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# Probability and Statistics for Engineers ST 370

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This 36 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 17 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 Quartiles IQR Suspected outliers 5number summary Boxplot Modified boxplot Homework 2 due Monday Jan 26 at 5PM From frequency histogram to median Relative Frequency Ages of Uninsured Senior Citizens Relative Frequency Histogram 6067 5576 7784 8592 Age ClassesRanges 93100 Another measure of spread Standard deviation Sample Variance 82 Deviation from mean the difference between an observation and the sample mean x1 x Sample Variance 822 the average of the squares of the deviations of the observations from their mean S2 x1 c2 x2 c2 xn c2 n l xi C2 n l Toy Examples Data 210 12 What is the sample variance How about this 40V 40V 40V 40V 40 Sample Standard Deviation s Sample Standard Deviation s the square root of the sample variance 2quot xi W 11 1 Remarks on the definition of sample SD The sum of the deviations of the obs from their mean is always 0 Why square the deviations rather than absolute deviations Mean is a natural center under the squaring SD is a natural measure of spread for the normal distributions Remarks on sample SD Why sample SD rather than sample variance SD is natural for measuring spread for normal dist SD is in the original scale Why n1 rather than lntuitively speaking SD is not defined for n1 Sum of deviations is always 0 which means if we know n1 of them we know the last one Only n1 deviations can change freely n1 degrees of freedom Properties of sample SD 3 measures the spread about the mean 3 should be used only when the mean is chosen to measure the center sO if and only if there is no spread When sgtO elsewhere increases when more spread 3 like the mean is not resistant Even less resistant Why Inclass exercise Calculate mean and variance First data 5 8 9 10 11 Second data 5 8 9 10 111 Examples 2001 Twoseater Cars The highway mileages of the 18 gasolinepowered twoseater cars 13131619 212123 23 24 26 26 27 27 27 28 28 3O 3O Mean 234 SD53 The highway mileages of the 19 twoseater cars 13131619 212123 23 24 26 26 27 27 2 w w 30 3O 68 Mean 258 SD114 Averaging reduces variability The standard deviation of batting averages of all teams in the American League IS 0008 The standard deviation of all players in the American League IS 002154 Three measures of spread The range is the spread of all the observations The interquartile range is the spread of roughly the middle 50 of the observations Sample 0D is a measure of the distance from sample mean Sample SD can be regarded as a typical distance of the observations from their mean The fivenumber summary vs sample Mean and SD The fivenumber summary is preferred for a skewed distribution or a distribution with strong outliers x and s are preferred for reasonably symmetric distributions that are free of outliers Alwa s lot our data rst Use boxplots Property of sample SD If the data are symmetric and unimodal one peak we can use the Empirical Rule OR 6895997 rule to construct interyals Roughly 68 of the data will fall between gc S and xs Roughly 95 of the data will fall between g ZS and x2s Roughly 997 of the data will Tall between x 39 and 3H3 These numbers come from a normal distribution which we ll see later Doesn t apply when data are skewed Example Assume we have a symmetric unimodal data set with 515 and s114 Then according to the empirical rule Roughly 68 of data between 515114401 and 515114629 Roughly 95 of data between 5152114287 and 515 1 14743 Roughly 997 almost all of data between 5153114173 and 5073114857 This sample mean and variance is for Math SAT data Pretty close 67 within 1 sd 95 within 2 sd really close So from jusr the mean and sd you can get a pretty good sense of the whole distribution Exercise for Empirical rule The distribution of the length of bolts has a bell shape with a mean of4 inches and a standard deviation of 0007 inch a About 68 of bolts manufactured Will be between wnat lengths b What ercenta39e of bolts will be between 3986 inches and 4014 inches c lfthe company discards any bolts less than 3986 inches and greater than 4014 inches what percentage or bolts manufactured will be discarded d What I ercentave of bolts manufactured will be between 4007 and 4021 inches e What percentage of bolts manufactured will be greater than 4021 inches f What percentage of bolts manufactured will be less than 3993 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 lnclass exercise 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 Reading assignment Read Sections 23 and 24 of Vardeman and Jobe

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