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STATISTICAL ANALYSES IM342 By Prof. Dr. Ahmed Farouk Abdul Moneim 1 INTRODUCTION STATISTICS is concerned with: 1. Collection of Data from Observations, Experiments,…etc Data Populations are the complete set of data about Systems attributes. Data Samples are Subsets of the Population. 2. Presenting the Collected data in the form of Histograms, Box Plots, Time series,…….etc 3. Processing these Data in order to render them amenable to practical applications in: • Engineering • Management and Decision-Making. 3 STATISTICAL ANALYSES are concerned with: 1. Inferences about POPULATIONS from data SAMPLES collected from Observations and Experiments (Induction approach of Science). المنهاج اإلســتقرائى 2. TESTING HYPOTHESES About PARAMETRS of Gven Probability Distributions of Observed Variables (Mean Life Time of Products), (Comparisons among different Products) 3. DESIGNING EXPERIMENTS and Devicing Methods for PROCESSING Data Collected from these Experiments STATISTICAL ESTIMATIONS 5 THE NEEDS FOR SAMPLING Economic and Social Indicators Material Physical and Chemical Properties Reliability and Safety Parameters Ergonomics and Human Factors 6 Economic Indicators • Per capita income as a measure of prosperity • Inflation Rate or Price Index Increase Rate • Unemployment Ratio Social Indicators • Illiteracy Rate • Age Pattern 7 Material Physical and Chemical Properties • Ultimate Tensile Strength and Yield Point Of Metals • Magnetic Permeability of Metals • Percentage of Alloying Elements • Resistance to Corrosion of Different Steels 8 Reliability and Safety Parameters Mean Time Between Failures MTBF Systems Availability Maintainability of Equipment Design Factor of Safety 9 Ergonomics and Human Factors Anthropometric Measures Time and Motion Studies Human Reliability 10 REMEMBER! Random Variables Type Discrete Continuous Range x1, x2 ,…, xn Continuous from x1 to xn Probability Distribution Mass Function f(x) Probability Density Function f(x) MEAN = Expected Value =E(X) Variance = Var(x) E( X ) n x i 1 Var ( X ) n x i 1 i f ( xi ) f ( xi ) 2 i xf ( x )dx x1 Var ( X ) xn x 2 f ( x ) dx x1 - E ( X ) - E ( X ) 2 2 E ( X 2 ) E ( X ) 2 Examples E( X ) xn Uniform, Bernoulli, Binomial, Geometric, Negative Binomial, Poisson E ( X 2 ) E ( X ) 2 Uniform, Normal, Exponential, Erlang, Weibull,… PROBABILITY DISTRIBUTIONS N DISCRETE E ( X ) (1 / N ) xi var ( X ) (1 / N ) xi2 E ( X ) f ( x) 1 / n UNIFORM BINOMIAL 2 i 1 f ( x ) C xN p x (1 p ) N x E ( X ) Np var( X ) Np(1 p) f ( x ) (1 p) x 1 p GEOMETRIC E ( X ) 1 / p var( X ) (1 p) / p 2 NEGATIVE BINOMIAL f ( x ) C x r (1 p) x r p r E ( X ) r / p var( X ) r(1 p) / p 2 r 1 POISSON f ( x) x x! e E ( X ) var ( X ) CONTINUOUS UNIFORM f ( x ) NORMAL 1 ba f (x) = EXPONENTIAL In the range from a to b 1 e s 2p æ x-m ö2 -ç ÷ è s ø f ( x ) e x E ( X ) (a b) / 2 var( X ) (b a ) 2 / 12 With mean µ and variance σ2 With mean 1/λ and variance 1/λ2 POINT ESTIMATORS Population Parameters Sample Estimators Mean X Variance 2 S2 and X and S2 Are CONSTANTS BUT Difficult and 2 Even Mostly Impossible to determine Because of Time and Cost Barriers Are RANDOM VARIABLES Why? They are obtained Easily from SAMPLE DATA 14 X Is a Random REQUIRED to FIND: • Its MEAN, VARIANCE and Variable • Its PROBABILITY DISTRIBUTION FUNCTION To be able to do that, we should consider the following Important Theorem CENTRAL LIMIT THEOREM Given X 1 , X 2 , X 3 ,......, X n N Random Variables with Different Arbitrary probability Distribution Functions Their Sum Y X 1 X 2 X 3 ........ X n n, Y is a Random Variable Distributed According to NORMAL DISTRIBUTION For large values of 15 The Proof of the CENTRAL LIMIT theorem is on an Excel sheet Find Mean and Variance of X 1 n 1 X X i X 1 X 2 X 3 ... X n n i 1 n A General Rule If Y = a X, then E(Y)= a E(X) Var(Y) = a2 Var(X) 1 E X 1 E X 2 E X 3 ... E X n n 1 n E X ... n n E X The n points of Sample are Taken from the same population Therefore, EX 1 EX 2 EX 3 EX n Population Mean 1 Var X 1 Var X 2 Var X 3 ... Var X n 2 n 1 2 n 2 2 Mean of X 2 2 2 Var X 2 ... 2 n n n Variance of X 2 Var X n Standard Deviation of X n 16 The Sample Mean Is used as an X ESTIMATOR of the population Mean Similarly The Sample Variance could be used as Generally, for each ESTIMATOR of the population Variance POPULATION PARAMETER There is A corresponding OR S2 SAMPLE ESTIMATOR SAMPLE STATISTIC 2 17 The selection of the most appropriate SAMPLE STATISTIC as an ESTIMATOR for a POPULATION PARAMETER Is governed by the 1. following Properties BIAS evaluated as follows: Bias E 2. STANDARD ERROR SE evaluated as follows: SE Var 3. MEAN SQUARE ERROR MSE MSE E 2 evaluated as follows: Var Bias 2 18 Example Evaluate the Bias and SE and MSE of the following ESTIMATORS: X , S 2 and S For the population parameters: , 2 , Bias X E X 0 Then X is an Unbiased Estimator of µ The Standard Error SE Var X n Find the Mean Square Error of Estimator MSE Var X Bias 2 2 n 0 X 2 n 19 SOLVED EXAMPLES Example 1 Rubber tires are known to have 20000 km as mean life time and 5000 km as standard deviation. A Sample of 16 rubber tires are selected randomly, what is the probability that the SAMPLE MEAN will be in the interval from 17000 to 22000. Find the standard error in using the sample mean as an ESTIMATOR for the population mean. P(17000 < X < 22000) X is normal variable with mean m = 20000 and standard deviation S S= s n = 5000 = 1250 16 17000 - 20000 X - m 22000 - 20000 P(17000 < X < 22000) = P( < < ) 1250 S 1250 P(-2.4 < Z < 1.6) = F(1.6) - F(-2.4) = 0.945 -.0082 = 0.937 SE SE Standard Error n 5000 16 1250 Example 2 Suppose that X is the number of observed “successes” in a sample of n observations where p is the probability of success on each observation. a) Show that p̂ = X/n is an unbiased estimator of p b) Show that the standard error of p is p(1 p) / n X is a Random Variable distributed in BINOMIAL distribution X is a Binomial Variable then, E ( X ) np Var ( X ) np (1 p ) X , then E ( pˆ ) E ( X ) / n np / n p n Bias ( pˆ ) E ( pˆ ) p p p 0 Estimator pˆ of p : pˆ SE ( pˆ ) Var ( pˆ ) Var ( X / n) (1 / n 2 )Var ( X ) SE ( pˆ ) (1 / n 2 ) * np (1 p ) Estimator p̂ p (1 p ) / n is an Unbiased Estimator of p Example 3 Let X be a random variable with mean μ and variance б2 Given two independent random samples of sizes n1 and n2, with sample means 2 2 X 1 , X 2 E( X 1 ) E( X 2 ) Var ( X 1 ) Var ( X 2 ) n1 n2 Show that X a X 1 (1 a ) X 2 0 a 1 is an unbiased estimator of µ If X 1 , X 2 are independent , Find the value of the standard error of a that minimizes X Var ( X ) a 2Var ( X 1 ) (1 a ) 2 Var ( X 2 ) a E ( X ) aE ( X 1 ) (1 a ) E ( X 1 ) a (1 a ) Bias ( X ) E ( X ) 0 2 2 n1 (1 a ) 2 2 n2 d 2 2 Var ( X ) 2a 2(1 a ) 0 da n1 n2 a 1 a 0 n1 n2 a n1 n1 n2 a( 1 1 1 ) n1 n2 n2