Chapter 4. Probability and Probability Distributions


 Amelia Strickland
 1 years ago
 Views:
Transcription
1 Chapter 4. robability and robability Distributions
2 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 degree of accuracy to which the sample mean, sample standard deviation, or sample proportion represent the corresponding population values. To decide at what point the result of the observed sample is not possible. This means that we need to know how to find the probability of obtaining a particular sample outcome. robability is the tool that enables us to make an inference.
3 Definition of robability 1 Classical definition Each possible distinct result is called an outcome; an event is identified as a collection of outcomes. The probability of an event E is computed by taking the ratio of the number of outcomes favorable to event E N e to the total number of N of possible outcomes: event E N N e
4 Definition of robability 2 Relative frequency If an experiment is conducted n different times and if event E occurs on n e of these trials, then the probability of event E is approximately event E ne n
5 Basic Event Relations and robability Laws 1 The probability of an event, say event A, will always satisfy the property: 0 A 1 Mutually exclusive Two events A and B are said to be mutually exclusive if they cannot occur simultaneously. A or B A + B
6 Basic Event Relations and Complement robability Laws 2 The complement of an event A is the event that A does not occur. The complement of A is denoted by the symbol A. Union The union of two events A and B is the set of all outcomes that are included in either A or B or both. Intersection A + A 1 The intersection of two events A and B is the set of all outcomes that are included in both A and B. A B A + B AI B
7 Basic Event Relations and robability Laws 3 Conditional robability When probabilities are calculated with a subset of the total group as the denominator, the result is called a conditional probability. Consider two events A and B with nonzero probabilities, A and B. The conditional probability of event A given event B is: A B A B B
8 Basic Event Relations and Independence robability Laws 3 The occurrence of event A is not dependent on the occurrence of event B or, simply, that A and B are independent event. A B A When events A and B are independent, it follows that: A B A B A A B
9 Bayes Formula 1 Let A 1, A 2,, A k be a collection of k mutually exclusive and exhaustive events with A i >0 for i1,, k. Then for any other B for which B >0 Example 1. k j A A B A A B B B A B A k i i i j j j j 1,..., 1
10 Bayes Formula 2 Sensitivity The sensitivity of a test or symptom is the probability of a positive test result or presence of the symptom given the presence of the disease. Specificity The specificity of a test or symptom is the probability of a negative test result or absence of the symptom given the absence of the disease. False positive The false positive of a test or symptom is the probability of a positive test result or presence of the symptom given the absence of the disease. False negative The false negative of a test or symptom is the probability of a negative test result or presence of the symptom given the presence of the disease.
11 Bayes Formula 3 redictive value positive The predictive value positive of a test or symptom is the probability that a subject has the disease given that the subject has a positive test result or has the symptom. redictive value negative The predictive value negative of a test or symptom is the probability that a subject does not have the disease, given that the subject has a negative test result or does not have the symptom. Example 2.
12 Discrete and Continuous Variables Discrete random variable When observation on a quantitative random variable can assume only a countable number of values, the variable is called a discrete random variable. Continuous random variable When observations on a quantitative random variable can assume any one of the uncountable number of values in a line interval, the variable is called a continuous random variable.
13 robability Distribution for Discrete Random Variables 1 For discrete random variables, we can compute the probability of specific individual values occurring. The probability distribution for a discrete random variable displays the probability y associated with each value of y. roperties of discrete random variables: The probability associated with every value of y lies between 0 and 1. The sum of the probabilities for all values of y is equal to 1. The probabilities for a discrete random variable are additive. Hence, the probability that y1, 2, 3,, k is equal to k. Example 3.
14 robability Distribution for Discrete Random Variables 2 Binomial distribution or experiment roperties: A binomial experiment consists of n identical trials. Each trial results is one of two outcomes. We will label one outcome a success and the other a failure. The probability of success on a single trial is equal to π and π remains the same from trial to trial. The trials are independent; that is, the outcome of one trial does not influence the outcome of any other trial. The random variability y is the number of successes observed during the n trials.
15 General Formula for Binomial robability The probability of observing y successes in n trials of a binomial experiment is: Example 3 n! y y! n y π 1 π y! n y where n number of trials π probability of success on a trial 1 π probability of failure on a trial y number of successes in n trials n! n n 1 n
16 Mean and Standard Deviation of the Binomial robability Distribution Mean µ µ nπ Standard deviation σ σ nπ 1 π whereπ is the probability of success in agiven trial and n is the number of trials in the binomial experiment. Example 6
17 robability Distributions for Continuous Random Variables Theoretically, a continuous random variable is one that can assume values associated with infinitely many points in a line interval. It is impossible to assign a small amount of probability to each value of y and retain the property that the probabilities sum to 1. To overcome this difficulty, for continuous random variables, the probability of an interval of values is the event of interest or the probability of y falling in a given interval.
18 Normal Distribution 1 Normal distribution distribution Normal Density µ Normal probability density function f y µ 1 2 2σ y e 2πσ 2
19 Normal Distribution 2 Area under a normal curve Normal Density Normal Density µ µ Normal Density Normal Density µ µ
20 Normal Distribution 3 Z score To determine the probability that a measurement will be less than some value y, we first calculate the number of standard deviations that y lies away from the mean by using the formula: The value of z computed using this formula is sometimes referred to as the z score associated with the yvalue. Using the computed value of z, we determine the appropriate probability by using the z table. Example 8 z y µ σ
21 Normal Distribution 4 100pth percentile The 100pth percentile of a distribution is that value, y p, such that 100p% of the population values fall below y p and 1001p% are above y p. To find the percentile, z p, we find the probability p in z table. To find the 100pth percentile, y p, of a normal distribution with mean µ and standard deviation σ, we need to apply the reverse of the standardization formula: Example 9 y µ + z pσ p
22 Random Sampling Random number table Random number generator
23 Sampling Distributions 1 A sample statistic is a random variable; it id subject to random variation because it is based on a random sample of measurements selected from the population of interest. Like any other random variable, a sample statistic has a probability distribution. We call the probability distribution of a sampling statistic the sampling distribution of that statistic. Example 10
24 Sampling Distributions 2 The sampling distribution of y has mean and standard deviation σ y, which are related to the population mean µ, and standard deviation σ, by the following relationship: µ y µ µ σ y y σ n The sampling deviations have means that are approximately equal to the population mean. Also, the sampling deviations have standard deviations that are approximately equal to σ n. If all possible values of y have been generated, then the standard deviation of y would equal to σ n exactly.
25 Central Limit Theorems 1 Let y denote the sample mean computed from a random sample of n measurements from a population having a mean, µ, and finite standard deviation, σ. Let µ y and σ y denote the mean and standard deviation of the sampling distribution of y, respectively. Based on repeated random samples of size n from the population, we can conclude the following: µ y µ σ n σ y When n is large, the sampling distribution of y will be with the approximation becoming more precise as n increases. distribution of approximately normal When the population distribution is normal, the sampling y is exactly nromal for any sample size n.
26 Central Limit Theorems 2 The Central Limit Theorems provide theoretical justification for our approximating the true sampling distribution of the sample mean with the normal distribution. Similar theorems exist for the sample median, sample standard deviation, and the sample proportion. For applying the Central Limit Theorems, no specific shape is required for the theorems to be validated. However, this is not true in general. If the population distribution had many extreme values or several modes, the sampling distribution of y would require n to be considerably larger in order to achieve a symmetric bell shape.
27 Central Limit Theorems 3 It is very unlikely that the exact shape of the population distribution will be known. Thus, the exact shape of the sampling distribution of y will not be known either. The important point to remember is that the sampling distribution of y will be approximately normally distributed with a mean µ, the y µ population mean, and a standard deviation σ y σ n. The approximation will be more precise as n, the sample size for each sample, increases and as the shape of the population distribution becomes more like the shape of a normal distribution. How large should the sample size be for the Central Limit Theorem to hold? In general, the Central Limit Theorem holds for n > 30. However, one should not apply this rule blindly. If the population is heavily skewed, the sampling distribution for y will still be skewed even for n > 30. On the other hand, if population is symmetric, the Central Limit Theorem holds for n < 30.
28 Central Limit Theorems 4 y Central Limit Theorem for : y Let denote the sum of a random sample of n measurements from a population having a mean µ and finite standard deviation σ. Let µ and σ denote the mean and standard y deviation of the sampling distribution y of y, respectively. Based on repeated random samples of size n from the population, we can conclude the following: µ nµ y σ y When nσ n is large, the sampling distribution of y will be normal with the approximation becoming more precise as approximately n increases. When the population distribution is normal, the sampling distribution of y is exactly nromal for any sample size n.
29 Normal Approximation to the Binomial 1 The binomial random variable y is the number of successes in the n trials. Let n random variables, I 1, I 2,, I n defined as: I i 1 0 if if the ith trial results in a success the ith trial results in a failure n To consider the sum of the random variables, I 1, I 2,, I n, I. i 1 i A 1 is placed in the sum for each success that occurs and a 0 n for each failure that occurs. Thus, I is the number of i 1 i successes that occurred during the n trials. Hence, we conclude n that y Ii. 1 i Because the binomial random variable y is the sum of independent random variables, each having the same distribution, we can apply the Central Limit Theorem for sums to y.
30 Normal Approximation to the Binomial 2 The normal distribution can be used to approximate the binomial distribution when n is of an appropriate size. The normal distribution that will be used has a mean and standard deviation given by the following formula: µ nπ σ nπ 1 π π the probability of success Example 11
31 Normal Approximation to the Binomial 3 The normal approximation to the binomial distribution can be unsatisfactory if nπ < 5 or n1 π < 5. If π is small and n is modest, the actual binomial distribution is seriously skewed to the right. In such a case, the symmetric normal curve will give an unsatisfactory approximation. If π is near 1, so n1 π < 5, the actual binomial will be skewed to the left, and again the normal approximation will not be very accurate. The normal approximation is quite good when nπ or n1 π exceed about 20. In the middle zone, nπ or n1 π between 5 and 20, a modification called continuity correction makes a substantial contribution to the quality of the approximation.
32 Normal Approximation to the Binomial 4 The point of the continuity correction is that we are using the continuous normal curve to approximate a discrete binomial distribution. The general idea of the contunity correction is to add or subtract 0.5 from a binomial value before using normal probabilities. A picture of the situation as the following: Instead of y 5 p[ z / ] p z use y 5.5 [ z / ] p z The actual binomial y 5 C probability is : C C C C C
33 Homework 4.39, 4.40 p , 4.96 p p.189
34 Example 1 A book club classifies members as heavy, medium, or light purchasers, and separate mailings are prepared for each of these groups. Overall, 20% of the members are heavy purchasers, 30% medium, and 50% light. A member is not classified into a group until 18 months after joining the club, but a test is made of the feasibility of using the first 3 months purchases to classify members. The following percentages are obtained from existing records of individuals classified as heavy, medium, or light purchasers: First 3 Group % Months Heavy Medium Light urchases If a member purchases no books in the first 3 months, what is the probability that the member is a light purchaser? Note: This table contains conditional percentages for each column.
35 Answer to Example formula, According to Bayes'? L H H M M L L L L L L
36 Example 2 A screening test for a disease shows the result as the following table. What are the sensitivity, specificity, false positive, false negative, predictive value positive, and predictive value negative? Test Result Disease resent D Absent D Total ositive T a b a + b Negative T c d c + d Total a + c b + d n
37 Answer to Example 2 negative value predictive positive value predictive positive false negative false specificity sensitivity D D T D D T D D T d c d T D D D T D D T D D T b a a T D d b b D T c a c D T d b d D T c a a D T
38 Example 3 An article in the March 5, 1998, issue of The New England Journal of Medicine discussed a large outbreak of tuberculosis. One person, called the index patient, was diagnosed with tuberculosis in The 232 coworker of the index patient were given a tuberculin screening test. The number of coworkers recording a positive reading on the test was the random variable of interest. Did this study satisfy the properties of a binomial experiment?
39 Answer to Example 3 Were there n identical trials? Yes Did each trial result in one of two outcomes? Yes Was the probability of success the same from trial to trial? Yes Were the trials independent? Yes Was the random variable of interest to the experimenter the number of successes y in the 232 screening tests? Yes All five characteristics were satisfied, so the tuberculin screening test represented a binomial experiment.
40 Example 4 An economist interview 75 students in a class of 100 to estimate the proportion of students who expect to obtain a C or better in the course. Is this a binomial experiment? Answer: Were there n identical trials? Yes Did each trial result in one of two outcomes? Yes Was the probability of success the same from trial to trial? No
41 Example 5 What is the probability distribution of the number of heads in tosses of 4 coins? Answer: Let y is the number of heads observed. Then the empirical sampling results for y: y Frequency Observed Relative Frequency Expected Relative Frequency
42 Answer to example 5 continued robability distribution for the number of heads when 4 coins are tossed. y Number of Heads
43 Example 6 Suppose that a sample of households is randomly selected from all the households in the city in order to estimate the percentage in which the head of the household in unemployed. To illustrate the computation of a binomial probability, suppose that the unknown percentage is actually 10% and that a sample of n5 is selected from the population. What is the probability that all five heads of the households are employed? What is the probability of one or fewer being unemployed? Answer: y 5 5! 0.9 5!5 5! ! 0.9 5!0! y 4 or
44 Example 7 A company producing the turf grass takes a sample of 20 seeds on a regular basis to monitor the quality of the seeds. According to the result from previous experiments, the germination rate of the seeds is 85%. If in a particular sample of 20 seeds there are only 12 had germinated, would the germination arte of 85% seem consist with the current results? Answer: µ nπ σ nπ 1 π Thus, y12 seeds is more than 3 standard deviation less than the mean number of seeds µ 17; it is not likely that in 20 seeds we would obtain only 12 germinated seeds if π really is equal to 0.85.
45 The binomial distribution for n 20 and π0.85 Count Number of Germinated Seeds
46 Example 8 The mean daily milk production of a herd of Guerney cows has a normal distribution with µ70 pounds and σ13 pounds. What is the probability that the milk production for a cow chosen at random will be less than 60 pounds? What is the probability that the milk production for a cow chosen at random will be greater than 90 pounds? What is the probability that the milk production for a cow chosen at random will be between 60 pounds and 90 pounds?
47 Answer to Example 8 1 To compute the z value corresponding to the value of 60 pounds. y µ z σ Normal Density µ
48 Answer to Example 8 2 To compute the z value corresponding to the value of 90 pounds. Then, check the z table to find out the corresponding probability of the values greater than 90 pounds. y µ z σ Normal Density µ
49 Answer to Example 8 3 The area between two values 60 and 90 is determine by finding the difference between the areas to left of the two values Normal Density µ
50 Example 9 The Scholastic Assessment Test SAT is an examination used to measure a person s readiness for college. The mathematics scores are used to have a normal distribution with mean 500 and standard deviation 100. What proportion of the people taking the SAT will score below 350? To identify a group of students needing remedial assistance, say, the lower 10% of all scores, what is the score on the SAT?
51 Answer to Example 9 To find the proportion of scores below 350: z y µ σ Normal Density µ To find the 10 th percentile, we first find z 0.1 in z table. Since is the value nearest and its corresponding z is 1.28, we take z and then compute: y0.1 µ + z0.1σ
52 Random Numbers
53 Example 10 1 The population consists of 500 pennies from which we compute the age of each penny: age 2000 date on penny. What are the distributions of y based on sample of sizes n 5, 10 and 25? Given the population mean µ and the population standard deviation σ Frequency Ages
54 Example 10 2 Frequency Frequency Mean Age Mean Age Sample Size Mean of y Standard Deviation of y n Frequency Mean Age Sampling distribution of y for n 5, 10 and 25
55 Example 11 Using the normal approximation to the binomial to compute the probability of observing 460 or fewer in a sample of 1000 favoring consolidation if we assume that 50% of the entire population favor the change. Answer : µ nπ σ z nπ 1 π y µ σ fy y
6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
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 informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationThe Normal distribution
The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bellshaped and described by the function f(y) = 1 2σ π e{ 1 2σ
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 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 informationStatistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013
Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.11.6) Objectives
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More informationNormal Distribution as an Approximation to the Binomial Distribution
Chapter 1 Student Lecture Notes 11 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 informationThe basics of probability theory. Distribution of variables, some important distributions
The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a
More informationChapter 5: Normal Probability Distributions  Solutions
Chapter 5: Normal Probability Distributions  Solutions Note: All areas and zscores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that
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 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 informationMT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...
MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationUnit 16 Normal Distributions
Unit 16 Normal Distributions Objectives: To obtain relative frequencies (probabilities) and percentiles with a population having a normal distribution While there are many different types of distributions
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+ Section 6.2 and 6.3
Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 15 scale to 0100 scores When you look at your report, you will notice that the scores are reported on a 0100 scale, even though respondents
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationSection 5 Part 2. Probability Distributions for Discrete Random Variables
Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability
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 3:304:30, Wed 45 Bring a calculator, and copy Tables
More information1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700
Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,
More informationModels for Discrete Variables
Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations
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 informationPROBABILITY AND SAMPLING DISTRIBUTIONS
PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi  0 0 seema@iasri.res.in. Introduction The concept of probability
More informationProbability distributions
Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.142.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,
More informationNormal and Binomial. Distributions
Normal and Binomial Distributions Library, Teaching and Learning 14 By now, you know about averages means in particular and are familiar with words like data, standard deviation, variance, probability,
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More informationMATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem
MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 111) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
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 35, 36 Special discrete random variable distributions we will cover
More informationMATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS
MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution
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 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 informationUnit 29 ChiSquare GoodnessofFit Test
Unit 29 ChiSquare GoodnessofFit Test Objectives: To perform the chisquare hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni
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 informationHypothesis Testing. Learning Objectives. After completing this module, the student will be able to
Hypothesis Testing Learning Objectives After completing this module, the student will be able to carry out a statistical test of significance calculate the acceptance and rejection region calculate and
More informationSAMPLING DISTRIBUTIONS
0009T_c07_308352.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 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 37, 38 The remaining discrete random
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 informationChapter 15 Multiple Choice Questions (The answers are provided after the last question.)
Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately
More informationNumerical Summarization of Data OPRE 6301
Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting
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 informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationP (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or nondeterministic experiments. Suppose an experiment can be repeated any number of times, so that we
More informationSession 1.6 Measures of Central Tendency
Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices
More informationIntroduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
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 informationThe Standard Normal distribution
The Standard Normal distribution 21.2 Introduction Massproduced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More information, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0
Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after SimeonDenis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationINTRODUCTORY STATISTICS
INTRODUCTORY STATISTICS Questions 290 Field Statistics Target Audience Science Students Outline Target Level First or Secondyear Undergraduate Topics Introduction to Statistics Descriptive Statistics
More informationBasics of Probability
Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.
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 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 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 information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
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 informationGCSE HIGHER Statistics Key Facts
GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information
More informationStatistics 104: Section 6!
Page 1 Statistics 104: Section 6! TF: Deirdre (say: Deardra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm3pm in SC 109, Thursday 5pm6pm in SC 705 Office Hours: Thursday 6pm7pm SC
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationDescriptive Statistics
Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2 (b) 1
Unit 2 Review Name Use the given frequency distribution to find the (a) class width. (b) class midpoints of the first class. (c) class boundaries of the first class. 1) Miles (per day) 12 9 34 22 56
More informationPeople have thought about, and defined, probability in different ways. important to note the consequences of the definition:
PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models
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 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 information
An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476
Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.
More informationLecture 2 : Basics of Probability Theory
Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different
More informationCurriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 20092010
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 20092010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
More informationChapter 3: The basic concepts of probability
Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording
More informationAssigning Probabilities
What is a Probability? Probabilities are numbers between 0 and 1 that indicate the likelihood of an event. Generally, the statement that the probability of hitting a target that is being fired at is
More informationProbabilities and Random Variables
Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationMCQ S OF MEASURES OF CENTRAL TENDENCY
MCQ S OF MEASURES OF CENTRAL TENDENCY MCQ No 3.1 Any measure indicating the centre of a set of data, arranged in an increasing or decreasing order of magnitude, is called a measure of: (a) Skewness (b)
More informationThe Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationChapter 5  Probability
Chapter 5  Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set
More informationLecture 4: Probability Spaces
EE5110: Probability Foundations for Electrical Engineers JulyNovember 2015 Lecture 4: Probability Spaces Lecturer: Dr. Krishna Jagannathan Scribe: Jainam Doshi, Arjun Nadh and Ajay M 4.1 Introduction
More informationEMPIRICAL FREQUENCY DISTRIBUTION
INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION  described by mathematical models 2 1 when some empirical distribution approximates
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
More informationMath 117 Chapter 7 Sets and Probability
Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a welldefined collection of specific objects. Each item in the set is called an element or a member. Curly
More informationSampling Distribution of a Normal Variable
Ismor Fischer, 5/9/01 5.1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationContinuous Random Variables Random variables whose values can be any number within a specified interval.
Section 10.4 Continuous Random Variables and the Normal Distribution Terms Continuous Random Variables Random variables whose values can be any number within a specified interval. Examples include: fuel
More informationReport of for Chapter 2 pretest
Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every
More informationThe Normal Curve. The Normal Curve and The Sampling Distribution
Discrete vs Continuous Data The Normal Curve and The Sampling Distribution We have seen examples of probability distributions for discrete variables X, such as the binomial distribution. We could use it
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 informationExploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More informationRecitation, Week 3: Basic Descriptive Statistics and Measures of Central Tendency:
Recitation, Week 3: Basic Descriptive Statistics and Measures of Central Tendency: 1. What does Healey mean by data reduction? a. Data reduction involves using a few numbers to summarize the distribution
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationDr. Peter Tröger Hasso Plattner Institute, University of Potsdam. Software Profiling Seminar, Statistics 101
Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam Software Profiling Seminar, 2013 Statistics 101 Descriptive Statistics Population Object Object Object Sample numerical description Object
More information32 Measures of Central Tendency and Dispersion
32 Measures of Central Tendency and Dispersion In this section we discuss two important aspects of data which are its center and its spread. The mean, median, and the mode are measures of central tendency
More informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationMATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables
MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,
More information