Engineering Investigation PDF

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Engineering Investigation...

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CIV3204 – Engineering Investigation Table of Contents 1. Basics ............................................................................................................................................................................................... 3 1.1 Definitions ................................................................................................................................................................................ 3 1.2 Statistics .................................................................................................................................................................................... 3 1.3 Unit Expectations ...................................................................................................................................................................... 4 2. Stats & Data .................................................................................................................................................................................... 5 2.1 Definitions ................................................................................................................................................................................ 5 2.2 Data .......................................................................................................................................................................................... 5 2.3 Sampling ................................................................................................................................................................................... 6 3. Exploratory Data Analysis ............................................................................................................................................................... 7 3.1 Motivation ................................................................................................................................................................................ 7 3.2 Organizing Numerical Data ....................................................................................................................................................... 7 3.3 Organizing Categorical Data ..................................................................................................................................................... 9 3.4 Organizing Bivariate Data ....................................................................................................................................................... 10 3.5 Principles of Graphical Excellence .......................................................................................................................................... 10 4. Descriptive Statistics ..................................................................................................................................................................... 11 4.1 Central Tendency vs. Variation ............................................................................................................................................... 11 4.2 Quartiles ................................................................................................................................................................................. 12 4.3 Measures of Variation ............................................................................................................................................................ 12 4.4 Shape of Distribution .............................................................................................................................................................. 13 5. Probability Theory & Discrete Probability Distributions ............................................................................................................... 14 5.1 Probability Theory .................................................................................................................................................................. 14 5.2 Mutually Exclusive Events ...................................................................................................................................................... 14 5.3 Types of Probabilities ............................................................................................................................................................. 15 5.4 Discrete Probability Distributions ........................................................................................................................................... 18 5.5 Binomial Distribution .............................................................................................................................................................. 19 5.5 Poisson Distribution ................................................................................................................................................................ 20 6. Continuous Probability Distributions ............................................................................................................................................ 21 6.1 Introduction ............................................................................................................................................................................ 21 6.2 Gaussian Distributions ............................................................................................................................................................ 21 6.3 Exponential Distribution ......................................................................................................................................................... 25 6.4 Log-Normal Distribution ......................................................................................................................................................... 26 6.5 Distribution Fitting .................................................................................................................................................................. 27 7. Sampling Distributions & Interval Estimates ................................................................................................................................. 28 7.1 Sampling Distributions ............................................................................................................................................................ 28 7.2 Interval Estimates ................................................................................................................................................................... 31 8. Hypothesis Formulations .............................................................................................................................................................. 34 8.1 Objectives ............................................................................................................................................................................... 34 8.2 Hypothesis Formulation ......................................................................................................................................................... 34 8.3 Motivation .............................................................................................................................................................................. 35 8.4 Hypothesis Test for Means ..................................................................................................................................................... 38 8.5 Potential Pitfalls & Ethical Considerations ............................................................................................................................. 40 9. Inferences for Means of Two Populations .................................................................................................................................... 41 9.1 Two-Sample Setup for Difference In Means ........................................................................................................................... 41 9.2 Difference between Two Means ............................................................................................................................................ 41 9.3 Hypothesis Tests for Variance ................................................................................................................................................ 44 9.4 Paired Difference test (Dependant Samples) ......................................................................................................................... 45 10. Analysis of Variance (ANOVA) ..................................................................................................................................................... 46 10.1 The ANOVA ........................................................................................................................................................................... 46 10.2 Post-Hoc Testing ................................................................................................................................................................... 49

11. Linear Regression ........................................................................................................................................................................ 50 11.1 Definitions ............................................................................................................................................................................ 50 11.2 Simple Linear Regression Model .......................................................................................................................................... 50 11.3 Simple Linear Regression Equation ...................................................................................................................................... 51 11.4 Measures of Variation .......................................................................................................................................................... 52 11.5 Meaning of Covariance ......................................................................................................................................................... 52 12. Analysis of Linear Regression ...................................................................................................................................................... 54 12.1 Introduction ..................................................................................................................................................................... 54 12.2 Residual Analysis .................................................................................................................................................................. 55 12.3 Meaning of Covariance ......................................................................................................................................................... 56 12.4 Inferences about the Slope ................................................................................................................................................... 57 12.5 t-Test for the Correlation Coefficient ................................................................................................................................... 60 12.6 Confidence interval for the mean of Y.................................................................................................................................. 60 12.7 Pitfalls of Regression Analysis ............................................................................................................................................... 61 13. Multivariate Linear Regression ................................................................................................................................................... 62 13.1 Introduction .......................................................................................................................................................................... 62 13.2 Multiple Linear Regression Equation Coefficients ................................................................................................................ 63 14. Analysis of Multivariate Linear Regression ................................................................................................................................. 65 14.1 Assumptions in Multiple Regression .................................................................................................................................... 65 14.2 Quadratic Regression............................................................................................................................................................ 68 15. Time Series Analysis .................................................................................................................................................................... 70 15.1 Introduction .......................................................................................................................................................................... 70 15.1 Autocovariance & Autocorrelation ....................................................................................................................................... 71 15.3 Fundamental Assumption ..................................................................................................................................................... 72 15.4 Definitions ............................................................................................................................................................................ 72 15.5 Application of Time Series Analysis ...................................................................................................................................... 73 15.6 Autoregressive (AR) Models ................................................................................................................................................. 74 15.7 Moving Average (MA) Models .............................................................................................................................................. 76 15.7 Autoregressive Moving Average ........................................................................................................................................... 78

CIV3204: Engineering Investigation

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1. Basics 1.1 De Definit finit finitions ions -

Statistics = collect/organise/analyse/interpret data System = no. of components logically (or physically) linked together for some purpose Process = set of activities operating a system that transforms inputs to outputs Data = Specific observations of measured numbers Information = processed & summarised data yielding facts/ideas Knowledge = Selected & organised information forming the basis for decisions

1.2 Sta Statisti tisti tistics cs ▪

Decision Making Process 1 = Get Data 2 = Find information based on data 3 = Apply theory to info to get knowledge 4 = Make decisions

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Importance of Statistics - Statistics are seen in climate data (ave. temp trends), sport (betting odds), traffic data (road deaths - Statistics allows us to get data, process it & make a decision. - Interpreting this data could involve skewness, sampling error, conditional probability etc.  we must take all this into consideration before arriving at a conclusion

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Risk & Uncertainty - Each scenario (decisions & conclusions drawn from statistics) typifies some level of risk & uncertainty - As engineers, we must quantify & define an acceptable level of risk, to make informed decisions & recommendations (critical for assessing liability)

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Statistics & Probability - Statistical concepts provide a means to make informed decisions in the presence of risk & uncertainty - Statistics are rooted in probability theory, which is the process of quantifying uncertainty, randomness, or fluctuation involved in a stochastic process - Why is an understanding of statistics important for Civil Engineers? ➢ Mission of a Civil Engineer: ”Design, build, operate &/or improve the physical system & products.” ➢ Independent assessment of what makes things ”tick” ➢ Evaluate new processes ➢ Forecast/predict effects of system changes ➢ Knowing when to ”call the plumber”  an engineer must use his knowledge to critically examine the work of others i.e “understand how quantitative methods are used to assist in the investigation and analysis of engineering problems”

CIV3204: Engineering Investigation

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2. Stats & Data 2.1 De Definit finit finitions ions -

Population = entire collection of things in consideration Sample = portion of population to analyse Parameter = measure of population (computed to describe a characteristic of population) Statistic = measure of sample

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Sample unit = Part of population we investigate Observation = information from a unit Variable = information about observation (Single observation) Variate = linear combination of variables (Combining variables)

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2.2 Dat Dataa ▪

Sources of Data Primary Data = Through data collection ➢ Direct observation (count) ➢ Experimentation (get results and understand significance) ➢ Survey (questionnaire) Secondary Data = Data compilation (use/search for data that already exists i.e not collected by you) ➢ Printed data (published historical data) ➢ Electronic data (searching internet)

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Types of Data - Categorical data = don’t put a number to it i.e yes/no, colour - Numerical data = numbers i.e height/weight ➢ Discrete random variables = counting process (no floating point i.e no. of kids you have) ➢ Continuous random variables = measuring process (has floating point i.e height/weight) no result is the same - Quantitative = data expressed numerically (i.e age, speed) - Qualitative = data you cant put a number on (i.e type of car) Can convert qualitative to quantitative data to make analysis easier

CIV3204: Engineering Investigation

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2.3 Sam Sampling pling ▪

Sampling Issues Why sample from an entire population? ➢ Impracticality of recording data for the entire population i.e polls before elections. ➢ Rare events ➢ Futility of testing ➢ Accuracy of testing: Becomes more difficult to achieve if too many objects need to be tested. ➢ Randomisation is critical

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Types of Survey Errors - Coverage error = part of population excluded from sample - Non response error = part of sample may not respond - Sampling error = Conclusions differ from sample to sample - Measurement error = wrong measurements lead to wrong conclusions

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Sampling error & Sampling Bias Sampling error = error when sample is used to estimate population parameter - Function of sample size & variability - Can only be minimized Sample bias = error by bad sampling - Error associated w/ instrument drift, incorrect sampling frame, non-response & choice of q’s - Can be eliminated through optimization/planning

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Statistical Methods Descriptive statistic = collecting & describing data 1. Collect data (i.e survey) 2. Present data (i.e tables/graphs) 3. Characterise data (i.e through sample mean) Inferential statistics = draw conclusion/make decision concerning a population based on sample results Estimation: I.e estimate population mean weight using sample mean weight Hypothesis testing: I.e test the claim that the population mean weight is 70kg

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Frequency Polygon Similar to a histogram, but can be used to compare sets of data (i.e male v. female) - Only class midpoint values listen on x-axis

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Cumulative Frequency

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The Ogive Reports the no. of values in the data set that are less than or equal to the given value - Frequency or percentage distribution - Class boundaries should be indic...