1 The Role of Statistics in Engineering ENM 500 Chapter 1 The adventure begins… A look ahead

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3 1-1 The Engineering Method and Statistical Thinking Figure 1.1 The engineering method

4 The Engineering or Scientific Method Figure 1-1 Describes the Scientific or Engineering Method. Several steps rely on statistical methods –Conduct experiments – how are efficient experiments designed? –Identify the important factors – how do we account for variability when we measure these factors? –Confirm the solution – how do we accept or reject a solution/hypothesis based on measurements? Variability complicates the task. Statistical methods help us understand and deal with variability.

5 Statistical techniques are useful for describing and understanding variability. By variability, we mean successive observations of a system or phenomenon do not produce exactly the same result. Statistics gives us a framework for describing this variability and for learning about potential sources of variability. 1-1 The Engineering Method and Statistical Thinking

6 Why is variability important to us? We want to predict results and control results with accuracy. Variability makes predictions and control more difficult and less accurate. If a particular part was required to be 1” ” and the actual standard deviation was 0.010”, almost one- third of the parts would be out of tolerance, even if their mean was exactly 1.000”! Would you rather work in a room that had a constant temperature of 70 o or one where the temperature alternated between 50 o and 90 o every 30 minutes?

7 Why Do We Study Probability & Statistics? Statistics – deals with the collection, presentation, analysis, and use of data to make decisions, solve problems, and design products & processes. –use statistics to draw inferences. Examples: quality, performance, or durability of a product, weather forecasts, utilization or loading of system. Probability – allows us to use information & data to make intelligent statements & forecasts about future events. –Probability helps quantify the risks associated with statistical inferences Prob & Stat are foundations for other coursework, e.g. reliability and quality courses, robust design, simulation, design of experiments, decision analysis, forecasting, time-series analysis, and operations research.

8 What do we want to know about our data? A measure of central tendency: Average or mean - A measure of variability: Sample variance – Sample Standard Deviation - We build models to explain this variability

9 An Example

10 Sample vs. Population Measures – Statistical Inference The sample mean ( ) estimates the population mean ( ) The sample variance ( ) estimates the population variance ( ) SAMPLE POPULATION MEAN: VARIANCE: The population can sometimes be conceptual and essentially have infinite size.

11 Sample vs. Population Measures We use sample measures ( ) to draw conclusions about the population measures ( ). The sample will be a (random) subset of the population The population may not yet exist, so the sample may be from a small set of prototypes (analytic) –There is an issue of stability – do the prototypes accurately reflect the prospective population?

12 Sample Data – May be obtained from: Observational Study – sample is drawn randomly from current process or system Designed experiment – deliberate changes are made to the controllable variables of a process or system. The system output is observed & inferences made about the effects of controlling the input. Retrospective Study – Historical observations. Were you fortunate enough that the needed variables were actually collected accurately!?

13 Concept of Models Common engineering/physical models: –F = ma –I = E/R –d = vt Mechanistic models: used when we understand the physical mechanism relating these variables. Empirical models: use our engineering & scientific knowledge of the phenomena, but are not built on first-principle understanding of the underlying mechanism. They are data driven. Let the data do the talking, right?

Mechanistic and Empirical Models A mechanistic model is built from our underlying knowledge of the basic physical mechanism that relates several variables. Example: Ohm’s Law Current = voltage/resistance I = E/R I = E/R + 

Mechanistic and Empirical Models An empirical model is built from our engineering and scientific knowledge of the phenomenon, but is not directly developed from our theoretical or first- principles understanding of the underlying mechanism.

Mechanistic and Empirical Models Example Suppose we are interested in the average molecular weight (M n ) of a polymer. Now we know that M n is related to the viscosity of the material (V), and it also depends on the amount of catalyst (C) and the temperature (T ) in the polymerization reactor when the material is manufactured. The relationship between M n and these variables is M n = f(V,C,T) say, where the form of the function f is unknown. where the  ’s are unknown parameters.

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Mechanistic and Empirical Models In general, this type of empirical model is called a regression model. The estimated regression line is given by

19 Figure 1-15 Three-dimensional plot of the wire and pull strength data.

20 Figure 1-16 Plot of the predicted values of pull strength from the empirical model.

21 Designing Engineering Experiments Experiments are often used to confirm theory or to evaluate various design options –Often, several factors may be important –Each factor may have more than one level of concern Full factorial design – considers all factors at all levels of interest –For K factors, each having two levels, a total of 2 K experiments are required –For K = 4, N = 16 –For K = 8, N = 256 Fractional factorial design – only a subset of factor combinations are actually tested

22 Design of Experiments (DOE) Assume you want to investigate the impact of three factors on the pull-off force of a connector: –Wall thickness (3/32” and 1/8”) –Cure times (1 hour and 24 hours) –Cure temperature (70 o F and 100 o F) We can now conduct an experiment to assess the impact of each of these variables (separately & interacting), each variable being assessed at two different levels Since other sources of variability may be present, we would do multiple experiments (replicate) at each design point.

23 Full Factorial Design Figure S1-1 The factorial experiment for the connector wall thickness problem.

24 Importance of Factor Interactions Figure S1-2 The two-factor interaction between cure time and cure temperature.

25 The Key Distinction The key difference between observational studies and experimental designs is this: –In a proper experiment you can eliminate confounding factors and isolate effects of interest. –In an observational study you take existing data. This may make it impossible to distinguish the effects of two factors that appear to explain observations equally well.

26 Time Series The correct analysis and interpretation of data collected over time is very important in assessing & controlling the performance of a system or process. –When is performance normal & when is it out of control? –What factors are driving a system out of control? –What corrections should be applied to regain control? –When has a change occurred – a fundamental shift in the process behavior?

Observing Processes Over Time Figure 1-11 Adjustments applied to random disturbances over control the process and increase the deviations from the target.

Observing Processes Over Time Figure 1-12 Process mean shift is detected at observation number 57, and one adjustment (a decrease of two units) reduces the deviations from target.

Observing Processes Over Time Figure 1-13 A control chart for the chemical process concentration data.

Probability and Probability Models Probability models help quantify the risks involved in statistical inference, that is, risks involved in decisions made every day. Probability provides the framework for the study and application of statistics.

31 Let’s Toss a Coin There are 1000 coins one of which contains two heads; the others are fair. A coin is selected at random and tossed 10 times. If heads appear on all ten tosses, what is the probability that the coin selected is the two-headed coin? P(two-headed is selected) =.001 P(toss 10 heads in a row – fair coin) = (1/2) 10 = 1/1024 .001 Therefore P(two-headed coin selected given 10 heads observed) .5

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