Chapter 2

# Chapter 2 - Chapter 2 Experimental Plans for More Than Two...

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Chapter 2 Experimental Plans for More Than Two Treatments STAT 332 Spring 2017

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In this chapter, we will focus on two basic designs: Complete Randomized Design (CRD) Completely randomized design Randomized Complete Block Design (RCBD) Randomized block design STAT 332 Spring 2017
Example A product developer is investigating the tensile strength of a new synthetic fiber that will be used to make cloth for men’s shirts. Strength is usually affected by the percentage of cotton used in the blend of materials for the fiber. The engineer is interested in 4 different levels of cotton content: 15%, 20%, 25% and 35%. She decided to test 5 shirts at each cotton content level. The objective of this experiment is to model the relationship between the tensile strength of the shirt and the cotton content. STAT 332 Spring 2017

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This is an example of a single-factor experiment with ? = 4 levels of factor and ? = 5 replicates. This experiment has 4 treatments In this experiment, the number of replicates (units) given to each treatment is the same, we say the design is balanced. STAT 332 Spring 2017
Complete Randomized Design The design of the experiment is called a complete randomized design if the 20 runs ( 4 treatments × 5 replicates) was made in random order In a complete randomized design, we randomly assign a number to each experimental run STAT 332 Spring 2017

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Back to the example Suppose the engineer runs the test that we have determined in a random order and recorded the results: STAT 332 Spring 2017 Cotton content 15% 20% 25% 30% 7 12 14 19 7 17 19 25 15 12 19 22 11 18 18 19 9 18 18 23
It is always a good idea to examine the experimental data graphically. Scatterplots and boxplots can be useful The scatterplot on the right showed that the strength increases as the cotton content increases. STAT 332 Spring 2017

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The boxplot showed similar trend: higher cotton content result in increased strength STAT 332 Spring 2017
Suppose we wish to be more objective in our analysis of the data and wish to test for differences between the mean strength at all 4 levels of cotton content We could perform t-test for all 6 possible pairs of means, but this is not the most efficient solution to this problem and doing so will inflates the Type I error. Suppose that all 4 means are equal, so if we select ? = 0.05, the probability of reaching the correct decision on any single comparison is 0.95. However, the probability of reaching the correct conclusion on all 6 comparisons is considerably less than 0.95, so the Type I error is inflated The appropriate procedure for testing the equality of several means is the analysis of variance (ANOVA) STAT 332 Spring 2017

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The Model Suppose we have a balanced experimental plan with r replicates (observations) for each of the t treatment levels of a single factor. We will find it useful to describe the observations from an experiment with a model: 𝑌 ??
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