Contents. An Overview of Statistical Applications CHAPTER 1. Contents (ix) Preface... (vii)

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1 Contents (ix) Contents Preface... (vii) CHAPTER 1 An Overview of Statistical Applications 1.1 Introduction Probability Functions and Statistics Discrete versus Continuous Functions Distributions Describing Randomness Data Organization Common Statistical Estimators Applications of Normal Distribution The Standard Normal Distribution Characteristics of the Normal Distribution Function Confidence Bounds Determination of Sample Size Random Variables Summation The Central Limit Theorem The Binomial Distributions Bernoulli and the Binomial Distribution Asking People Questions Survey Results The Binomial and the Normal Distributions The Poisson Distribution Testing of Hypotheses Before-and-After Tests with Two Distinct Choices Before-and-After Tests with Generalized Alternative Hypothesis Other Useful Statistical Tests Summary... 1

2 (x) Contents CHAPTER Preliminaries.1 Introduction.... Basic Concepts Characteristics Attributes Variables Numeric Variables Categorical Variables Data Classification and Tabulation Tabulation of Data Frequency Distribution Simple Frequency Distribution Grouped Frequency Distribution Cumulative Frequency Table Less than Cumulative Frequency Table More than Cumulative Frequency Table Measures of Central Tendency Arithmetic Mean Simple Arithmetic Average Weighted Arithmetic Mean Merits of Arithmetic Mean Demerits of Arithmetic Mean Properties of Mean Statistical Applications to Transportation Engineering Median Merits of Median Demerits of Median Mode Merits of Mode Demerits of Mode... 54

3 Contents (xi).10 Geometric Mean Merits of Geometric Mean Demerits of Geometric Mean Harmonic Mean Merits of Harmonic Mean Demerits of Harmonic Mean Relation between A.M, G.M and H.M Partition Values (Quartiles, Deciles and Percentiles) Quartiles Deciles Percentiles Measures of Dispersion Characteristics of an Ideal Measure of Dispersion Types of Measures of Dispersion Range Coefficient of Range Merits of Range Demerits of Range Uses of Range Inter-Quartile Range Quartile Deviation Coefficient of Quartile Deviation Mean Deviation Coefficient of Mean Deviation Merits of Mean Deviation Demerits of Mean Deviation Standard Deviation Coefficient of Standard Deviation Merits of Standard Deviation Demerits of Standard Deviation... 83

4 (xii) Contents CHAPTER 3 Probability 3.1 Introduction Classical Probability Properties of Classical Probability Probability of Failure Relative Frequency Approach of Probability Symbolic Notation Axiomatic Theory of Probability Independent and Dependent Events Conditional Probability Multiplication Theorem on Probability Baye s Theorem (Statement) CHAPTER 4 Random Variables 4.1 Introduction Discrete Random Variable Probability Distribution for a Discrete Random Variable Probability Mass Function Distribution Function Additional Properties of Distribution Function Mean and Variance of a Discrete Distribution Continuous Random Variable Probability Density Function Cumulative Distribution Function Mean and Variance of a Continuous Random Variable Joint Distributions Joint Probability Function Joint Probability Distribution of Discrete Random Variables Marginal Probability Function of a Discrete Random Variables Joint Distributive Function of Discrete Random Variables... 13

5 Contents (xiii) 4.10 Conditional Probability Distribution Independent Random Variables Joint Probability Function of Continuous Random Variables Joint Probability Distribution Function of Continuous Random Variables Marginal Distribution Function Marginal Density Functions Conditional Probability Density Functions Mathematical Expectation and Moments Properties of Mathematical Expectation Variance Properties of Variance Covariance Moments Moments about an Arbitrary Number Moments about Origin Skewness and Kurtosis Moment Generating Function Properties of Moment Generating Function Discrete Probability Distributions Binomial Distribution Expected Frequencies and Filling of a Binomial Distribution Recurrence Relation Moments, Skewness and Kurtosis of the Binomial Distribution Moment Generating Function of a Binomial Distribution Characteristics of a Binomial Distribution Poisson Distribution Conditions under which Poisson Distribution is used Poisson Probability Function Poisson Frequency Distribution Moment of a Poisson Distribution Recurrence Relation... 01

6 (xiv) Contents Characteristics of Poisson Distribution Moment Generating Function of the Poisson Distribution Reproductive Property of the Poisson Distribution Discrete Uniform Distribution The Negative Binomial and Geometric Distribution Geometric Distribution Continuous Probability Distributions Uniform Distribution Moments of the Uniform Distribution Mean of Uniform Distribution Variance of Uniform Distribution Moment Generating Function of the Uniform Distribution Exponential and Negative Exponential Distribution Normal Distribution Standard Normal Variable Distribution Function (z) of Standard Normal Variate Area under Normal Curve Area under Standard Normal Curve Properties of Normal Curve Mean of Normal Distribution Variance of Normal Distribution Mode of Normal Distribution Median of the Normal Distribution Moment Generating Function of Normal Distribution with Respect to Origin Mean Deviation of Normal Distribution Fitting a Normal Distribution Linear Combination of Independent Normal Variables Fitting a Normal Distribution Normal Approximation to Binomial Distribution Characteristic Function Gamma Distribution Mean and Variance of Gamma Distribution Gamma Distribution of Second Kind... 58

7 Contents (xv) 4.31 Beta Distribution of First Kind Beta Distribution of Second Kind Weibull Distribution CHAPTER 5 Curve Fitting 5.1 Introduction The Method of Least Squares The Least-Squares Line Fitting a Parabola by the Method of Least Squares CHAPTER 6 Correlation and Regression 6.1 Introduction Correlation Types of Correlation Coefficient of Correlation Properties of Coefficient of Correlation Methods of Finding Coefficient of Correlation Scatter Diagram Direct Method Spearman s Rank Correlation Coefficient Calculation of r (Correlation Coefficient) (Karl Pearson s Formula) Regression Regression Equation Curve of Regression Types of Regression Regression Equations (Linear Fit) Linear Regression Equation of y on x Regression Equation of x and y Angle between Two Lines of Regression Coefficient of Determination... 91

8 (xvi) Contents 6.16 Coefficient non-determination Coefficient of Alienation Multi Linear Regression Uses of Regression Analysis CHAPTER 7 Sampling 7.1 Introduction Population Sample Sampling Random Sampling Simple Random Sampling Stratified Sampling Systematic Sampling Sample Size Determination Sampling Distribution CHAPTER 8 Hypothesis Testing 8.1 Introduction Hypothesis Hypothesis Testing Types of Hypothesis Null Hypothesis Alternative Hypothesis Computation of Test Statistic Level of Significance Critical Region One Tailed Test and Two Tailed Tests One Tailed Test Two-Tailed Test

9 Contents (xvii) 8.9 Errors Procedure for Hypothesis Testing Important Tests of Hypothesis Critical Values Test of Significance Large Samples Test of Significance for Single Mean Test of Significance for Difference of Means of Two Large Samples Test of Significance for the Difference of Standard Deviations of Two Large Samples Test of Significance for Single Proportion Testing of Significance for Difference of Proportions CHAPTER 9 Chi-Square Distribution 9.1 Introduction Contingency Table Calculation of Expected Frequencies Chi-Square-Distribution Characteristic Function of χ distribution Mean and Variance of χ (Chi-Square) Additive Property of Independent Chi-Square Variate Degrees of Freedom Conditions for Using 9.9 Uses of χ (Chi-Square) Test χ (Chi-Square) Test χ (Chi-Square) Test as a Test of Goodness of Fit Test for Independence of Attributes Homogeneity Chi-Square χ (Chi-Square) Distribution of Sample Variance Testing a Hypothesis about the Variance of Normally Distributed Population Decision Rule... 38

10 (xviii) Contents CHAPTER 10 Test of Significance Small Samples 10.1 Introduction Moments about Mean Properties of Probability Curve Assumptions for t-test Uses of t-distribution Interval Estimate of Population Mean Types of t-test Significant Values of t Test of Significance of a Single Mean Student s t-test for Difference of Means Paired t-test F-Distribution CHAPTER 11 ANOVA (Analysis of Variance) 11.1 Introduction Assumptions One Way ANOVA Working Rule CHAPTER 1 Analysis of Time Series 1.1 Introduction Purpose of Time Series Study Editing of Data Components of Time Series Mathematical Model for a Time Series Methods of Measuring Trend Free-Hand Method Semi-Average Method

11 Contents (xix) Moving Average Method Method of Least Square Non-Linear Trend Conversions of Trend Equations CHAPTER 13 Index Numbers 13.1 Introduction Definitions and Characteristics Definition Characteristics Uses Types of Index Numbers Problems in the Construction of Index Numbers Method of Constructing Index Numbers Tests for Consistency of Index Numbers Time Reversal Test Factor Reversal Test Circular Test Quantity Index Numbers Consumer Price Index Number Utility of Consumer Price Index Number Formulas for Constructing Consumer Price Index Chain Base Method Base Conversion Base Conversion Base Shifting Splicing Deflation Index

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