Introduction 509

Introduction 509 - Introduction to Engineering Statistics...

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Unformatted text preview: Introduction to Engineering Statistics We must draw conclusions from partial information. What What time should I leave to make it on time to class? How How many hours do we guarantee a light bulb will operate? Will Will the pen caps fit the pen barrels? Is Is the grocery bag strong enough to hold a 20 pound bag of potatoes? Where Where should we target a filing machine so that no more than 1% of “16 oz” bottles of Pepsi have less than 16 oz? We Draw Conclusions about Populations Populations Examples: Examples: – All possible driving times from home to class. – Lives in hours of a certain type of light bulb manufactured by a certain company. – Outside diameters of all pen barrels produced and inside diameters of all pen caps produced. – Tensile strength of paper used in the manufacture of grocery bags. – Fill on all bottles of Pepsi coming off a filling machine. 1 Population – a Definition A population is the set of all possible population observations of interest. A population has two parts to it: population – The individual being measured. – The measurement being taken. Example Example – Life in hours of every light bulb manufactured. – Inside diameter in mm of all pen caps produced. Our Conclusions are Based on Samples Examples: Examples: – Random sample of 10 possible driving times from home to class. – Life in hours of 50 randomly chosen light bulbs manufactured by a certain company. – Outside diameters of 25 randomly chosen pen barrels and 25 randomly chosen pen caps. – Tensile strength of 20 random samples of paper used in the manufacture of grocery bags. – Fill in 100 randomly chosen bottles of Pepsi. Sample – a Definition A sample is a subset of the population of sample interest. It It also has two parts: – The individual being measured. – The measurement being taken. Example Example – Life in hours of 50 randomly chosen light bulbs. bulbs. – Inside diameter in mm of 25 randomly chosen pen caps. 2 Why Use Sampling? Time Time Money Money Accuracy Accuracy Destructive Destructive testing Simple Random Sample A sample of size n is a simple random sample sample if every possible sample of size n taken from the population has the same chance of being selected. Choosing Choosing randomly means we can use the laws of probability to draw our conclusions. We We can use the laws of probability to know the chance we are wrong. Other Sampling Schemes Stratified Stratified random sample – Divide population into several non-overlapping nonpopulations and then take a simple random sample from each subpopulations. – Stratification is an engineering decision based on sources of variation. Systematic Systematic random sample – We sample every mth item, starting the process with a randomly selected item for the first m. 3 An engineer applies basic scientific principles to real world problems. Develop Develop models to explain real world phenomena. Collect Collect data to test the models. The The collected data is a sample (partial information). – Does the collected data support the model? – What is the chance we are wrong? Statistical Thinking & Engineering All All work occurs in a system of interconnected processes. Variation Variation exists in all processes. Understanding Understanding and reducing variation are the keys to success. It takes engineering and statistics to improve a process. Process Process improvement generally comes from simplifying a process and removing unwanted sources of variation. “Six “Six Sigma” – Define the problem. – Measure important characteristics with validated measurement systems. – Analyze potential sources of variation. – Improve process through experimentation and data analysis. – Control the process to maintain improvements. 4 Deterministic Models Deterministic Deterministic models make no attempt to explain variability. Ideal Ideal Gas Law: P= nRT V Ohm’s Ohm’s Law: V = IR Probabilistic or Statistical Models If If we add variation to a deterministic model we get a probabilistic or statistical model. P= nRT +ε V V = IR + ε Where ε is random error or the variation we inherently expect in nature. A Probabilistic Model Describes a Population If If 20 students measure the voltage for a simple circuit with known resistance and current, we will get an array of voltages, all close to the voltage predicted by Ohm’s Law. These These 20 voltages are a sample of the population represented by V = IR + ε 5 Consider the Pen Barrel Outside Diameter Let Let yi be the observed outside diameter for the ith pen barrel inspected. Let represent Let represent the true average or mean outside barrel diameter. Deterministic Deterministic Model: y = µ i Probabilistic Probabilistic Model: yi = µ + ε i Where εi represents the variation from the mean associated with the ith pen barrel. Strength (S) of a Polymer Filament depends on amount of catalyst (C), polymerization temperature (T), and polymerization pressure (P). Deterministic Deterministic Model: S = β 0 + β1C + β 2T + β 3 P Probabilistic Probabilistic Model: yi = β 0 + β1C + β 2T + β 3 P + ε i Data Collection – A Retrospective Study Analyze Analyze previously collected data. Example: Example: – Use previously collected tensile strength data for paper samples. Draw Draw backs: – Conditions may change over time. – Data often missing or hard to coordinate with other data. 6 Data Collection – Observational Study Observe Observe a population through sampling. Example: Example: – Pen caps are not fitting onto pen barrels. – We are bottling Pepsi and want to be sure that virtually all of our product will pass FDA approval which requires a minimum of 16 oz in each bottle Observational Study Population (select) Sample (describes) (calculate) µ ,σ 2 x, s2 Parameter (estimate) Statistic Statistics and Parameters A statistic is calculated from a random sample. It statistic describes the sample and is used to estimate a parameter. Statistics change from sample to sample. x, s2 A parameter describes a population. It is fixed parameter and usually unknown. 2 µ ,σ 7 Data Collection - Designed Experiments Manipulate Manipulate factors according to a well defined strategy in order to obtain a desired result. Factors Factors are the variables we manipulate in order to produce a desired result. – Catalyst, temperature and pressure are factors for controlling the strength of a polymer filament. The The values a factor is set to are called levels. levels Treatment Treatment is a given combination of the levels of each factor. Examples Examples where You might use an Experimental Design Developing Developing a process to produce polymer filaments of a certain strength. – Process Development Changing Changing a process to produce stronger polymer filaments. – Process Improvement Experimental & Observational Units The Experimental The Experimental Unit is the smallest unit to which we apply a treatment combination. Experimental Experimental error is a measure of the variability among the experimental units. 8 Experimental Unit A given treatment (C,T,P) is given to a given batch of polymer. The batch of polymer is the experimental unit. experimental Variability Variability from batch to batch of polymer produced at the same C,T,P is the experimental experimental error. Experimental & Observational Units The Observational The Observational Unit is the unit upon which we make the measurement. Observational Observational error is a measure of the variability among the observational units. Observational Units We We measure the strength of a filament made from the polymer. The filament is the observational unit. observational Variability Variability from filament to filament from within the same batch is the observational observational error. 9 Experimental & Observation Units Another Example A manufacturer of clay pots is interested in their manufacturer strength. Firing temperature determines strength. He fires 10 pots at a time in the oven. Experimental Experimental Unit: 10 pots in an oven Experimental Experimental Error: variability that results from trying to reproduce the firing. Observational Observational Unit: a pot Observational Observational error: variability among the 10 pots in a single firing. Basic Principles of Experimental Design Replication means Replication means that each treatment combination is applied to multiple experimental units. Replication allows us to estimate experimental error. Randomization Randomization means that the experimental runs are performed in random order. This minimizes the effects of any systematic changes that occur during the experiment. Local Local Control of Error seeks to control anything other than the factors that might affect the response. This reduces random error among experimental units. A study was carried out on the coating thickness of a panel produced by a paint operation. operation. Viscosity is thought to impact the coating coating thickness. Only two levels of viscosity are used (low and high). The experimenters produced in random order 6 batches of paint, 3 at each level of viscosity. 5 panels are painted at a time with each batch of paint. The goal of the experiment was to determine if higher viscosity leads to thicker coating. 10 Identify Response Response of Interest Factor(s) Factor(s) Factor Factor Levels Treatments Treatments Replications Replications Randomization Randomization Local Local control of error Experimental Experimental unit and error Observational Observational unit and error Experimental Designs A Completely Randomized Design randomly Completely allocates all the treatment combinations to the experimental units. A Randomized Complete Block Design Randomized randomly allocates treatment combinations to the experimental units within the blocks. Blocks are determined using known sources of variation such as vendors. Paired Paired Designs are a variation of Blocked Designs in which two treatment combinations are applied to one experimental unit. Generally, we are comparing two processes or methods. Examples Examples of Engineering Experiments Screening Screening Factors to determine which actually affect the response. We would consider C,T,P,X,Y,Z’s effect on polymer filament strength. Predict Predict the behavior of a response over a specified range of factors. Use responses to test a model. If data supports the model, we can use the model to predict the average responses. Optimization Optimization is an attempt to find the best combination of factor levels to produce a desired response. Robust Robust Design attempts to make products and processes robust to known sources of variation. We are trying to achieve a target over a wide range of operating conditions. 11 ...
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