Probability & Statistics Syllabus PDF

Probability & Statistics Dr. Dalia Abdel Massih, Ph.D  Syllabus Catalogue Description Set theory; probability axioms; random variables (RV); continuous and discrete probability density functions; distributions; operations on RV’s, sampling distributions; confiden...

Probability & Statistics Dr. Dalia Abdel Massih, Ph.D  Syllabus Catalogue Description Set theory; probability axioms; random variables (RV); continuous and discrete probability density functions; distributions; operations on RV’s, sampling distributions; confidence intervals (single variable); hypothesis testing (single variable); linear regression (single variable); non-linear regression. Prerequisite: MTH 201 Calculus III. Topics Probability, random variables and generic probability distributions, mathematical expectation, discrete probability distributions, continuous probability distributions, sampling distributions and data description, one-sample estimation problems, one-sample hypothesis testing, and simple linear regression and correlation.1 - 2  Syllabus Textbook “Probability and Statistics for Engineers and Scientists”, Walpole, R.E., Myers, R.H., Myers, S.L. and Ye, K., 9th edition, Prentice Hall, 2012. Additional References “Probability and Statistics for Scientists and Engineers” Q, Rao V. Dukkipati, New Academic Science Ltd, 2011. Any other probability and statistics textbook.  Syllabus Course Outline Week Relevant Course Content Chapters 1 Introduction to Statistics and Data Analysis Chapter 1 1 Probability Chapter 2 2 Random Variables and Probability Distributions Chapter 3 2,3 Mathematical Expectation Chapter 4 3,4 Discrete Probability Distributions Chapter 5 Exam I 4 Continuous Probability Distributions Chapter 6 5 Fundamental Sampling Distributions and Data Chapter 8 Description 5 One Sample Estimation Problems Chapter 9 Exam II 6 One Sample tests of Hypotheses Chapter 10 7 Simple Linear Regression and Correlation Chapter 11 1-4  Syllabus Learning Outcomes 1. Understand basic probability theory and its rules 2. Employ random variables and their distributions to solve engineering problems 3. Analyze mathematical expectations of random variables 4. Identify various well-known discrete and continuous probability distribution functions 5. Select an appropriate probability distribution function for a given problem 6. Understand the fundamentals of sampling distributions 7. Understand the one-sample estimation problem 8. Perform hypothesis testing 9. Apply linear regression and compute correlation 1-5  Syllabus Grading Quizzes and Attendance 10% Exam I 25% or 30% * Exam II 25% or 30% * Final Exam 35% *: 30% on max(Exam I, Exam II) and 25% on min(Exam I, Exam II). 1-6  Syllabus Course Policies Students are expected to regularly attend the sessions. Missing any session does not excuse students from their responsibility for the work performed or for any announcement made during their absence. Students are expected to arrive to the sessions on time, otherwise, they will not be allowed to enter the classroom. Mobile phone use is forbidden during the sessions. Students using their mobile phones will be asked to leave the session. Failure to take a test or the final exam during the assigned class period will result in a grade of zero being recorded for that test unless the student has a valid reason. The student should communicate with the instructor or the department office within 24 hours of the exam date to present the excuse. No makeup exams will be given for the two exams. In case a student fails to take an exam, the student will lose 5% of the total course grade and the remaining 25% will be distributed in the following manner: 10% for the other exam (that will now count for 35% of the final grade) and 15% for the final exam (that will now count for 50% of the final grade). Note that this applies if there is a valid reason otherwise a 0 is recorded as the exam grade. Makeup exams will be given for students who fail to attend the final exam for a valid reason. Programmable calculators are not allowed during exams. 1-7 Chapter 1 Introduction to Statistics and Data Analysis Copyright © 2017 Pearson Education ,Ltd. All rights reserved.  Introduction What Engineers Do?  An engineer is someone who solves problems of interest to society with the efficient application of scientific principles by: Refining existing products Designing new products or processes – CREATIVE PROCESS 1-9  Introduction Statistics Supports The Creative Process The field of probability and statistics deals with the collection, presentation, analysis and use of data to: Make decisions Solve problems Design products and processes Statistics is the Science of Data 1 - 10  Introduction Definition of Statistics  It is a science which helps us to collect, analyze and present data systematically.  It is the process of collecting, processing, summarizing, presenting, analyzing and interpreting of data in order to study and describe a given problem.  Statistics is the art of learning from data.  Statistics may be regarded as (i) the study of populations, (ii) the study of variation, and (iii) the study of methods of the reduction of data.  Statistics gives us a framework for describing variability and learning about potential sources of variability 1 - 11  Introduction Importance of Statistics  It simplifies mass of data (condensation);  Helps to get concrete information about any problem;  Helps for reliable and objective decision making;  It presents facts in a precise & definite form;  It facilitates comparison (Measures of central tendency and measures of dispersion);  It facilitates Predictions (Time series and regression analysis are the most commonly used methods towards prediction.);  It helps in formulation of suitable conclusions and recommendations; 1 - 12  Introduction Application areas of probability and statistics:  Engineering: Improving product design, testing product performance, determining reliability and maintainability, working out safer systems of flight control for airports, study the effect of variability on responses, etc.  Business: Estimating the volume of retail sales, designing optimum inventory control system, producing auditing and accounting procedures, improving working conditions in industrial plants, assessing the market for new products. 1 - 13  Introduction Application areas of probability and statistics:  Quality Control: Determining techniques for evaluation of quality through adequate sampling, in process control, consumer survey and experimental design in product development etc. Realizing its importance, large organizations are maintaining their own Statistical Quality Control Department. Economy, Health and Medecine, Biology, Psychology, Sociology,… 1 - 14  Introduction There are two main branches of statistics: Descriptive statistics Inferential statistics Involves organizing, Involves using sample summarizing and data to draw conclusions Displaying data. about a population. e.g. Tables, charts, averages,… 1 - 15  Introduction Descriptive statistics:  It is the first phase of Statistics;  involves any kind of data processing designed to the collection, organization, presentation, and analyzing the important features of the data with out attempting to infer/conclude any thing that goes beyond the known data. (Bar graphs, histograms, Pie charts, …Measures of Central Tendency: Mean, Median, Mode,…), Measures of Variability: Variance, Standard, deviation,…)  Describes the nature or characteristics of the observed data (usually a sample) without making conclusion or generalization. 1 - 16  Introduction The following are some examples of descriptive Statistics:  The daily average temperature range was 25oC last week.  The maximum amount of coffee export (as observed from the last 20 years) was in the year 2004.  The average age of athletes participated in London Marathon was 25 years.  50% of the instructors at LAU are female.  The scores of 50 students in a GEN 331 exam are found to range from 20 to 90. 1 - 17  Introduction Inferential statistics (Inductive Statistics):  It is the second phase of Statistics. It focuses on making generalizations about a larger population based on a representative sample of that population. Because inferential statistics focuses on making predictions (rather than stating facts) its results are usually in the form of a probability.  It is concerned with the process of drawing conclusions (inferences) about specific characteristics of a population based on information obtained from samples;  It is a process of performing hypothesis testing, determining relationships among variables, and making predictions.  The area of inferential statistics entirely needs the whole aims to give reasonable estimates of unknown population parameters. 1 - 18  Introduction The following are some examples of Inferential statistics (Inductive Statistics):  Calculation of the probability of exceeding a certain performance level,  Calculation of the probability of having a certain number of defects objects in manufacturing  Errors calculation  Fitting data using regression  Find correlations between variables 1 - 19  Introduction Main terms in statistics Data: Certainly known facts from which conclusions may be drawn. Statistical data: Raw material for a statistical investigation which are obtained when ever measurements or observations are made. i. Quantitative data: data of a certain group of individuals which is expressed numerically. Example: Heights, Weights, Ages and, etc of a certain group of individuals. ii. Qualitative data: data of a certain group of individuals that is not expressed numerically. Example: Colors, Languages, Nationalities, Religions, health, poverty etc of a certain group of individuals. 1 - 20  Introduction Variable: A variable is a factor or characteristic that can take on different possible values or outcomes. A variable can be qualitative or quantitative (numeric). Example: Income, height, weight, properties, etc of a certain group of individuals/materials are examples of variables. Population: A complete set of observation (data) of the entire group of individuals under consideration. A population can be finite or infinite. Example: The number of students in this class, the population in Lebanon etc. Sample: A set of data drawn from population containing a part which can reasonably serve as a basis for valid generalization about the population. A sample is a portion of a population selected for further analysis. 1 - 21  Introduction Sample size: The number of items under investigation in a sample. Survey (experiment): it is a process of obtaining the desired data. Two types of survey: Census Survey: A way of obtaining data referring the entire population including a total coverage of the population. Sample Survey: A way of obtaining data referring a portion of the entire population consisting only a partial coverage of the population. 1 - 22  Introduction STEPS/STAGES IN STATISTICAL INVESTIGATION 1. Collection of Data: Data collection is the process of gathering information or data about the variable of interest. Data are inputs for Statistical investigation. Data may be obtained either from primary source or secondary source. 2. Organization of Data Organization of data includes three major steps. Editing: checking and omitting inconsistencies, irrelevancies. Classification : task of grouping the collected and edited data. Tabulation: put the classified data in the form of table. 1 - 23  Introduction STEPS/STAGES IN STATISTICAL INVESTIGATION 3. Presentation of Data The purpose of presentation in the statistical analysis is to display what is contained in the data in the form of Charts, Pictures, Diagrams and Graphs for an easy and better understanding of the data. 4. Analyzing of Data In a statistical investigation, the process of analyzing data includes finding the various statistical constants from the collected mass of data such as measures of central tendencies (averages), measures of dispersions and … It merely involves mathematical operations: different measures of central tendencies (averages), measures of variations, regression analysis etc. 1 - 24  Introduction STEPS/STAGES IN STATISTICAL INVESTIGATION 5. Interpretation of Data  Involve interpreting the statistical constants computed in analyzing data for the formation of valid conclusions and inferences.  It is the most difficult and skill requiring stage.  It is at this stage that Statistics seems to be very much viable to be misused.  Correct interpretation of results will lead to a valid conclusion of the study and hence can aid in taking correct decisions.  Improper (incorrect) interpretation may lead to wrong conclusions and makes the whole objective of the study useless 1 - 25  Role of Probability The Role of Probability Without some formalism of probability theory, the student cannot appreciate the true interpretation from data analysis through modern statistical methods. The discipline of probability, then, provides the transition between descriptive statistics and inferential methods. 1 - 26  Samples, Populations Samples from populations Samples are collected from populations, which are collections of all individuals or individual items of a particular type. A population = scientific system. Example A manufacturer of computer boards may wish to eliminate defects. Sampling process = collecting information on 50 computer boards (to check their effectiveness The population = all computer boards manufactured by the firm over a specific period of time. Any conclusions drawn should extend to the entire population of computer boards 1 - 27  Relationship between probability and inferential statistics Thus for a statistical problem, the sample along with inferential statistics allows us to draw conclusions about the population, with inferential statistics making clear use of elements of probability. 1 - 28  Sampling Procedures; Collection of Data How sampling is done? Each member of the population has an equal chance Selection follows a and probability of certain pattern being selected Dividing a population Dividing the population into subgroups with the into subpopulations or same characteristics as homogeneous the whole sample. subgroups (strata) Then, subgroups based on specific themselves are characteristics.. randomly selected. 1 - 29  Examples of Some Measurements from Samples Measures of Location The Sample Mean and Median The mean is simply a numerical average. The median is the middle value of a set of data containing an odd number of values or the average of 2 middle values of a set of data with an even number of values.  Examples of Some Measurements from Samples Measures of Location The Sample Mean and Median As an example, suppose the data set is the following: 1.7, 2.2, 3.9, 3.11, and 14.7. The sample mean and median are, respectively, Clearly, the mean is influenced considerably by the presence of the extreme observation, 14.7, whereas the median places emphasis on the true “center” of the data set. The purpose of the sample median is to reflect the central tendency of the sample in such a way that it is uninfluenced by extreme values. From an engineering view, the sample mean is the centroid of the data in a sample 1 - 31 Importance of Variability 1 - 32  Importance of Variability The measures of Mean and Median are enough to represent the variability of the data 1 - 33  Measures of Variability There are many measures of spread or variability. Perhaps the simplest one is the sample range Xmax − Xmin. The sample measure of spread that is used most often is the sample standard deviation. Sample standard deviation is, in fact, a measure of variability. Large variability in a data set produces relatively large values of (x − x)2 and thus a large sample variance. The quantity n − 1 is often called the degrees of freedom associated with the variance estimate. 1 - 34  Exercises Exercise 1 In a study conducted by the Department of Mechanical Engineering at Virginia Tech, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company, and a measure of flexibility was recorded for each (cm/N). The data are as follows: a)Calculate the sample mean and median for the data for the two companies. b)Plot the data for the two companies on the same line and give your impression regarding any apparent differences between the two companies. c)Compute the variance in “flexibility” for both company A and company B. Does there appear to be a difference in flexibility between company A and company B? 1 - 35  Exercises Exercise 1 SOLUTION 1 - 36  Measures of Location The Sample Mean and Median Exercise 2 The tensile strength of silicone rubber is thought to be a function of curing temperature. A study was carried out in which samples of 12 specimens of the rubber were prepared using curing temperatures of 20◦C and 45◦C. The data below show the tensile strength values in megapascals a)Show a dot plot of the data with both low and high temperature tensile strength values. b)Compute sample mean tensile strength for both samples. c)Does it appear as if curing temperature has an influence on tensile strength, based on the plot? Comment further. d)Does anything else appear to be influenced by an increase in curing temperature? Explain. e)Compute the sample standard deviation in tensile strength for the samples separately for the two temperatures. Does it appear as if an increase in temperature influences the variability in tensile strength? Explain. 1 - 37  Exercises Exercise 2 SOLUTION 1 - 38  Discrete and Continuous Data Continuous Discrete A set of data is said to be A set of data is said to be discrete if continuous if the values belonging the values belonging to the set are to the set can take on any value distinct and separate within a finite or infinite interval Measured Counted Examples Examples Speed of a car The number of tickets sold in a day Length of a leaf The number of students in your class Daily wind speed The number of employees in a company Freezer temperature The number of computers in each Weight of newborn babies department Product box measurements and weight 1 - 39  Discrete and Continuous Data Continuous Discrete 1 - 40  DATA DISPLAY Some simple but often effective displays that complement the study of statistical populations. 1. Scatter Plot 2. Histogram 1 - 41  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Scatter plot A textile manufacturer who designs an experiment where cloth specimen that contain various percentages of cotton are produced. Five cloth specimens are manufactured for each of the four cotton percentages. 1 - 42  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Scatter plot of tensile strength and cotton percentages 1 - 43  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Histogram Consider the data of Table below, which specifies the “life” of 40 similar car batteries recorded to the nearest tenth of a year. The batteries are guaranteed to last 3 years. 1 - 44  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Histogram Relative Frequency Distribution of Battery Life 1 - 45  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Histogram Relative Frequency Distribution of Battery Life 1 - 46  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Histogram Estimating Frequency Distribution of Battery Life 1 - 47  Statistical Modeling, Scientific, Inspection, and Graphical Diagnostics Skewness of data A distribution is said to be symmetric if it can be folded along a vertical axis so that the two sides coincide. A distribution that lacks symmetry with respect to a vertical axis is said to be skewed. skewed to the right Symmetric skewed to the left 1 - 48