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# Probability and Statistics for Engineers ST 370

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This 38 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 370 at North Carolina State University taught by Yichao Wu in Fall. Since its upload, it has received 24 views. For similar materials see /class/223949/st-370-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

North Carolina State University STAT 370 Probability and Statistics for Engineers Yichao Wu Lass class Sample standard deviation 11 2 xi W Homework 3 due Monday Feb 2 at 5PM Effect of adding a constant to all data points adds constant to sample mean and median doesn t change sample variance Effect of multiplying all data points by a constant multiplies sample mean and median by the constant multiplies sample variance by square of the constant Effect of adding to max increases sample mean but sample median the same Statistics and parameter Numerical summarizations of sample data are called sample statistics Numerical summarizations of population and theoretical distributions are call parameters Roman letters are used as symbols for statistics and Greek letters are used to stand for parameters Statcrunch httpstatcrunchstatncsuedu Tutorial httpwww4statncsued uwoodardstatcrunch Summary of Chapter 3 Examine distributions Overall pattern Shape Symmetric or skewed How many modes Bellshaped Outliers Graphical tools for quantitative data Stemplot Histograms Boxplot Measuring center Mean median mode Spread IQR range standard deviation Read Chapter 3 of Vardeman and Jobe Homework 3 due Monday February 2 at 5PM Design of Experiments So far we have discussed a little bit about data collection SR8 and summarizing data collected from surveys We have been mainly dealing with observational studies We selected a sample that was representative on average of the population so that we can generalize results to the population Design of Experiment cont However observational studies are not good for determining CAUSEEFFECT relationships Why If we observe a certain effect there could be many causes To show that A causes B not only would we need to see that when A happens then B also happens but that if not A then not B no other cause Design of Experiment cont Think of a tutoring program and we want to show that it helps students to do better on standardized tests We would need to show that if an individual oes throuh our roram the do better and if they don t go through our program they don t Now we can t make each individual both go through our program and not go through the program What we would want is two groups that are similar except for whether or not they went through our program We would take a group and randomly assign them to either our progra the control group and study differences between the groups Design of Experiment A designed experiment is a controlled study conducted to determine the effect that varying one or more explanatory variables factors the treatments has on a response variable The goal is to see if different treatments CAUSE different values Treatment and response variable Treatment is a combination of the values of each factor oet of conditions of interest Think drugs brands locations etc Response Variable output of interest what we monitor to determine how system is working May have more than one response variable We ll focus on numeric and mostly continuous response variables Experimental units Experimental Units units to which a treatment is applied or units created under certain treatment conditions don t hit this too hard A patient that will receive an assigned drug A concrete block tha will be created from an assigned mixture of concrete A paper towel of a certain brand lnclass exercise 1 Describe the treatments experimental units and res onse variables for the ex eriment described below A field experiment is conducted to compare the yield of three varieties of com A field containing 30 plots of land is used for the experiments Each variety is planted on 10 randomly selected plots The yield for each plot is measured at the time of harvest lnclass exercise 2 Describe the treatments experimental units and res onse variables for the ex eriment described below An experiment is conducted to compare the effect of three drugs on the lean percentage of hogs A total of 30 hogs are assigned to the three drugs in a completely randomized fashion so that 10 hogs will receive weekly injections with each drug The lean percentage of each hog is recorded at the time of slaughter Designing an experiment 1 Identify the problem to be solved statement of I roblem should give direction as to how to conduct the experiment identify the population to be studied and the response variable etc 2 Determine the factors that affect the outcome usually an expert does this 3 Determine the number of experimental units as a general rule choose as many as time and money will allow 4 Determine how the factors will be handled which will be controlled manipulated and not controlled 5 Collect and process data 6 Draw conclusions from experiment inferential statistics Example A farmer wishes to determine the optimal level of a new fertilizer on his soyoean crop He has an acre of land in one location and 300 soypean plants that were bought at the same place Design an experiment that will assist him Objective Objective of experimental design to determine the effect of treatments thus we want conditions to be as close as possible for experimental units getting different treatments Reasons for variability of responses Treatment effect Experimental error noise Experimental error Variability among values of the response variable for experimental units that receive the same treatment Ixpermental error does not mean you did anything wrong Sources of experimental error Inherent variability in experimental units Measurement error Variations in applyingcreating treatment Effects from any other Extraneousor lurking variables Inherent variability in experimental units No two people paper towels concrete blocks or even lab rats are exactly tne same Give two people the same drug and they won t respond the same to it Measurement error lf same experimental unit is measured more than once will the value be the same If two different researchers measure the same experimental vv we always get exactly the same measurement Variations in applyingcreating the treatment When two experimental units are assigned the same treatment do they get EXACTLY the same treatment Example If two different researchers mix the concrete will it come out exactly the same Extraneous or lurking variables Effects from any other Extraneous or lurking Variables Extraneous variables are those variables that are not intended to be part of the treatment but they may affect the response This is the catchall category What else could cause differences in observed values of the response variable Reduce experimental error We are trying to reduce experimental error to make it easier to detect treatment effects Inherent variability remedy Choose experimental units that are as similar as possible Measurement error remedy Try to define a consistent measurement protocol mod 9 a CLEAR definition of what the response variable is Good measuring eluir ment Variability of treatments remedy Try to define a consistent treatment applicationcreation protocol Extraneous variables remedy Try to use an isolated or closely controlled environment Reduce experimental errorcont Controlled variables kept constant across experimental units This does not mean simply that you have control of the variable but that it is kept constant for all experimental units Don t want to see the effects of these variables on the response Doesn t give information on what happens at levels other than the fixed one Any extraneous variables that are NO controlled are lurking variables Experimental error No matter how hard we try some sources of experimental error will remain Main techniques randomization blocking Randomization different than random sampling which is selecting the sample so that each sample has the same likelihood of being include in randomization we have selected the experimental units possibly by a convenience sample and want to now assign randomly to treatments Randomization Randomize the assignment of experimental units to each treatment Example Patients are randomly assigned to drugs We can still restrict the randomization such that w Ufr xm h m nm rof replicates in each treatment group Randomize the order of applying the treatment order of measurements etc Example randomize the order in which slabs of concrete are mixed and poured Do this when feasible and practical Purpose of randomization protect against a systematic influence of lurking variables Does NOT prevent them from affecting the response hopefully prevents them from affecting the results Does NOT reduce experimental error but we average out effects over treatments Randomization method 1 completely randomized designCRD Suppose we do r replicates per treatment From the experimenta units available randomly select r units to receive treatment 1 From what s left randomly select r units to receive treatment 2 Continue until all treatments have been assigned to r experimental units Advantage of CRD over something that is not randomized To balance out the effects of lurking variables identified variables or unidentified variables Disadvantage of CRD over other types of randomization It is not always practical think of a medical study may not be practical for a doctor to randomly assign patients to receive treatment or placeboactually could be unethical Randomization method 2 Randomized Complete Block Design RCBD We will discuss it later Blocking A block is a subset of experimental units or experimental runs Choose blocks to be fairly homogeneous Multiple blocks of experimental units or runs are uSeu in the experiment Experiment is run over several days dayblock Experiment is run at multiple locations locationblock Plane is thrown by different people throwerblock Paint different types of surfaces surface typeblock Why blocking There is inherent variability in experimental units that cannot be avoided but unitsruns can be grouped into fairly homogeneous subgroups That is an extraneous variable that cannot be controlled fixed but we can manage it in some way Ex Don t have timeresources to do the entire experiment in one day or at one place By reducing variability among bloc we ter determine treatment effects for the block then hopefully we can combine information from blocks Want results to apply to a population of blocks Ex Don t want paint results to apply only to one type of surface Ex Don twant results to on a to one thrower Want evidence that the results are repeatable on different days or at different locations etc How to block Randomized Complete Block Design RCBD Divide experimental units or runs into blocks Conduct a miniComplete randomized design in each group block We ll be looking for consistent treatment effects across blocks If we see the same results in each block great If not then things are more difficult Example paint experimentCRD 4 treatments paint types 60 experimental units CRD 60 experimental units and 4 treatments r 604 15 Randomly select 15 out of 60 units to receive treatment 1 Out of the 45 left we randomly select 15 out of 45 units to receive treatment 2 and so on Example paint experiment RCBD Instead suppose 3 types of experimental units 20 units of each waml Wood Brick RCBD r 15 5 drywall 5 wood 5 brick Randomly select 5 out of 20 units of drywall 5 out of 20 units of wood and 5 out of 20 units of brick to receive treatment 1 Continue random assignment to other treatments in this wav Replication Multiple experimental units per treatment notjust multiple measurements of experimental units which only captures measurement error To be able to see trends vs flukes remember we want to generalize to population To quantify the amount of experimental errorThis is very important Gives us a way of describing variability in responses determining significance and quantifying the quality of different estimates To be able to generalize the results to a population of experimental units that might receive that treatment if we Only had one patient on Lhc Ulug VVOUId dIIyUIIc believe uslNFERENCEwhat we will do Summary Design of experiment factors treatments response Experimental error Variables affecting the response Factors treatments controlled variables lurking variables Techniques for handling lurking varialbes Randomization Blocking Reproducibility of the result Replication Reading assignment Read Sections 23 and 24 of Vardeman and Jobe Form teams for course project

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