DOE L-02 C-01 Introduction C02 Basic Statistical Methods.pptx

dfgbhjhhhhhhhhhhhhhgvgfhfjdtyesdcfhvbjnhgxhdgcfbhjnjmkh

1. Design of Experiments Lecture-2 1 Varinder Singh @Goa.

2. Today’s session 2 Chapter # Topic Dates 1 Introduction to Designed Experiments 2 Basic Statistical Methods 3 Analysis of Variance 4 Experiments with Blocking Factors 5 Factorial Experiments 6 Two-Level Factorial Designs 7 Blocking and Confounding Systems for Two-Level Factorials 8 Two-Level Fractional Factorial Designs 9 Other Topics on Factorial and Fractional Factorial Designs 10 Regression Modeling 11 Response Surface Methodology 12 Robust Design 13 Random Effects Models 14 Experiments with Nested Factors and Hard-to-Change Factors Not in previous semester’s Course Handout. Syllabus yet to be revised. Design and Analysis of Experiments- Douglas C. Montgomery, Eighth edition, 2013, Wiley India.

3. Trial and error/Best guess. One factor at a time (OFAT). Factorial- Full and Fractional. Three approaches… 1/2 3

4. 4 Designed Experiments designed experiment is a test or series of tests in which purposeful changes are made to the input variables of a process so that we may observe and identify corresponding changes in the output response. The objectives of the experiment may include 1. Determining which variables are most influential on the response, y. 2. Determining where to set the influential x’s so that y is near the nominal requirement. 3. Determining where to set the influential x’s so that variability in y is small. 4. Determining where to set the influential x’s so that the effects of the uncontrollable variables are minimized. If a process is in statistical control but still has poor capability, then to improve process capability it will be necessary to reduce variability. Designed experiments may offer a more effective way to do this than SPC.

5. 5 EXPERIMENTAL DESIGN FUNDAMENTALS An experiment is a planned method of inquiry conducted to support or refute a hypothesized belief or to discover new information about a product, process, or service. It is an active method of obtaining information, as opposed to passive observation. It involves inducing a response to determine the effects of certain controlled parameters. Examples of controllable parameters, often called factors, are the cutting tool, the temperature setting of an oven, the selected vendor, the amount of a certain additive, or the type of decor in a restaurant. The factors may be quantitative or qualitative (discrete). response variable is generally the resulting effect or output of system . a treatment is a certain combination of factor levels whose effect on the response variable is of interest. The variation in experimental units that have been exposed to the same treatment is attributed to experimental error. This variability is due to uncontrollable factors, or noise factors.

6. 6 Effect of one factor at a time Experiments with two or more variables usually avoid changing only one variable at a time—that is, keeping the other variables constant. There are reasons for this. Figure (a) shows the design points for such an approach, with value of factor A is fixed, the two design points are ( 0 , - 1 ) and (0,1). Figure (b) shows the response function value at the design points. Note that to maximize the response function in this experiment, we would move in the direction of the arrow shown along the major axis of factor A. This, however, does not lead to the proper point for maximizing the response function. Another drawback to the one-variable-at-a-time approach is that it cannot detect interactions

7. 7 FACTORIAL EXPERIMENT 2 level Experiment Each factor can be set at two levels, denoted by - 1 and 1. The point (0,0) represents the current setting of the factor levels. In this notation, the first coordinate represents the level of factor A, and the second coordinate represents the level of factor B. A level of - 1 represents a value below the current setting of 0, and a level of 1 represents a value above the current setting. Thus, four new design points, (—1, —1), (1, —1), (—1,1), and (1,1), can be investigated. to maximize the response function, we will consider the direction in which the gradient is steepest. Our next experiment could be conducted in a region with the center at (2,2) with respect to the current origin. These same principles apply to more than two variables as well.

8. 8 Interaction effects Interactions exist when the nature of the relationship between the response variable and a certain factor is influenced by the level of some other factor. factor A stays the same regardless of the level of factor B. The rate of change of the response as a function of factor A's level changes as B's level changes from -1 to 1 Interactions depict the joint relationship of factors on the response function. Such effects should be accounted for in multifactor designs because they better approximate real-world events. For example, a new motivational training program for employees (factor B) might impact productivity (response variable) differently for different departments (factor A) within the organization.

9. 9 Another way to represent interaction is through contour plots

10. Replication: Replication involves a repetition of the experiment under similar conditions (that is, similar treatments). allows us to obtain an estimate of the experimental error, the variation in the experimental units under identically controlled conditions. The experimental error forms a basis for determining whether differences in the statistics found from the observations are significant. Randomization Randomization means that the treatments should be assigned to the experimental units in such a way that every unit has an equal chance of being assigned to any treatment. Such a process eliminates bias and ensures that no particular treatment is favored. Random assignments of the treatments and the order in which the experimental trials are run ensures that the observations, and therefore the experimental errors, will be independent of each other. An experimental unit is the quantity of material (in manufacturing) to which one trial of a single treatment is applied. Features of experimentation (experiments to ensure replication, randomization and if needed blocikng ) 10

11. Blocking Using this concept, variability between blocks is eliminated from the experimental error, which leads to an increase in the precision of the experiment. Variability of the response function within a block can be attributed to the differences in the treatments because the impact of other extraneous variables has been minimized. Treatments are assigned at random to the units within each block. Features of experimentation 11

12. Randomization. Blocking. Replication. Randomization: Accuracy. Blocking: Impact of variables. Replication: Precision. Three main DOEs- Completely Randomized design. Randomized Block design. Matched-Pair design. Three principles of DOE 12 Accuracy and Precision

13. What factors to vary? No. of experiments? Sequence of experiments? Designs- Full factorial designs. Fractional factorial designs (Screening designs). Response surface designs. Mixture designs. Taguchi array designs. Split plot designs. Different design in different stage of Experimentation. DOE: Designs 13

14. Factors and Levels… 1/2 14 Speed (10, 20) Surface finish One Factor (Speed), Two Levels (10 and 20). Two Factors (Speed and Feed rate), Two (10 and 20)/Three Levels (60, 70 and 80). Speed (10, 20) Feed rate (60, 70, 80) Surface finish

15. Factors and Levels…. 2/2 15 Four Factors & Two Levels of each Sunlight (4, 8 hrs ) Plant Food (2, 20 g) Water (100 ml, 500ml) Brand of plant (A, B)

16. Basic Statistical Methods (Comparative Experiments) Chapter-2 16

17. 17 Sampling Distribution Every time we take a random sample and calculate a statistic, the value of the statistic changes ( remember, a statistic is a random variable ). If we continue to take random samples and calculate a given statistic over time, we will build up a distribution of values for the statistic. This distribution is referred to as a sampling distribution . A sampling distribution is a distribution that describes the chance fluctuations of a statistic calculated from a random sample.

18. 18 Sampling Distribution of the Mean The probability distribution of is called the sampling distribution of the mean . The distribution of , for a given sample size n , describes the variability of sample averages around the population mean μ .

19. 19 Sampling Distribution of the Mean If a random sample of size n is taken from a normal population having mean μ x and variance , then is a random variable which is also normally distributed with mean μ x and variance . Further, is a standard normal random variable .

20. 20 Sampling Distribution of the Mean n(100,5) n(100,3.54) 5/sqrt(2)=3.54 1 2 3 4 Original population

21. 21 Sampling Distribution of the Mean Example : A manufacturer of steel rods claims that the length of his bars follows a normal distribution with a mean of 30 cm and a standard deviation of 0.5 cm. Assuming that the claim is true, what is the probability that a given bar will exceed 30.1 cm? (b) Assuming the claim is true, what is the probability that the mean of 10 randomly chosen bars will exceed 30.1 cm? (c) Assuming the claim is true, what is the probability that the mean of 100 randomly chosen bars will exceed 30.1 cm?

22. 22 Sampling Distribution of the Mean Example : A manufacturer of steel rods claims that the length of his bars follows a normal distribution with a mean of 30 cm and a standard deviation of 0.5 cm. Assuming that the claim is true, what is the probability that a given bar will exceed 30.1 cm? (z=30.1-30)/0.5=0.2 p=0.42) (b) Assuming the claim is true, what is the probability that the mean of 10 randomly chosen bars will exceed 30.1 cm? (z=30.1-30)/(0.5/sqrt(10)=0.63 p=0.26) (c) Assuming the claim is true, what is the probability that the mean of 100 randomly chosen bars will exceed 30.1 cm? (z=30.1-30)/(0.5/sqrt(100)=2 p=0.02)

23. 23 Sampling Distribution of the Mean .42 .02 . 26

24. 24 Expected value of sample mean is population mean The mean has variance Variance = Properties of Sample Mean as Estimator of Population Mean As n increase, decrease.  x =   x _ _ standard error:

25. 25

26. 26 n =16   X = 2.5 n = 4   X = 5 When the Population is Normal then the Sampling Distribution is Also Normal Central Tendency Variation Population Distribution Sampling Distributions  x  =   x = _ _

27. 27 Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes almost normal regardless of shape of population

28. 28 n =30   X = 1.8 n = 4   X = 5 When The Population is Not Normal Population Distribution Sampling Distributions  = 50  = 10 X Variation  x  =   x = _ _ Central Tendency

29. 29 Central Limit Theorem Rule of thumb: normal approximation for will be good if n > 30. If n < 30, the approximation is only good if the population from which you are sampling is not too different from normal.

30. 30 t-Distribution So far, we have been assuming that we knew the value of σ. This may be true if one has a large amount of experience with a certain process. However, it is often true that one is estimating σ along with μ from the same set of data.

31. 31 t-Distribution To allow for such a situation, we will consider the t statistic: which follows a t-distribution.  standard error of the mean

32. 32 t-Distribution t(n= ) = Z t(n=6) t(n=3)

33. 33 t-Distribution If is the mean of a random sample of size n then is a random variable following the t- distribution with parameter ν = n – 1 , where ν is degrees of freedom .

34. 34 t-Distribution The t-distribution has been tabularized. t α represents the t-value that has an area of α to the right of it. Note, due to symmetry, t 1-α = -t α t .05 t .80 t .20 t .95

35. 35

36. 36 Example: t-Distribution The resistivity of batches of electrolyte follow a normal distribution. We sample 5 batches and get the following readings: 1400, 1450, 1375, 1500, 1550. Does this data support or refute a population average of 1400?

37. 37 1.71 Refute Refute Support t=2.78 p=0.025 Example: t-Distribution

38. A cement company wants to know whether the average bond strength of Modified mortar (more water) is the same or different from that of Unmodified mortar (standard water). The company tested 10 samples of Modified mortar and 10 samples of Unmodified mortar. Bond strengths recorded by the tests are- Comparative experiment-1: Problem Textbook : p-26. 38 Cement Formulation (Modified, Unmodified) Strength One Factor, Two Levels Goal of the experiment- Average bond strength of Modified mortar (more water) is the same or different from that of Unmodified mortar (standard water).

39. Comparative experiment-1: Solution Textbook: p-26, 28, 40. 39 Modified . Unmodified Is average bond strength of Modified mortar the same as that of Unmodified mortar? Ho: Modified = Unmodified. Average strength of mortar… H1: Modified ≠ Unmodified. Alpha =0.05. Level of Significance. p-value= 0.042. Excel function- 0.042 = T.Test (Array1,Array2,2Tail,EqualVariance) p-value<0.05. Hence reject Ho. . That is, average bond strengths of two mortars are different . Bond strengths recorded by the tests are-

40. Previous problem was- Two-sample t-test.

41. Next problem: First as Two sample t-test, same as the previous problem. Then as Two- paired sample t-test