Statistics

• Provide an introduction to the probability theory

• Demonstrate the concepts probability and conditional probability
• Apply tree Diagrams for the problems that involve conditional probability and Bayes’ theorem
• Define random variables
• Apply the knowledge of basic probability distributions in real world issues

• Introduction to Probability: Permutations, combinations, Venn diagram, events, sample space, equally likely events, mutually exclusive events, axioms of probability, laws of probability, conditional probability, independence.
• Bayes’ Theorem and Applications: Partition, total probability theorem, Bayes’ theorem, tree diagram.
• Random Variable: Discrete and continuous random variables, probability mass function, probability density function, expectation, moments, mean and variance, moment generating functions, probability generating functions.
• Probability Distributions: Discrete uniform, Bernoulli, binomial, Poisson, geometric, uniform, exponential and normal distributions, applications of the normal distribution,

sampling distribution of the sample means.

• Joint Distributions: Joint distributions, marginal distribution, conditional distributions, conditional expectation and variance.

• Lecture demonstration by Lecturer and Tutorial discussions by Instructors

• In-course assessments            30%
• End of course Examination   70%

• Fundamentals of Probability with Stochastic Processes, Saeed Ghahramani, 3^rd Edition, 2005.
• Probability and Statistics for Engineers and Scientists, Walpole R.E., Myers R.H., Myers S.L., Ye K.E., 9^th Edition, 2010.
• Schaum’s outline of Statistics, Murray R. Spiegel, 5^th Edition, 2014.

• Provide fundamental knowledge in basic statistical concepts

• Explain basic concepts and principles of statistics
•  Examine data sets using summary statistics and graphical methods
• Apply simple random sampling method in real-world issues

• Fundamentals of Statistics: Types of data, population and sample, descriptive and inferential statistics, parameter and statistic, data collection methods.
• Descriptive Statistics: Frequency distribution, pie chart, bar chart, histogram, ogive, frequency polygon and curve, measures of central tendency and dispersion, measures of relative standings, skewness and kurtosis, box-plot, five number summary statistics, outliers.
• Introduction to Sampling Methods: Introduction to sampling; sampling unit, sampling frame, sampling and non-sampling errors, probability and non-probability sampling. simple random sampling; estimation of mean, total and proportions and its variance in samples from finite population, calculation of sample size.

•  Lecture demonstration by Lecturer, Tutorial discussions and laboratory practical by Instructors

• In-course assessments         30%
• End of course Examination  70%

• Introduction to the Practice of Statistics, Moore, McCabe, Craig, 6^th Edition, 2009.
• Exploratory Data Analysis in Business and Economics, Thomas Cleff, 2014.
• Statistical Methods, Rudolf J. Freund, William J. Wilson, 2^nd Edition, 2003.

• Provide fundamental knowledge in Inferential Statistics

• Demonstrate the concept of Inferential Statistics
• Develop the knowledge in different sampling distributions
• Outline the different methods of parameter estimation in Statistics and interpret confidence intervals
• Explain the principles of hypothesis testing with applications
• Determine alternative statistical methods when normality assumption is not met

• Introduction: Define: population; sample; parameter; and statistic, distinguish between descriptive statistics and inferential statistics.
• Sampling Distributions: Distributions of sample means, sample variances and sample proportions, unbiasedness, normal distribution, central limit theorem, theory of Student- t, , and F distributions.
• Point and Interval Estimation: Method of moments, maximum likelihood estimation, confidence intervals for one-sample, two-sample population characteristics, sample size calculation for parameter estimation, interpretation of confidence intervals.
• Testing Hypotheses: Steps in hypothesis testing, level of significance, Type-I and Type – II errors, p-value, power of test, Z-test, t –test,test, and F-test, goodness of fit test, sample size calculation for hypothesis testing.
• Non-parametric Tests: Sign test, Wilcoxon Signed-Rank test, Wilcoxon Rank-Sum test (Mann-Whitney U test), contingency tables.

• Lecture demonstration by Lecturer, Tutorial discussions and, Laboratory practical by Instructors

• In-course assessments            30%
• End of course Examination     70%

• Schaum’s outline of Statistics, Murray R. Spiegel, 5^th Edition, 2014.
• Probability and Statistics for Engineers and Scientists, Walpole R.E., Myers R.H., Myers S.L., Ye K.E., 9^th Edition, 2010.
• Applied Statistical Inference with Minitab, Sally A. Lesik, 2009.

• Provide knowledge in the application of statistics

• Describe correlation between the two variables and develop simple linear regression models
• Identify components of time series and apply basic time series models for forecasting
• Construct index numbers for real world issues

• Correlation and Regression: Correlation, simple linear regression, least square estimation, interpretation of regression parameters.
• Time Series Analysis: Construction of time series plots and interpretation, components of time series, decomposition of time series components, additive and multiplicative models, moving average, exponential smoothing, forecasting.
• Index Numbers: Simple and weighted averages of price relative indices, construction of Paache, Laspeyres and Fisher indices, consumer price index, applications of index numbers.

•  Lecture demonstration by lecturer, Tutorial discussions and Laboratory practical by Instructors.

•  In-course assessments             30%
• End of course Examination    70%

• Applied Regression Analysis: A Research Tool, Rawlings, J. O., Wadsworth. 1988.
•  Linear regression analysis: Theory and computing, Yan, X., Su, X., & World Scientific (Firm). Singapore: World Scientific Pub. Co., 2009.
• Business Statistics: For Contemporary Decision Making Ken Black, 9th Edition: For Contemporary Decision Making: Wiley Global Education, 2009.

Hourly Breakdown

Provide a sound knowledge in general theory of statistical distributions and its applications

•  Recall fundamental knowledge of probability concepts
• Apply theory of various discrete and continuous distributions in real world issues
• Determine moments of random variables using the knowledge of generating functions and characteristic functions
• Derive probability distributions of function of random variables using transformation technique

Probability Distributions: Discrete Distributions: Uniform, Bernoulli and Binomial, Poisson, Poisson approximation to Binomial, Geometric, Negative binomial, Poisson approximation to Negative binomial, Hypergeometric, Binomial approximation to Hypergeometric, Multinomial distribution.

Continuous Distributions: Uniform, Exponential,  Normal, Gamma ,  Beta , Chi-Square, Weibull, Lognormal, Student-t, F distribution, Cauchy.

Functions of Random variables: Probability generating function, Moment generating function, Cumulant generating function,  Characteristic function , Convolution, Distributions of functions of random variables

Joint Distributions: Joint Distribution, Marginal Distribution, Conditional Distribution, Conditional Expectation and Variance, Independence, Correlation, Bivariate Normal distribution, Transformations of random variables, Order statistics.

• In-course assessments             30%
• End of course Examination     70%

• R.E.Walpole, R.H.Myers, S.L.Myers, K.E.Ye, “Probability and Statistics for Engineers and Scientists”, 9^th Edition, 2010.
• John E. Freund, “Mathematical Statistics with applications”, 8^th edition,2014.
• R.V. Hogg &A.T.Craig, “Introduction to Mathematical Statistics”, 4^thedition, 1978.

Hourly Breakdown

• Introduce the concept of methods of sampling
• Gain knowledge in ratio and regression estimators

• Illustrate the properties of stratification, stratified random sampling, proportional allocation and optimum allocation
• Apply the stratified random sampling in real world data
• Discuss the ratio and regression estimators
• Utilize the ratio and regression methods for estimating population parameters
• Discuss the concept of systematic sampling and cluster sampling
• Apply the systematic sampling and cluster sampling in real world data
• Evaluate the efficiency of estimators

Stratified Sampling:  stratification, advantages and disadvantages, stratified random sampling with associated mathematical background, estimation of mean, total and proportions and its variance in samples from finite population, confidence interval of estimators, models for cost function, allocation of sample size: proportional allocation, optimum allocation, Neyman allocation

Ratio and Regression methods: estimation of population ratio in simple random sampling and stratified random sampling, ratio and regression estimators of population mean and total in simple random sampling and stratified random sampling, variance, bias and mean square error of estimators, confidence interval of estimators, efficiency of estimators

Systematic sampling:  design of systematic sampling, estimation of mean, total and its variance in sample from finite population, inter class correlation coefficient

Cluster sampling: Introduction of cluster sampling, estimation of mean, total and its variance in sample from finite population

• Lecture demonstration and tutorial discussions

•  In-course assessments         30%
• End of course Examination  70%

• William G. Cochran,  Sampling Techniques, Third Edition, 2008.
• Sharon L. Lohr, Sampling: Design and Analysis, Second Edition, 2010.
•  R. Lyman Ott, Elements Survey Sampling. Sixth Edition,2006.
• S.R.S Rao, Sampling methodologies with Application, 2000.

Hourly Breakdown

• Analyze the experimental data
• Interpret the results of a statistical experiment
• Explain the difference between the experimental designs
•  Examine the suitability of the models for different experimental situations

Factorial Experiments: Two-way classification, Interaction, Diagrammatic explanation of interaction, Three-way classification, fixed, random and mixed models, Completely Randomized Design(CRD), Randomized Complete Block Design(RCBD), Latin Square Design(LSD), Nested Design, Nested and Crossed Design, Split Plot Design.

• In-course assessments             30%
• End of course Examination     70%

• Douglas C. Montgomery, Design and Analysis of Experiments, Wiley WSeries, 2012.
• H.R. Lindman, Analysis of Variance in Experimental Design, Springer Series, 1992.

Hourly Breakdown

• Introduce the fundamental concepts of statistical inference
• Acquire and apply the Bayesian inferential procedure in rigorous way
• Enable to solve the statistical inference problems

• Determine the sufficiency and minimal sufficiency
• Apply the factorization criterion to find the sufficient statistic
• Determine whether a distribution belongs to an exponential family
• Derive the methods of moment estimator and the maximum likelihood estimator
• Apply the ideas of bias, unbiased and minimum variance unbiased estimators
• Prove the Rao- Blackwell theorem and Cramer-Rao inequality
• Evaluate Cramer-Rao lower bounds
• Utilize the estimation criteria to select the best estimators
• Determine the asymptotic distributions of given estimators
• Recall the simple and composite hypotheses and confidence intervals
• Apply Neyman-Pearson lemma to find the most powerful and uniformly most powerful tests
• Utilize likelihood ratio tests and maximum likelihood ratio tests to determine critical regions
• Formulate the Bayesian analysis for a range of standard statistical problems
• Distinguish classical and Bayesian inferential paradigms

• Sufficiency principle: Sufficient statistics and factorization criterion, minimal sufficient statistics, exponential families of distributions, complete sufficient statistics.

• Point estimation: Methods of estimation;method of moment, maximum likelihood. Estimation criteria; bias, mean squared error, unbiasedness, relative efficiency, minimum variance unbiased estimators, Rao- Blackwell theorem, Cramer-Rao lower bound, efficiency, consistency.Asymptotic behaviour of MLEs.

• Interval estimation:Methods of estimation; inverting test statistics, pivots and approximate maximum likelihood.

• Hypothesis testing: –Simple and composite hypotheses, types of error, power, most powerful tests, uniformly most powerful tests, Neyman-Pearson lemma, likelihood ratio tests, maximumlikelihood ratio tests, sequential analysis, sequential likelihood ratio tests.
• Introduction to Bayesian theory: Prior and posterior distributions; Bayesian estimators, tests and intervals for parameters.

• Teaching methods consist of lectures and tutorial exercises.

• In-course Assessments          30%
• End-of-course Examination 70%

•  G. Casella, and R.L.Berger, Statistical Inference. 2nd ed. Belmont, CA: Duxbury Press, 2001.
•  R.V. Hoggand E.A.Tanis, Probability and Statistical Inference, 5th ed. Upper Saddle River, NJ: Prentice Hall, 1997.
•  R.V. Hogg, J.W.McKean, and A.T.Craig, Introduction to Mathematical Statistics, 6th ed. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.
•  D.D.Wackerly, W. Mendenhall, and R.L.Scheaffer, Mathematical Statistics with Applications, 5th ed. Belmont, CA: Duxbury Press, 1996.

Provide knowledge and techniques in fitting regression models to real world data

• Compare deterministic and stochastic relationships
• Distinguish  linear and nonlinear models
• Estimate the parameters of  linear regression models
•  Construct Analysis of Variance table to make inference about linear regression model
•  Apply diagnostic checks to identify possible violations of the model assumptions
• Choose the best fitting model for prediction

• Introduction: Response variable, Explanatory variable, deterministic relationship, stochastic/probabilistic relationship, scatter plot, linear and non-linear relationship, correlation coefficient
• Simple Linear Regression: Simple linear regression model, Assumptions of Simple linear regression, Linear and non-linear models, Estimation of parameters: Least squares and Maximum likelihood estimation, Properties of least square estimators, Statistical Inference on regression coefficients, Diagnostic checking of simple linear regression model and possible remedies, measure of goodness of fit, prediction, Analysis of Variance approach, Lack of fit, simple linear regression model using matrix approach, Polynomial regression models.
• Multiple Linear Regression: Multiple linear regression model  and its parameter estimation, Assumptions of multiple regression model, Matrix approach of multiple linear regression , Analysis of variance approach, Use of Dummy variables, Choice of variables,  Transformation of variables, Gauss Markov Theorem,  Sequential and partial regression sum of squares; Regression residuals diagnostics and model selection procedures, Prediction, Multicollinearity and its impacts .

•  In-course assessments              30%
• End of course Examination      70%

• Norman R. Draper, Harry Smith., “Applied Regression Analysis”, 3^rd Edition, Willey, 1998.
• Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining .,“Introduction to Linear Regression Analysis”, 5^th Edition, Willey, 2013.
• C.R. Rao, H. Toutenburg, Shalabh, and C. Heumann.,” Linear Models and Generalizations – Least Squares and Alternatives”, Springer, 2008.

•  Introduce basic concepts and theory of stochastic processes
• Familiarize with standard stochastic processes and their properties
• Acquaint with stochastic modelling and applications

• Explain the basic characteristics of stochastic processes
• Evaluate the important properties of random variables
•  Apply relevant stochastic models for randomly varying dynamic systems
•  Determine the waiting time and confidence interval for mean of Poisson processes
• Test for the Markov property
• Apply Chapman-Kolmogorov equation to determine the transition probabilities
• Find the stationary distribution of a Markov chain
• Classify the states of a Markov chain
• Evaluate the probability distribution of a random walk

• Introduction – Basic properties and examples of stochastic processes, Stationary process, Independent increments, Expectation and Covariance functions.
• Standard Stochastic Processes – The Bernoulli, Normal and the Wiener Processes, Counting Processes; Poisson process, Non-homogeneous, Generalized and Compound Poisson processes. Inter-arrival times and waiting times distributions. Filtered Poisson processes, Renewal processes.
• Markov Processes – Markov property, Markov chains, Transition probability matrices, Chapman-Kolmogorov equation, Classification of states, Decomposition of Markov chains, Stationary distribution, Limiting distribution.
• Random walk – Unrestricted random walk, Symmetric random walk.

• Lectures, Tutorials, Handouts, Problem solving, e-resources

• In-course Assessments              30%
• End-of-course Examination     70%

• E. Parzen, “Stochastic Processes”, SIAM Edition; Society for Industrial and Applied Mathematics Philadelphia, 1999.
•  Sheldon M. Ross, “Introduction to Probability Models”, 10^th edition, Academic Press Elsevier, 2013.
• P. W. Jones & P. Smith, “Stochastic Processes An Introduction”, 1^st  edition, Oxford University Press Inc, New York, 2001.

• Introduce the fundamentals of statistical quality control
• Provide the quality control methods and tools

• Explain key concepts in statistical process control
• Construct control charts to improve the process quality
• Distinguish  variable charts  and attribute charts
• Design appropriate acceptance sampling plans

• Introduction to the Concept of Quality Control: Fundamental concepts of quality and quality improvement
• Methods of Statistical Process  and Product Control: Process control, Product control, Causes of quality variation, Control charts: basic principles, choices of control limits, sample size and sampling frequency, analysis of pattern on control charts.
• Control Charts for Variables: Chart (Mean Chart), -chart (Range chart), Estimation of population mean and population standard deviation, Relationship between population standard deviation and range, Lack of control, Tests for lack of control, Interpretation of  and  charts.
•  Control Charts for Attributes: Control Chart for Nonconforming items; -chart and  – chart, Control Chart of Nonconformities; -chart and -chart
• Acceptance Sampling For Attributes: Operating characteristic curve, Type A and Type B OC-curve, Average Outgoing Quality Limit (AOQL), Average Total Inspection, Acceptable Quality Level (AQL), Lot Tolerance Fraction Defective (LTFD), Producer’s Risk and Consumer’s Risk, Single sampling plan, Double sampling plan, Average Sample Number (ASN).

• In-course assessments             30%
• End of course Examination     70%

• D.C. Montogomery, “Introduction to Statistical Quality Control”, 6^th edition, John Wiley & Sons, 1993.
• E. L. Grant, S. Richard & S. Leavenworth, “Statistical Quality Control”, 7^th edition, 1996.

• Provide profound knowledge in statistical modeling
• Introduce hypothesis testing procedures for non-normal data

• Recall the standard probability distributions
• Identify appropriate probability distribution for a given real life problem
• Examine the suitability of the model for a given data set
• Apply appropriate non-parametric statistical test for non-normal data

• Statistical Modelling: Standard distributions; Binomial, Poisson, Geometric, Hypergeometric, Negative Binomial, Multinomial, Exponential, Gamma, Weibull, Normal, Chi-square, Beta, Pareto, and their use in modelling, Fitting parametric models, Assessing Goodness of fit
• Non-parametric tests:Tests for normality, Contingency tables, single sample and paired samples non-parametric tests,  Run test,  Kruskal-Wallis test, Rank correlation

• Lecture demonstration, Tutorial discussions and laboratory practical

• In-course Assessments               30%
• End-of-course Examination       70%

• Lee. J. Bain and Max Engelhardt, “Introduction to Probability and Mathematical Statistics”, Kent Publishing Company, 2014.
• R. E. Walpole, R. H. Myers, S. L. Myers and K.E. Ye, “Probability and Statistics for Engineers and Scientists”, Prentice Hall, 2014.

• Analyze data sets using statistical software
• Describe data types and data structures
• Apply build-in functions to import and manipulate data sets
• Calculate summary statistics for given data sets
• Utilize build-in functions for probability distributions and simulation
• Make use of statistical software to write simple functions for given statistical problems

• Introduction to the software: History, user community, online helps and resources, saving datasets.
• Data types and data structures: Numerical and text values, vectors, factors, matrices and arrays, lists and data frame.
• Importation and manipulation: Read text, Excel, SPSS and Minitab datasets into software and build-in datasets, operations, manipulation and extraction.
• Descriptive statistics: Tables, summary statistics, graphical representation.
• Probability distributions and simulation: Build-in functions for probability distributions, simulation, plotting probability distributions.
• Inferential statistics: Confidence intervals and hypothesis tests, simple and multiple linear regressions, analysis of variance.
• Simple function: Writing simple functions to automate regular statistical tasks.

• In-course practical assessments             30%
• End of course practical Examination     70%

• P. L. Micheaux, R. Drouilhet, & B. Liquet, B, “The R software: Fundamentals of programming and statistical analysis”, Springer, 2013.
• J. Braun & D. J. Murdoch, “A first course in statistical programming with R”. Second edition, Cambridge University Press, 2007.