 # design and analysis experiment Anonymous

### Question Description

this class is design and analysis experiment

see the files attached

only the highlighted questions

and i attached slides for chapter4 2017_09_27_photo_00000576.jpg 2017_09_27_photo_00000575.jpg 2017_09_27_photo_00000574.jpg
Design of Engineering Experiments – The Blocking Principle • Text Reference, Chapter 4 • Blocking and nuisance factors • The randomized complete block design or the RCBD • Extension of the ANOVA to the RCBD • Other blocking scenarios…Latin square designs Chapter 4 1 The Blocking Principle • Blocking is a technique for dealing with nuisance factors • A nuisance factor is a factor that probably has some effect on the response, but it’s of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized • Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units • Many industrial experiments involve blocking (or should) • Failure to block is a common flaw in designing an experiment (consequences?) Chapter 4 2 The Hardness Testing Example • Text reference, pg 139, 140 • We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester • Gauge & measurement systems capability studies are frequent areas for applying DOE • Assignment of the tips to an experimental unit; that is, a test coupon • Structure of a completely randomized experiment • The test coupons are a source of nuisance variability • Alternatively, the experimenter may want to test the tips across coupons of various hardness levels • The need for blocking Chapter 4 3 The Hardness Testing Example • Suppose that we use b = 4 blocks: • Notice the two-way structure of the experiment • Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks) Chapter 4 4 Extension of the ANOVA to the RCBD • Suppose that there are a treatments (factor levels) and b blocks • A statistical model (effects model) for the RCBD is  i  1, 2,..., a yij     i   j   ij   j  1, 2,..., b • The relevant (fixed effects) hypotheses are H 0 : 1  2  Chapter 4  a where i  (1/ b) j 1 (    i   j )    i b 5 Extension of the ANOVA to the RCBD ANOVA partitioning of total variability: a b  ( y i 1 j 1 ij a b  y.. )   [( yi.  y.. )  ( y. j  y.. ) 2 i 1 j 1 ( yij  yi.  y. j  y.. )]2 a b i 1 j 1  b ( yi.  y.. ) 2  a  ( y. j  y.. ) 2 a b   ( yij  yi.  y. j  y.. ) 2 i 1 j 1 SST  SSTreatments  SS Blocks  SS E Chapter 4 6 Extension of the ANOVA to the RCBD The degrees of freedom for the sums of squares in SST  SSTreatments  SSBlocks  SSE are as follows: ab  1  a  1  b  1  (a  1)(b  1) Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means Chapter 4 7 ANOVA Display for the RCBD Manual computing (ugh!)…see Equations (4-9) – (4-12), page 124 Design-Expert analyzes the RCBD Chapter 4 8 Manual computing: Chapter 4 9 Chapter 4 10 Vascular Graft Example (pg. 126) • To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin • Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures Chapter 4 11 Chapter 4 12 Vascular Graft Example Design-Expert Output Chapter 4 13 Residual Analysis for the Vascular Graft Example Chapter 4 14 Residual Analysis for the Vascular Graft Example Chapter 4 15 Residual Analysis for the Vascular Graft Example • Basic residual plots indicate that normality, constant variance assumptions are satisfied • No obvious problems with randomization • No patterns in the residuals vs. block • Can also plot residuals versus the pressure (residuals by factor) • These plots provide more information about the constant variance assumption, possible outliers Chapter 4 16 Multiple Comparisons for the Vascular Graft Example – Which Pressure is Different? Also see Figure 4.2 Chapter 4 17 Other Aspects of the RCBD See Text, Section 4.1.3, pg. 132 • The RCBD utilizes an additive model – no interaction between treatments and blocks • Treatments and/or blocks as random effects • Missing values • What are the consequences of not blocking if we should have? • Sample sizing in the RCBD? The OC curve approach can be used to determine the number of blocks to run..see page 133 Chapter 4 18 Random Blocks and/or Treatments σ2β Chapter 4 Design & Analysis of Experiments 8E 2012 Montgomery 19 Chapter 4 Design & Analysis of Experiments 8E 2012 Montgomery 20 Chapter 4 Design & Analysis of Experiments 8E 2012 Montgomery 21 The Latin Square Design • Text reference, Section 4.2, pg. 158 • These designs are used to simultaneously control (or eliminate) two sources of nuisance variability • A significant assumption is that the three factors (treatments, nuisance factors) do not interact • If this assumption is violated, the Latin square design will not produce valid results • Latin squares are not used as much as the RCBD in industrial experimentation Chapter 4 22 The Rocket Propellant Problem – A Latin Square Design • • • • This is a 5  5 Latin square design Page 159 shows some other Latin squares Table 4-12 (page 162) contains properties of Latin squares Statistical analysis? Chapter 4 23 Statistical Analysis of the Latin Square Design • The statistical (effects) model is  i  1, 2,..., p  yijk    i   j   k   ijk  j  1, 2,..., p k  1, 2,..., p  • The statistical analysis (ANOVA) is much like the analysis for the RCBD. • See the ANOVA table, page 160 (Table 4.10) • The analysis for the rocket propellant example follows Chapter 4 24 Chapter 4 25 Chapter 4 26 Graeco-Latin Squares = = This question has not been answered.

Create a free account to get help with this and any other question! Brown University

1271 Tutors California Institute of Technology

2131 Tutors Carnegie Mellon University

982 Tutors Columbia University

1256 Tutors Dartmouth University

2113 Tutors Emory University

2279 Tutors Harvard University

599 Tutors Massachusetts Institute of Technology

2319 Tutors New York University

1645 Tutors Notre Dam University

1911 Tutors Oklahoma University

2122 Tutors Pennsylvania State University

932 Tutors Princeton University

1211 Tutors Stanford University

983 Tutors University of California

1282 Tutors Oxford University

123 Tutors Yale University

2325 Tutors