n) i1 For the # of H when we flip a coin twice discrete distribution: μ = ( 1 ) P( 1 ) + ( ) P( ) + ( 3 ) P( 3 ) μ = (0) P(0) + (1) P(1) + () P() μ = (0) (1/4) + (1) (1/) + () (1/4) μ = 0 + 1/ + 1/ μ = P( ) P ( 1) P ( ) P ( 3) 50
41 5: Probability Distributions (Discrete Variables) 5. Mean and Variance of a Discrete Random Variable Variance of a discrete random variable: The variance, σ, of a discrete random variable is found by multiplying each possible value of the squared deviation, ( μ), by its own probability, P(), and then adding all of the products together: variance of : sigma squared = sum of (squared deviation times probability) n i i1 equivalent formula [( ) P( )] i n [ i P( i)] i1 (5.) (5.3b) 51
42 5: Probability Distributions (Discrete Variables) 5. Mean and Variance of a Discrete Random Variable n i P i 1 P 1 P n P n i1 [( ) ( )] ( ) ( ) ( ) ( )... ( ) ( ) For the # of H when we flip a coin twice discrete distribution: σ = ( 1 -μ) P( 1 ) + ( -μ) P( ) + ( 3 -μ) μ = 1 P( 3 ) σ = (0-1) P(0) + (1-1) P(1) + (-1) P() σ = (-1) (1/4) + (0) (1/) + (1) (1/4) σ = 1/ /4 σ = 1/ 1/ P( ) P ( 1) P ( ) P ( 3) 5
43 5: Probability Distributions (Discrete Variables) 5.3 The Binomial Probability Distribution An eperiment with only two outcomes is called a Binomial ep. Call one outcome Success and the other Failure. Each performance of ept. is called a trial and are independent. Prob of eactly successes n! P( ) p (1 p)!( n )! n = number of trials or times we repeat the eperiment. = the number of successes out of n trials. p = the probability of success on an individual trial. n num( successes) P( successes and n- failures) Bi means two like bicycle 0,..., n (5.5) n n!!( n )! 53
44 5: Probability Distributions (Discrete Variables) 5.3 The Binomial Probability Distribution Flip coin ten times. n! P( ) p (1 p)!( n )! n=10, =7, p= ! 1 1 P(7) 7!3! n ! 1 1 P(7) 1 7!(10 7)! 10 = # of Heads n()= ways to get Heads P(7) 10 3 P(7)
45 5: Probability Distributions (Discrete Variables) 5.3 The Binomial Probability Distribution Flip coin ten times. = # of Heads n()= ways to get Heads n ( ) P ( ) n=10 =0,,10 p=1/ n ( ) n!!( n )! n p (1 p) 1/104 n! P( ) p (1 p)!( n )! n Note: 1. 0 P() 1. ΣP()=1 55
46 5: Probability Distributions (Discrete Variables) 5.3 The Binomial Probability Distribution page page n=10, p=1/ n! P( ) p (1 p)!( n )! n Figure from Johnson & Kuby, 01. P ( )
47 5: Probability Distributions (Discrete Variables) 5.3 The Binomial Probability Distribution Eample: n=10, p=1/ What is the probability of getting 4, 5, or 6 heads? P(4 6)=P(4)+P(5)+P(6) P(4 6)=10/104+5/104+10/104 P(4 6)=67/ P ( )
48 5: Probability Distributions (Discrete Variables) 5.3 Mean and Standard Deviation of the Binomial Distribution The formula for the mean μ and variance σ of Binomial is n 0 np n! p (1 p)!( n )! n 0 n n! ( ) p (1 p)!( n )! np(1 p) n np(1 p) (5.7) (5.8) 58
49 5: Probability Distributions (Discrete Variables) 5.3 Mean and Standard Deviation of the Binomial Distribution Eample: n Before, using, we found 1. Now using, we get. Before, using 1/ found. [ P( )] 0 np () (1/ ) 1 n 0 Now using, np(1 p) [( ) P( )] () (1/ ) (1/ ) 1/, we we get. 0 1 P( ) n= =1 p=1/ 59
Class 11. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 11 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 5.3 continued Lecture 6.1-6.2 Go over Eam 2. 2 5: Probability
More informationMarquette University MATH 1700 Class 8 Copyright 2018 by D.B. Rowe
Class 8 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 208 by D.B. Rowe Agenda: Recap Chapter 4.3-4.5 Lecture Chapter 5. - 5.3 2 Recap Chapter 4.3-4.5 3 4:
More informationClass 16. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 16 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 013 by D.B. Rowe 1 Agenda: Recap Chapter 7. - 7.3 Lecture Chapter 8.1-8. Review Chapter 6. Problem Solving
More informationClass 12. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 12 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 2017 by D.B. Rowe 1 Agenda: Recap Chapter 6.1-6.2 Lecture Chapter 6.3-6.5 Problem Solving Session. 2
More informationThe 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
More informationConsider the following examples: ex: let X = tossing a coin three times and counting the number of heads
Overview Both chapters and 6 deal with a similar concept probability distributions. The difference is that chapter concerns itself with discrete probability distribution while chapter 6 covers continuous
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 2 Discrete Distributions The binomial distribution 1 Chapter 2 Discrete Distributions Bernoulli trials and the
More informationCIVL Learning Objectives. Definitions. Discrete Distributions
CIVL 3103 Discrete Distributions Learning Objectives Define discrete distributions, and identify common distributions applicable to engineering problems. Identify the appropriate distribution (i.e. binomial,
More informationNo, because np = 100(0.02) = 2. The value of np must be greater than or equal to 5 to use the normal approximation.
1) If n 100 and p 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? Why or why not? No, because np 100(0.02) 2. The value of np must be greater than or equal to 5 to
More informationLean Six Sigma: Training/Certification Books and Resources
Lean Si Sigma Training/Certification Books and Resources Samples from MINITAB BOOK Quality and Si Sigma Tools using MINITAB Statistical Software A complete Guide to Si Sigma DMAIC Tools using MINITAB Prof.
More informationx is a random variable which is a numerical description of the outcome of an experiment.
Chapter 5 Discrete Probability Distributions Random Variables is a random variable which is a numerical description of the outcome of an eperiment. Discrete: If the possible values change by steps or jumps.
More informationMAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
More informationLecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances
Physical Principles in Biology Biology 3550 Fall 2018 Lecture 9: Plinko Probabilities, Part III Random Variables, Expected Values and Variances Monday, 10 September 2018 c David P. Goldenberg University
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationChapter 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
More informationMATH 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
More informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More informationReview of the Topics for Midterm I
Review of the Topics for Midterm I STA 100 Lecture 9 I. Introduction The objective of statistics is to make inferences about a population based on information contained in a sample. A population is the
More informationVersion 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
More informationMA 1125 Lecture 12 - Mean and Standard Deviation for the Binomial Distribution. Objectives: Mean and standard deviation for the binomial distribution.
MA 5 Lecture - Mean and Standard Deviation for the Binomial Distribution Friday, September 9, 07 Objectives: Mean and standard deviation for the binomial distribution.. Mean and Standard Deviation of the
More informationLecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
More information7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4 - Would the correlation between x and y in the table above be positive or negative? The correlation is negative. -
More informationHomework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More informationStatistics Class 15 3/21/2012
Statistics Class 15 3/21/2012 Quiz 1. Cans of regular Pepsi are labeled to indicate that they contain 12 oz. Data Set 17 in Appendix B lists measured amounts for a sample of Pepsi cans. The same statistics
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationShifting 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
More informationChapter 7. Sampling Distributions
Chapter 7 Sampling Distributions Section 7.1 Sampling Distributions and the Central Limit Theorem Sampling Distributions Sampling distribution The probability distribution of a sample statistic. Formed
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationSTAT 111 Recitation 2
STAT 111 Recitation 2 Linjun Zhang October 10, 2017 Misc. Please collect homework 1 (graded). 1 Misc. Please collect homework 1 (graded). Office hours: 4:30-5:30pm every Monday, JMHH F86. 1 Misc. Please
More informationMidterm 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
More informationDiscrete Probability Distribution
1 Discrete Probability Distribution Key Definitions Discrete Random Variable: Has a countable number of values. This means that each data point is distinct and separate. Continuous Random Variable: Has
More informationSTAT 111 Recitation 3
STAT 111 Recitation 3 Linjun Zhang stat.wharton.upenn.edu/~linjunz/ September 23, 2017 Misc. The unpicked-up homeworks will be put in the STAT 111 box in the Stats Department lobby (It s on the 4th floor
More informationStat511 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
More information5: Several Useful Discrete Distributions
: Several Useful Discrete Distributions. Follow the instructions in the My Personal Trainer section. The answers are shown in the tables below. The Problem k 0 6 7 P( k).000.00.0.0.9..7.9.000 List the
More informationMidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationMidterm Test 1 (Sample) Student Name (PRINT):... Student Signature:... Use pencil, so that you can erase and rewrite if necessary.
MA 180/418 Midterm Test 1 (Sample) Student Name (PRINT):............................................. Student Signature:................................................... Use pencil, so that you can erase
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationLecture 12. Some Useful Continuous Distributions. The most important continuous probability distribution in entire field of statistics.
ENM 207 Lecture 12 Some Useful Continuous Distributions Normal Distribution The most important continuous probability distribution in entire field of statistics. Its graph, called the normal curve, is
More informationLecture 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:
More informationMath 14 Lecture Notes Ch. 4.3
4.3 The Binomial Distribution Example 1: The former Sacramento King's DeMarcus Cousins makes 77% of his free throws. If he shoots 3 times, what is the probability that he will make exactly 0, 1, 2, or
More information. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
More informationReview: 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
More informationBinomal and Geometric Distributions
Binomal and Geometric Distributions Sections 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 7-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationMath 14 Lecture Notes Ch The Normal Approximation to the Binomial Distribution. P (X ) = nc X p X q n X =
6.4 The Normal Approximation to the Binomial Distribution Recall from section 6.4 that g A binomial experiment is a experiment that satisfies the following four requirements: 1. Each trial can have only
More informationchapter 13: Binomial Distribution Exercises (binomial)13.6, 13.12, 13.22, 13.43
chapter 13: Binomial Distribution ch13-links binom-tossing-4-coins binom-coin-example ch13 image Exercises (binomial)13.6, 13.12, 13.22, 13.43 CHAPTER 13: Binomial Distributions The Basic Practice of Statistics
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationContents. The Binomial Distribution. The Binomial Distribution The Normal Approximation to the Binomial Left hander example
Contents The Binomial Distribution The Normal Approximation to the Binomial Left hander example The Binomial Distribution When you flip a coin there are only two possible outcomes - heads or tails. This
More informationMLLunsford 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
More informationMATH 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,
More informationThe Binomial distribution
The Binomial distribution Examples and Definition Binomial Model (an experiment ) 1 A series of n independent trials is conducted. 2 Each trial results in a binary outcome (one is labeled success the other
More informationGETTING STARTED. To OPEN MINITAB: Click Start>Programs>Minitab14>Minitab14 or Click Minitab 14 on your Desktop
Minitab 14 1 GETTING STARTED To OPEN MINITAB: Click Start>Programs>Minitab14>Minitab14 or Click Minitab 14 on your Desktop The Minitab session will come up like this 2 To SAVE FILE 1. Click File>Save Project
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationStatistics 511 Supplemental Materials
Gaussian (or Normal) Random Variable In this section we introduce the Gaussian Random Variable, which is more commonly referred to as the Normal Random Variable. This is a random variable that has a bellshaped
More informationHomework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
More informationCHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS
CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.. The weighted average of the possible
More informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationChapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
More informationChapter 5 Basic Probability
Chapter 5 Basic Probability Probability is determining the probability that a particular event will occur. Probability of occurrence = / T where = the number of ways in which a particular event occurs
More informationIn a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
More information4.1 Probability Distributions
Probability and Statistics Mrs. Leahy Chapter 4: Discrete Probability Distribution ALWAYS KEEP IN MIND: The Probability of an event is ALWAYS between: and!!!! 4.1 Probability Distributions Random Variables
More informationInverse Normal Distribution and Approximation to Binomial
Inverse Normal Distribution and Approximation to Binomial Section 5.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 16-3339 Cathy Poliak,
More informationSTATISTICS and PROBABILITY
Introduction to Statistics Atatürk University STATISTICS and PROBABILITY LECTURE: PROBABILITY DISTRIBUTIONS Prof. Dr. İrfan KAYMAZ Atatürk University Engineering Faculty Department of Mechanical Engineering
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More informationguessing Bluman, Chapter 5 2
Bluman, Chapter 5 1 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question.
More informationChapter 8 Homework Solutions Compiled by Joe Kahlig. speed(x) freq 25 x < x < x < x < x < x < 55 5
H homework problems, C-copyright Joe Kahlig Chapter Solutions, Page Chapter Homework Solutions Compiled by Joe Kahlig. (a) finite discrete (b) infinite discrete (c) continuous (d) finite discrete (e) continuous.
More informationDiscrete 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
More informationElementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
More informationA Derivation of the Normal Distribution. Robert S. Wilson PhD.
A Derivation of the Normal Distribution Robert S. Wilson PhD. Data are said to be normally distributed if their frequency histogram is apporximated by a bell shaped curve. In practice, one can tell by
More information4.3 Normal distribution
43 Normal distribution Prof Tesler Math 186 Winter 216 Prof Tesler 43 Normal distribution Math 186 / Winter 216 1 / 4 Normal distribution aka Bell curve and Gaussian distribution The normal distribution
More informationAMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationStatistics 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
More informationStandard Normal, Inverse Normal and Sampling Distributions
Standard Normal, Inverse Normal and Sampling Distributions Section 5.5 & 6.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 9-3339 Cathy
More informationRandom Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES
Random Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES Essential Question How can I determine whether the conditions for using binomial random variables are met? Binomial Settings When the
More informationThe Central Limit Theorem. Sec. 8.2: The Random Variable. it s Distribution. it s Distribution
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 How Should You Think of a Random Variable? Imagine a bag with numbers
More informationPart 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.
More informationChapter 4 and 5 Note Guide: Probability Distributions
Chapter 4 and 5 Note Guide: Probability Distributions Probability Distributions for a Discrete Random Variable A discrete probability distribution function has two characteristics: Each probability is
More information4: Probability. What is probability? Random variables (RVs)
4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationMath Tech IIII, Mar 13
Math Tech IIII, Mar 13 The Binomial Distribution III Book Sections: 4.2 Essential Questions: What do I need to know about the binomial distribution? Standards: DA-5.6 What Makes a Binomial Experiment?
More informationChapter 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,
More informationChapter 8: Binomial and Geometric Distributions
Chapter 8: Binomial and Geometric Distributions Section 8.1 Binomial Distributions The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Section 8.1 Binomial Distribution Learning Objectives
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 Fall 216 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / Fall 216 1
More informationThe Binomial Distribution
The Binomial Distribution Properties of a Binomial Experiment 1. It consists of a fixed number of observations called trials. 2. Each trial can result in one of only two mutually exclusive outcomes labeled
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationMATH 118 Class Notes For Chapter 5 By: Maan Omran
MATH 118 Class Notes For Chapter 5 By: Maan Omran Section 5.1 Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Ex1: The test scores
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationProb and Stats, Nov 7
Prob and Stats, Nov 7 The Standard Normal Distribution Book Sections: 7.1, 7.2 Essential Questions: What is the standard normal distribution, how is it related to all other normal distributions, and how
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More informationMATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Tommy Khoo Your friendly neighbourhood graduate student. Midterm Exam ٩(^ᴗ^)۶ In class, next week, Thursday, 26 April. 1 hour, 45 minutes. 5 questions of varying lengths.
More informationChapter 7 presents the beginning of inferential statistics. The two major activities of inferential statistics are
Chapter 7 presents the beginning of inferential statistics. Concept: Inferential Statistics The two major activities of inferential statistics are 1 to use sample data to estimate values of population
More informationSection Sampling Distributions for Counts and Proportions
Section 5.1 - Sampling Distributions for Counts and Proportions Statistics 104 Autumn 2004 Copyright c 2004 by Mark E. Irwin Distributions When dealing with inference procedures, there are two different
More informationStatistical Tables Compiled by Alan J. Terry
Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative
More informationHOMEWORK: 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
More informationStats CH 6 Intro Activity 1
Stats CH 6 Intro Activit 1 1. Purpose can ou tell the difference between bottled water and tap water? You will drink water from 3 samples. 1 of these is bottled water.. You must test them in the following
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More information