Management Programme. MS-08: Quantitative Analysis for Managerial Applications
|
|
- Myles Patrick
- 6 years ago
- Views:
Transcription
1 MS-08 Management Programme ASSIGNMENT SECOND SEMESTER 2013 MS-08: Quantitative Analysis for Managerial Applications School of Management Studies INDIRA GANDHI NATIONAL OPEN UNIVERSITY MAIDAN GARHI, NEW DELHI
2 ASSIGNMENT QUESTIONS Course Code : MS - 8 Course Title : Quantitative Analysis for Managerial Applications Assignment Code : MS-8/SEM - II /2013 Coverage : All Blocks Note: Attempt all the questions and submit this assignment on or before 31 st October, 2013 to the coordinator of your study centre. 1. Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. Explain the statement. 2. Find the standard deviation and coefficient of skewness for the following distribution Variable Frequency A salesman has a 60% chance of making a sale to any one customer. The behaviour of successive customers is independent. If two customers A and B enter, what is the probability that the salesman will make a sale to A or B. 4. To verify whether a course in Research Methodology improved performance, a similar test was given to 12 participants before and after the course. The original marks and after the course marks are given below: Original Marks Marks after the course Was the course useful? Consider these 12 participants as a sample from a population. 5. Write short notes on a) Bernoulli Trials b) Standard Normal distribution c) Central Limit theorem
3 ASSIGNMENT ANSWERS 1. Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. Explain the statement. ANS. o Statistics may be defined as the science of collection, presentation analysis and interpretation of numerical data from the logical analysis. The word statistic is used to refer to - Numerical facts, such as the number of people living in particular area. - The study of ways of collecting, analyzing and interpreting the facts. o Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. o It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. According to this statement there are four stages: 1) Collection of Data: o It is the first step and this is the foundation upon which the entire data set. o Careful planning is essential before collecting the data. 2) Presentation of data: o The mass data collected should be presented in a suitable, concise form for further analysis. o The collected data may be presented in the form of tabular or diagrammatic or graphic form. 3) Analysis of data: o The data presented should be carefully analyzed for making inference from the presented data such as measures of central tendencies, dispersion, correlation, regression etc. 4) Interpretation of data: o The final step is drawing conclusion from the data collected. o A valid conclusion must be drawn on the basis of analysis. A high degree of skill and experience is necessary for the interpretation. Statistical Methods broadly fall into three categories as shown in the following chart.
4 o Fact becomes knowledge, when it is used in the successful completion of a decision process. o Once you have a massive amount of facts integrated as knowledge, then your mind will be superhuman in the same sense that mankind with writing is superhuman compared to mankind before writing. o The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties. o The above figure depicts the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. o That's why we need statistical data analysis. o Statistical data analysis arose from the need to place knowledge on a systematic evidence base. o This required a study of the laws of probability, the development of measures of data properties and relationships, and so on. o Statistical inference aims at determining whether any statistical significance can be attached that results after due allowance is made for any random variation as a source of error. o Intelligent and critical inferences cannot be made by those who do not understand the purpose, the conditions, and applicability of the various techniques for judging significance. o Statistics is a science assisting you to make decisions under uncertainties (based on some numerical and measurable scales). o Decision making process must be based on data neither on personal opinion nor on belief. o It is already an accepted fact that "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." So, let us be ahead of our time. o A body of techniques and procedures dealing with the collection, organization, analysis, interpretation, and presentation of information that can be stated numerically. o Descriptive statistics are used to organize or summarize a particular set of measurements. o In other words, a descriptive statistic will describe that set of measurements.
5 o Inferential statistics use data gathered from a sample to make inferences about the larger population from which the sample was drawn. o Initially derided by some mathematical purists, it is now considered essential methodology in certain areas. In number theory, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses. Methods of statistics including predictive methods in forecasting are combined with chaos theory and fractal geometry to create video works that are considered to have great beauty. The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed.[citation needed] With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art. Methods of statistics may be used predicatively in performance art, as in a card trick based on a Markov process that only works some of the time, the occasion of which can be predicted using statistical methodology. Statistics can be used to predicatively create art, as in the statistical or stochastic music invented by Iannis Xenakis, where the music is performance-specific. Thus we can say Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. ==================================================================== ====================================================================
6 2. Find the standard deviation and coefficient of skewness for the following distribution Variable Frequency ANS. FOR STANDARD DEVVIATION Variable f M.P. X d = (X-22.5)/5 fd fd 2 C.F N=75 fd=-9 f d 2 =193 Now, For, standard Deviation, σ = fd2 N fd N 2 i
7 Now putting the values from above table in to above equation σ = σ = σ = Standard Deviation σ = COEFFICIENT OF SKEWNESS c.f. SK = Q3Q12Mean Q3 Q1 Q 1 = Size of N/4 th observation = 75/4= th observation lies in the class Now, Q 1 = L + N 4 pcf i f
8 Now putting the values in to above equation Q = = x 5 Q 1 = Now, Q 2 = Size of N/2 th observation = 75/2= 37.5 th observation lies in the class between So Q 2 = Median = L + N 2 pcf i f Now putting the values in to above equation Q = = X 5 Q 2 = Now,
9 Q 3 = Size of 3N/4 th observation = 3(75)/4= th observation lies in the class between So, Q 3 = L + 3N 4 pcf i f Now putting the values in to above equation Q = = X 5 Q 3 = Now, For Coefficient of skewness: c.f. SK = Q3Q12Mean Q3 Q1 Now putting the values in to above equation SK = (133.44) = =
10 So, C.F of SK = ==================================================================== ====================================================================
11 3. A salesman has a 60% chance of making a sale to any one customer. The behaviour of successive customers is independent. If two customers A and B enter, what is the probability that the salesman will make a sale to A or B. ANS. The probability that the salesman make sale to any one customer = 60% = 0.60 The probability that the salesman fails to make sales to any one customer = 1-60 = 0.40 The behavior of successive customers is independent event therefore, 1) The probability that the salesman fails to make sales to any one customer in A & B = 0.40 X 0.40 = 0.16 = 16% 2) The probability that the salesman make sale to at least any one customer from A & B = 1 Probability that salesman sale to none of customer = = = 84%
12 4. To verify whether a course in Research Methodology improved performance, a similar test was given to 12 participants before and after the course. The original marks and after the course marks are given below: Original Marks Marks after the course Was the course useful? Consider these 12 participants as a sample from a population. ANS. Let us take the hypothesis that there is no difference in the marks obtained before and after the course, i.e. the course has not been useful Applying t-test (difference formula):
13 ==================================================================== ====================================================================
14 ANS. 5. Write short notes on a) Bernoulli Trials o In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure". o Repeated independent trials in which there can be only two outcomes are called Bernoulli trials in honor of James Bernoulli ( ). Bernoulli trials lead to the binomial distribution. If a number of trials are large, then the probability of k successes in n trials can be approximated by the Poisson distribution. The binomial distribution and the Poisson distribution are closely approximated by the normal (Gaussian) distribution. These three distributions are the foundation of much of the analysis of physical systems for detection, communication and storage of information. o In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment, is conducted. o The mathematical formalization of the Bernoulli trial is known as the Bernoulli process. o Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example: Did the coin land heads? Was the newborn child a girl? Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result meeting specified conditions, not in any moral judgment. Examples of Bernoulli trials include: Flipping a coin. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A fair coin has the probability of success 0.5 by definition. Rolling a die, where a six is "success" and everything else a "failure". In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum. The Assumptions of Bernoulli Trials i. Each trial results in one of two possible outcomes, denoted success (S) or failure (F). ii. The probability of S remains constant from trial-to-trial and is denoted by p. Write q = 1 p for the constant probability of F. iii. The trials are independent. o Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes "success" and the other outcome "failure".
15 o Let p be the probability of success in a Bernoulli trial. Then the probability of failure q is given by q = 1 - p. o Random variables describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure". o Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. o A random variable corresponding to a binomial is denoted by B (n, p), and is said to have a binomial distribution. o The probability of exactly k successes in the experiment B (n, p) is given by: o Bernoulli trials may also lead to negative binomial distributions (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions. o When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials.
16 ANS. b) Standard Normal distribution o The standard normal distribution is a special case of the normal distribution. o It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one. o The normal random variable of a standard normal distribution is called a standard score or a z score. o Every normal random variable X can be transformed into a z score via the following equation: z = (X - μ) / σ o Where X is a normal random variable, μ is the mean, and σ is the standard deviation. o In other words, A standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Areas under this curve can be found using a standard normal table (Table A in the Moore and Moore & McCabe textbooks). All introductory statistics texts include this table. o Some do format it differently. From the rule we know that for a variable with the standard normal distribution, 68% of the observations fall between -1 and 1 (within 1 standard deviation of the mean of 0), 95% fall between -2 and 2 (within 2 standard deviations of the mean) and 99.7% fall between -3 and 3 (within 3 standard deviations of the mean). o No naturally measured variable has this distribution. o However, all other normal distributions are equivalent to this distribution when the unit of measurement is changed to measure standard deviations from the mean. (That's why this distribution is important--it's used to handle problems involving any normal distribution.) o Recall that a density curve models relative frequency as area under the curve. o Assume throughout this document then that we are working with a variable Z that has a standard normal distribution. o The letter Z is usually used for such a variable, the small letter z is used to indicate the generic value that the variable may take.
17 For Example- Question: What is the relative frequency of observations above -1.48? Identify the range of values described by "above -1.48" (shaded green). Identify the area you need to find (shaded blue). It appears to be about 95%. Use the value to look up an area in your table. However, be careful. Doing so gives you this is nowhere near our initial guess. That's because the table is oriented to find areas under the curve to the left of.. So, in fact, looking up has found the answer to the question What is the relative frequency of measurements falling below This range, z < (in gray) and the associated area (in purple) are shown below. There are two ways to proceed. They are, of course, equivalent.
18 Since of the observations fall below -1.48, the remaining = must fall above Since the total area under the curve is exactly 1, and the purple area is , the blue area must be = In other words, subtraction from 1 is necessary % of the observations fall above (For any normal distribution, or 93.06% of the observations fall above 1.48 times the standard deviation below the mean.) Answer: or 93.06%. SAI ASSIGNMENTS:
19 ANS. c) Central Limit theorem o In probability theory, the central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed. o The central limit theorem has a number of variants. o In its common form, the random variables must be identically distributed. o In variants, convergence of the mean to the normal distribution also occurs for nonidentical distributions, given that they comply with certain conditions. o The Central Limit Theorem describes the characteristics of the "population of the means" which has been created from the means of an infinite number of random population samples of size (N), all of them drawn from a given "parent population". o The Central Limit Theorem predicts that regardless of the distribution of the parent population: 1) The mean of the population of means is always equal to the mean of the parent population from which the population samples were drawn. 2) The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N). 3) The distribution of means will increasingly approximate a normal distribution as the size N of samples increases. o Thus, the Central Limit Theorem explains the ubiquity of the famous bell-shaped "Normal distribution" (or "Gaussian distribution") in the measurements domain. o If x 1, x 2,..., x n are n random variables which are independent and having the same distribution with mean p. and standard deviation σ, then if, the limiting distribution of the standardised mean n SAI ASSIGNMENTS:
20 o In practice, if the sample size is sufficiently large, we need not know the population distribution because the central limit theorem assures us that the distribution of x can be approximated by a normal distribution. o A sample size larger than 30 is generally considered to be large enough for this purposes. o Many practical samples are of size higher than 30. o In all these cases, we know that the sampling distribution of the mean can be approximated by a normal distribution with an expected value equal to the population mean and a variance which is equal to the population variance divided by the sample size n. o We need to use the central limit theorem when the population distribution is either unknown or known to be non-normal. ================================================================= ================================================================= SAI ASSIGNMENTS:
Unit 4 Probability. Dr Mahmoud Alhussami
Unit 4 Probability Dr Mahmoud Alhussami Probability Probability theory developed from the study of games of chance like dice and cards. A process like flipping a coin, rolling a die or drawing a card from
More informationProbability and Statistics
Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT
More informationRandom Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
Random Variable Theoretical Probability Distribution Random Variable Discrete Probability Distributions A variable that assumes a numerical description for the outcome of a random eperiment (by chance).
More informationCS 361: Probability & Statistics
February 26, 2018 CS 361: Probability & Statistics Random variables The discrete uniform distribution If every value of a discrete random variable has the same probability, then its distribution is called
More informationV. Probability. by David M. Lane and Dan Osherson
V. Probability by David M. Lane and Dan Osherson Prerequisites none F.Introduction G.Basic Concepts I.Gamblers Fallacy Simulation K.Binomial Distribution L.Binomial Demonstration M.Base Rates Probability
More informationStatistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018
Statistics Boot Camp Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 March 21, 2018 Outline of boot camp Summarizing and simplifying data Point and interval estimation Foundations of statistical
More informationProbability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin,
Chapter 8 Exercises Probability, For the Enthusiastic Beginner (Exercises, Version 1, September 2016) David Morin, morin@physics.harvard.edu 8.1 Chapter 1 Section 1.2: Permutations 1. Assigning seats *
More informationProbability Distributions.
Probability Distributions http://www.pelagicos.net/classes_biometry_fa18.htm Probability Measuring Discrete Outcomes Plotting probabilities for discrete outcomes: 0.6 0.5 0.4 0.3 0.2 0.1 NOTE: Area within
More informationSociology 6Z03 Topic 10: Probability (Part I)
Sociology 6Z03 Topic 10: Probability (Part I) John Fox McMaster University Fall 2014 John Fox (McMaster University) Soc 6Z03: Probability I Fall 2014 1 / 29 Outline: Probability (Part I) Introduction Probability
More informationFRANKLIN UNIVERSITY PROFICIENCY EXAM (FUPE) STUDY GUIDE
FRANKLIN UNIVERSITY PROFICIENCY EXAM (FUPE) STUDY GUIDE Course Title: Probability and Statistics (MATH 80) Recommended Textbook(s): Number & Type of Questions: Probability and Statistics for Engineers
More informationIntroduction to Statistical Data Analysis Lecture 4: Sampling
Introduction to Statistical Data Analysis Lecture 4: Sampling James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis 1 / 30 Introduction
More informationDiscrete Distributions
Discrete Distributions STA 281 Fall 2011 1 Introduction Previously we defined a random variable to be an experiment with numerical outcomes. Often different random variables are related in that they have
More informationChapter Three. Hypothesis Testing
3.1 Introduction The final phase of analyzing data is to make a decision concerning a set of choices or options. Should I invest in stocks or bonds? Should a new product be marketed? Are my products being
More informationASSIGNMENT BOOKLET. Mathematical Methods (MTE-03) (Valid from 1 st July, 2011 to 31 st March, 2012)
ASSIGNMENT BOOKLET MTE-03 Mathematical Methods (MTE-03) (Valid from 1 st July, 011 to 31 st March, 01) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira
More informationProbability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur
Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Lecture No. # 36 Sampling Distribution and Parameter Estimation
More informationMath 10 - Compilation of Sample Exam Questions + Answers
Math 10 - Compilation of Sample Exam Questions + Sample Exam Question 1 We have a population of size N. Let p be the independent probability of a person in the population developing a disease. Answer the
More informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
More informationSTAT:5100 (22S:193) Statistical Inference I
STAT:5100 (22S:193) Statistical Inference I Week 3 Luke Tierney University of Iowa Fall 2015 Luke Tierney (U Iowa) STAT:5100 (22S:193) Statistical Inference I Fall 2015 1 Recap Matching problem Generalized
More informationREPEATED TRIALS. p(e 1 ) p(e 2 )... p(e k )
REPEATED TRIALS We first note a basic fact about probability and counting. Suppose E 1 and E 2 are independent events. For example, you could think of E 1 as the event of tossing two dice and getting a
More informationSampling Distributions
Sampling Error As you may remember from the first lecture, samples provide incomplete information about the population In particular, a statistic (e.g., M, s) computed on any particular sample drawn from
More informationBinomial random variable
Binomial random variable Toss a coin with prob p of Heads n times X: # Heads in n tosses X is a Binomial random variable with parameter n,p. X is Bin(n, p) An X that counts the number of successes in many
More informationHuman-Oriented Robotics. Probability Refresher. Kai Arras Social Robotics Lab, University of Freiburg Winter term 2014/2015
Probability Refresher Kai Arras, University of Freiburg Winter term 2014/2015 Probability Refresher Introduction to Probability Random variables Joint distribution Marginalization Conditional probability
More informationDiscrete probability distributions
Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter
More informationRandom processes. Lecture 17: Probability, Part 1. Probability. Law of large numbers
Random processes Lecture 17: Probability, Part 1 Statistics 10 Colin Rundel March 26, 2012 A random process is a situation in which we know what outcomes could happen, but we don t know which particular
More informationGlossary. The ISI glossary of statistical terms provides definitions in a number of different languages:
Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the
More informationLecture 7: Confidence interval and Normal approximation
Lecture 7: Confidence interval and Normal approximation 26th of November 2015 Confidence interval 26th of November 2015 1 / 23 Random sample and uncertainty Example: we aim at estimating the average height
More informationClass 26: review for final exam 18.05, Spring 2014
Probability Class 26: review for final eam 8.05, Spring 204 Counting Sets Inclusion-eclusion principle Rule of product (multiplication rule) Permutation and combinations Basics Outcome, sample space, event
More informationMath 140 Introductory Statistics
5. Models of Random Behavior Math 40 Introductory Statistics Professor Silvia Fernández Chapter 5 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Outcome: Result or answer
More informationSenior Math Circles November 19, 2008 Probability II
University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 9, 2008 Probability II Probability Counting There are many situations where
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Professor Silvia Fernández Lecture 8 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. 5.1 Models of Random Behavior Outcome: Result or answer
More informationBINOMIAL DISTRIBUTION
BINOMIAL DISTRIBUTION The binomial distribution is a particular type of discrete pmf. It describes random variables which satisfy the following conditions: 1 You perform n identical experiments (called
More informationAP Statistics Semester I Examination Section I Questions 1-30 Spend approximately 60 minutes on this part of the exam.
AP Statistics Semester I Examination Section I Questions 1-30 Spend approximately 60 minutes on this part of the exam. Name: Directions: The questions or incomplete statements below are each followed by
More informationQUANTITATIVE TECHNIQUES
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (For B Com. IV Semester & BBA III Semester) COMPLEMENTARY COURSE QUANTITATIVE TECHNIQUES QUESTION BANK 1. The techniques which provide the decision maker
More informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 5 Prof. Hanna Wallach wallach@cs.umass.edu February 7, 2012 Reminders Pick up a copy of B&T Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
More informationLecture 20 Random Samples 0/ 13
0/ 13 One of the most important concepts in statistics is that of a random sample. The definition of a random sample is rather abstract. However it is critical to understand the idea behind the definition,
More informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park April 3, 2009 1 Introduction Is the coin fair or not? In part one of the course we introduced the idea of separating sampling variation from a
More informationAnnouncements. Lecture 5: Probability. Dangling threads from last week: Mean vs. median. Dangling threads from last week: Sampling bias
Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure to
More informationMarquette University Executive MBA Program Statistics Review Class Notes Summer 2018
Marquette University Executive MBA Program Statistics Review Class Notes Summer 2018 Chapter One: Data and Statistics Statistics A collection of procedures and principles
More informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park August 19, 2008 1 Introduction There are three main objectives to this section: 1. To introduce the concepts of probability and random variables.
More information1 Normal Distribution.
Normal Distribution.. Introduction A Bernoulli trial is simple random experiment that ends in success or failure. A Bernoulli trial can be used to make a new random experiment by repeating the Bernoulli
More informationExpected Value - Revisited
Expected Value - Revisited An experiment is a Bernoulli Trial if: there are two outcomes (success and failure), the probability of success, p, is always the same, the trials are independent. Expected Value
More information3 PROBABILITY TOPICS
Chapter 3 Probability Topics 135 3 PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction It is often necessary
More informationChapter 18: Sampling Distributions
Chapter 18: Sampling Distributions All random variables have probability distributions, and as statistics are random variables, they too have distributions. The random phenomenon that produces the statistics
More informationLecture 8 Sampling Theory
Lecture 8 Sampling Theory Thais Paiva STA 111 - Summer 2013 Term II July 11, 2013 1 / 25 Thais Paiva STA 111 - Summer 2013 Term II Lecture 8, 07/11/2013 Lecture Plan 1 Sampling Distributions 2 Law of Large
More informationSampling distributions and the Central Limit. Theorem. 17 October 2016
distributions and the Johan A. Elkink School of Politics & International Relations University College Dublin 17 October 2016 1 2 3 Outline 1 2 3 (or inductive statistics) concerns drawing conclusions regarding
More informationChapter 14. From Randomness to Probability. Copyright 2012, 2008, 2005 Pearson Education, Inc.
Chapter 14 From Randomness to Probability Copyright 2012, 2008, 2005 Pearson Education, Inc. Dealing with Random Phenomena A random phenomenon is a situation in which we know what outcomes could happen,
More informationSampling: A Brief Review. Workshop on Respondent-driven Sampling Analyst Software
Sampling: A Brief Review Workshop on Respondent-driven Sampling Analyst Software 201 1 Purpose To review some of the influences on estimates in design-based inference in classic survey sampling methods
More informationChapter 5. Means and Variances
1 Chapter 5 Means and Variances Our discussion of probability has taken us from a simple classical view of counting successes relative to total outcomes and has brought us to the idea of a probability
More informationProbability Rules. MATH 130, Elements of Statistics I. J. Robert Buchanan. Fall Department of Mathematics
Probability Rules MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Introduction Probability is a measure of the likelihood of the occurrence of a certain behavior
More information18.05 Practice Final Exam
No calculators. 18.05 Practice Final Exam Number of problems 16 concept questions, 16 problems. Simplifying expressions Unless asked to explicitly, you don t need to simplify complicated expressions. For
More informationPSY 305. Module 3. Page Title. Introduction to Hypothesis Testing Z-tests. Five steps in hypothesis testing
Page Title PSY 305 Module 3 Introduction to Hypothesis Testing Z-tests Five steps in hypothesis testing State the research and null hypothesis Determine characteristics of comparison distribution Five
More informationTopic 3 Populations and Samples
BioEpi540W Populations and Samples Page 1 of 33 Topic 3 Populations and Samples Topics 1. A Feeling for Populations v Samples 2 2. Target Populations, Sampled Populations, Sampling Frames 5 3. On Making
More informationLecture 2 Binomial and Poisson Probability Distributions
Binomial Probability Distribution Lecture 2 Binomial and Poisson Probability Distributions Consider a situation where there are only two possible outcomes (a Bernoulli trial) Example: flipping a coin James
More informationFourier and Stats / Astro Stats and Measurement : Stats Notes
Fourier and Stats / Astro Stats and Measurement : Stats Notes Andy Lawrence, University of Edinburgh Autumn 2013 1 Probabilities, distributions, and errors Laplace once said Probability theory is nothing
More informationMATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Ramesh Yapalparvi Week 2 Chapter 4 Bivariate Data Data with two/paired variables, Pearson correlation coefficient and its properties, general variance sum law Chapter 6
More informationIntroduction to Statistical Data Analysis Lecture 7: The Chi-Square Distribution
Introduction to Statistical Data Analysis Lecture 7: The Chi-Square Distribution James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis
More informationIntroduction to probability
Introduction to probability 4.1 The Basics of Probability Probability The chance that a particular event will occur The probability value will be in the range 0 to 1 Experiment A process that produces
More informationSTATISTICS OF OBSERVATIONS & SAMPLING THEORY. Parent Distributions
ASTR 511/O Connell Lec 6 1 STATISTICS OF OBSERVATIONS & SAMPLING THEORY References: Bevington Data Reduction & Error Analysis for the Physical Sciences LLM: Appendix B Warning: the introductory literature
More informationCS 361: Probability & Statistics
February 19, 2018 CS 361: Probability & Statistics Random variables Markov s inequality This theorem says that for any random variable X and any value a, we have A random variable is unlikely to have an
More informationIntroductory Probability
Introductory Probability Bernoulli Trials and Binomial Probability Distributions Dr. Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK February 04, 2019 Agenda Bernoulli Trials and Probability
More informationACMS Statistics for Life Sciences. Chapter 13: Sampling Distributions
ACMS 20340 Statistics for Life Sciences Chapter 13: Sampling Distributions Sampling We use information from a sample to infer something about a population. When using random samples and randomized experiments,
More informationSTA Module 4 Probability Concepts. Rev.F08 1
STA 2023 Module 4 Probability Concepts Rev.F08 1 Learning Objectives Upon completing this module, you should be able to: 1. Compute probabilities for experiments having equally likely outcomes. 2. Interpret
More informationProbably About Probability p <.05. Probability. What Is Probability?
Probably About p
More informationFundamental Statistical Concepts and Methods Needed in a Test-and-Evaluator s Toolkit. Air Academy Associates
Fundamental Statistical Concepts and Methods Needed in a Test-and-Evaluator s Toolkit Mark Kiemele Air Academy Associates mkiemele@airacad.com ITEA 010 Symposium Glendale, AZ 13 September 010 INTRODUCTIONS
More informationpsychological statistics
psychological statistics B Sc. Counselling Psychology 011 Admission onwards III SEMESTER COMPLEMENTARY COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY.P.O., MALAPPURAM, KERALA,
More informationA SHORT INTRODUCTION TO PROBABILITY
A Lecture for B.Sc. 2 nd Semester, Statistics (General) A SHORT INTRODUCTION TO PROBABILITY By Dr. Ajit Goswami Dept. of Statistics MDKG College, Dibrugarh 19-Apr-18 1 Terminology The possible outcomes
More informationCarolyn Anderson & YoungShil Paek (Slide contributors: Shuai Wang, Yi Zheng, Michael Culbertson, & Haiyan Li)
Carolyn Anderson & YoungShil Paek (Slide contributors: Shuai Wang, Yi Zheng, Michael Culbertson, & Haiyan Li) Department of Educational Psychology University of Illinois at Urbana-Champaign 1 Inferential
More informationRandom variables (section 6.1)
Random variables (section 6.1) random variable: a number attached to the outcome of a random process (r.v is usually denoted with upper case letter, such as \) discrete random variables discrete random
More informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Introduction The markets can be thought of as a complex interaction of a large number of random
More information7.1 What is it and why should we care?
Chapter 7 Probability In this section, we go over some simple concepts from probability theory. We integrate these with ideas from formal language theory in the next chapter. 7.1 What is it and why should
More informationEach trial has only two possible outcomes success and failure. The possible outcomes are exactly the same for each trial.
Section 8.6: Bernoulli Experiments and Binomial Distribution We have already learned how to solve problems such as if a person randomly guesses the answers to 10 multiple choice questions, what is the
More informationConditional Probability
Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the
More informationProbability and Inference. POLI 205 Doing Research in Politics. Populations and Samples. Probability. Fall 2015
Fall 2015 Population versus Sample Population: data for every possible relevant case Sample: a subset of cases that is drawn from an underlying population Inference Parameters and Statistics A parameter
More informationDS-GA 1002 Lecture notes 11 Fall Bayesian statistics
DS-GA 100 Lecture notes 11 Fall 016 Bayesian statistics In the frequentist paradigm we model the data as realizations from a distribution that depends on deterministic parameters. In contrast, in Bayesian
More informationRandom Variables Example:
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More information1. Discrete Distributions
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 1. Discrete Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space Ω.
More informationELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random
More information1 What are probabilities? 2 Sample Spaces. 3 Events and probability spaces
1 What are probabilities? There are two basic schools of thought as to the philosophical status of probabilities. One school of thought, the frequentist school, considers the probability of an event to
More informationProbability Experiments, Trials, Outcomes, Sample Spaces Example 1 Example 2
Probability Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application. However, probability models underlie
More informationFundamentals of Probability CE 311S
Fundamentals of Probability CE 311S OUTLINE Review Elementary set theory Probability fundamentals: outcomes, sample spaces, events Outline ELEMENTARY SET THEORY Basic probability concepts can be cast in
More informationwhere Female = 0 for males, = 1 for females Age is measured in years (22, 23, ) GPA is measured in units on a four-point scale (0, 1.22, 3.45, etc.
Notes on regression analysis 1. Basics in regression analysis key concepts (actual implementation is more complicated) A. Collect data B. Plot data on graph, draw a line through the middle of the scatter
More informationCOVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS
COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COURSE: CBS 221 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the undergraduate
More information41.2. Tests Concerning a Single Sample. Introduction. Prerequisites. Learning Outcomes
Tests Concerning a Single Sample 41.2 Introduction This Section introduces you to the basic ideas of hypothesis testing in a non-mathematical way by using a problem solving approach to highlight the concepts
More informationIntroduction and Overview STAT 421, SP Course Instructor
Introduction and Overview STAT 421, SP 212 Prof. Prem K. Goel Mon, Wed, Fri 3:3PM 4:48PM Postle Hall 118 Course Instructor Prof. Goel, Prem E mail: goel.1@osu.edu Office: CH 24C (Cockins Hall) Phone: 614
More informationRandom Variables. Definition: A random variable (r.v.) X on the probability space (Ω, F, P) is a mapping
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More informationStatistics and Data Analysis in Geology
Statistics and Data Analysis in Geology 6. Normal Distribution probability plots central limits theorem Dr. Franz J Meyer Earth and Planetary Remote Sensing, University of Alaska Fairbanks 1 2 An Enormously
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics 5.1 Models of random behavior Outcome: Result or answer obtained from a chance process. Event: Collection of outcomes. Probability: Number between 0 and 1 (0% and 100%).
More informationIENG581 Design and Analysis of Experiments INTRODUCTION
Experimental Design IENG581 Design and Analysis of Experiments INTRODUCTION Experiments are performed by investigators in virtually all fields of inquiry, usually to discover something about a particular
More informationIntroduction: MLE, MAP, Bayesian reasoning (28/8/13)
STA561: Probabilistic machine learning Introduction: MLE, MAP, Bayesian reasoning (28/8/13) Lecturer: Barbara Engelhardt Scribes: K. Ulrich, J. Subramanian, N. Raval, J. O Hollaren 1 Classifiers In this
More informationProbability Distribution
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationOverview. Confidence Intervals Sampling and Opinion Polls Error Correcting Codes Number of Pet Unicorns in Ireland
Overview Confidence Intervals Sampling and Opinion Polls Error Correcting Codes Number of Pet Unicorns in Ireland Confidence Intervals When a random variable lies in an interval a X b with a specified
More informationSTAT/SOC/CSSS 221 Statistical Concepts and Methods for the Social Sciences. Random Variables
STAT/SOC/CSSS 221 Statistical Concepts and Methods for the Social Sciences Random Variables Christopher Adolph Department of Political Science and Center for Statistics and the Social Sciences University
More information18.05 Final Exam. Good luck! Name. No calculators. Number of problems 16 concept questions, 16 problems, 21 pages
Name No calculators. 18.05 Final Exam Number of problems 16 concept questions, 16 problems, 21 pages Extra paper If you need more space we will provide some blank paper. Indicate clearly that your solution
More informationQuantitative Methods for Economics, Finance and Management (A86050 F86050)
Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge
More informationIntroduction to Statistical Data Analysis Lecture 3: Probability Distributions
Introduction to Statistical Data Analysis Lecture 3: Probability Distributions James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis
More informationTheoretical Foundations
Theoretical Foundations Sampling Distribution and Central Limit Theorem Monia Ranalli monia.ranalli@uniroma3.it Ranalli M. Theoretical Foundations - Sampling Distribution and Central Limit Theorem Lesson
More informationIDAHO EXTENDED CONTENT STANDARDS MATHEMATICS
Standard 1: Number and Operation Goal 1.1: Understand and use numbers. K.M.1.1.1A 1.M.1.1.1A Recognize symbolic Indicate recognition of expressions as numbers various # s in environments K.M.1.1.2A Demonstrate
More informationUniversity of Jordan Fall 2009/2010 Department of Mathematics
handouts Part 1 (Chapter 1 - Chapter 5) University of Jordan Fall 009/010 Department of Mathematics Chapter 1 Introduction to Introduction; Some Basic Concepts Statistics is a science related to making
More informationSurvey on Population Mean
MATH 203 Survey on Population Mean Dr. Neal, Spring 2009 The first part of this project is on the analysis of a population mean. You will obtain data on a specific measurement X by performing a random
More informationHarvard University. Rigorous Research in Engineering Education
Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected
More information