This is a design and experiment Anova project. The purpose of his project is to apply the ANOVA approaches that covered in the textbookMontgomery-Design-and-analysis-of-experimentsto real-life situations. You are encouraged to identify problems or interests in your real life and apply the approaches in the textbook.Please select a proper dataset from real-life situation.The followings are steps suggested as an example: (Please refer to the guidelines of designing of experiments in the Table 1.1 of the textbook (P 14)
- Identify your interest/problem in your real-life situation
- Goal setting for the (simple) research
- Set up your design of experiment
- Determine response variable
- Factor(s) & levels
- Perform experiment
- Data collection
- Statistical Analysis of the data
- Select analytic tools (e.g., R or JMP or others, use R preferably)
- Analyze data
- Conclusions and recommendations
- Documentation & Submission
- The textbook link are :
- http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/502_06_Montgomery-Design-and-analysis-of-experiments-2012.pdf
- Please use factorial design for this project
- Please write up a 5-6 pages report on the project(no word count and format requirement).The project example report is attached below for reference.
Example Project Report Experimental Design and Analysis Exploring Efficient Heating Containers For Use in Microwave Ovens 1. Executive Summary The objective of our experiment is to seek the combination of container factors that minimizes time consumed to heat the substance inside to reach desired temperature. Our choice of factors: material type, container shape, material color, and cover status, composed the treatments investigated in our experiment. The nature of the experiment resulted in thermometer-measured observations of temperature as response variables which are obtained by heating the substance in the container in the same time interval. The experimental design chosen was a factorial 24 Randomized Complete Block Design (RCBD) with one block factor. The design matrix for the experiment was generated through a Custom Design in JMP and Design-Expert and the Fit Model analysis tool provided the output to support the documented conclusions. 2. Problem Recognition 2.1 Introduction For many decades now since the early 1950’s, microwave ovens have been commonly used in household kitchens to heat up and cook a variety of food and liquids. The microwave oven started off as being a giant 6 ft 750 lb machine but after many years of research and design improvements, it now finds itself being able to be placed in almost any kitchen décor. However, despite all the research that has gone into improving the design of the microwave oven, little thought has been given to the selection of the most efficient choice of material used as a container for heating. In this experiment we will be attempting to optimize the conditions for the ideal heating container that can be used in a microwave and provide the best statistical results. 2.2 Motivation Since currently most people do not follow any good design for heating substances in the microwave, a lot of energy and time is wasted which has a huge negative impact on the environment. Through the implementation of our designed experiment, we can potentially save a lot of energy and revolutionize the way people heat up their food and drinks. 2.3 Objective The objective of our experiment is to determine which factors significantly affect the efficiency of heating inside a microwave. Once those factors are recognized, then we want to optimize the factors such that the amount of time used to heat a substance is minimized and thereby the amount of energy consumed in heating is reduced. 3. Choice of Factors The four container factors we hypothesize are most important with respect to overall optimization include: material type, container shape, material color, and cover status. These four factors compose the experimental factors we wish to study in our experiment. Table 1 list these factors along with the chosen levels for each. The potential factors that we are considering for our design are as follows: Table 1: Experimental factors and levels Factor Level 1 Level 2 Type of Material Plastic Glass Shape Cylinder Rectangle Color Clear Opaque Cover Open Closed 4. Description of Factors 4.1 Type of Material The two types of materials we chose to test in this experiment are plastic and glass. These materials were chosen because they are most commonly used as containers among housewives. We think the type of material makes a significant difference in the efficiency of heating because of the difference in material absorption of microwaves. 4.2 Shape The shapes of container we chose to test are cylinder and rectangle. These shapes were chosen due to them being used in drinking containers and in sandwich boxes. We believe the shape will have a significant difference in heating efficiency because since they are very distinct in form and open area, the directions by which the microwaves will reach the substance will differ significantly, potentially affecting the rate of heating. 4.3 Color We will be using two types of colored containers – clear and opaque. The color of the container should make a significant difference in heating efficiency because of the relationship between the color of a material and the wavelengths it absorbs or transmits. Therefore it is possible that a certain type of color tends to absorb microwaves better than others. 4.4 Cover The last factor chosen was whether or not the container will be covered. It is known that when a container is covered, it will keep heat from escaping from the container. But at the same time, it may also impede the flow of additional microwaves into the container. Since the relative rates of both are not known, it is imperative to test this factor and determine which one plays a more important role in heating efficiency. 5. Selection of Response Variable Temperature is the response variable selected that will be used to determine which combination of factors allow for most efficient heating inside of a microwave. The initial temperature will be held constant for all runs and then immediately after the samples are processed in the microwave, the final temperature will be measured for all samples using a digital thermometer to ensure accuracy of the measurements. The experimental design chosen was a factorial 24 Randomized Complete Block Design (RCBD) with one block factor, the type of substance being heated. The substances chosen in the block were water and milk to ensure that the type of substance does not interact with the container and the change in temperature are truly due to the effect of the factors relating to the container. The design matrix for the experiment was generated through a Custom Design in JMP. The experimental worksheet is provided below: Table 2: JMP Experimental Design Worksheet Run Block Material Shape Color Cover Temperature 1 Water Glass Cylinder Opaque Closed . 2 Water Glass Rectangle Clear Closed . 3 Water Plastic Rectangle Clear Open . 4 Water Plastic Cylinder Opaque Open . 5 Water Plastic Rectangle Opaque Closed . 6 Water Plastic Cylinder Opaque Closed . 7 Water Plastic Rectangle Clear Closed . 8 Water Plastic Cylinder Clear Closed . 9 Water Glass Cylinder Clear Closed . 10 Water Glass Cylinder Clear Open . 11 Water Plastic Rectangle Opaque Open . 12 Water Plastic Cylinder Clear Open . 13 Water Glass Rectangle Opaque Open . 14 Water Glass Cylinder Opaque Open . 15 Water Glass Rectangle Clear Open . 16 Water Glass Rectangle Opaque Closed . 17 Milk Glass Cylinder Clear Open . 18 Milk Glass Rectangle Opaque Open . 6. Performing the experiment We prepared eight categories of containers which are listed in Table 2. The measuring tool we used is an electronic infrared thermometer which has a decimal accuracy. Please see the Appendix for the pictures of containers and microwave oven and electronic infrared thermometer. There are two blocks, one is water, and the other is milk. We performed the experiment in the same day. First, we did experiment with water which is contained in a big plastic container to ensure unique water resource and to minimize the variation of the water temperature. Moreover, we measured the temperature of water before each run so we can keep records of how much the temperature differs after being heated instead of a single temperature value after heating which makes the experiment more precise. Therefore, response in this experiment is the temperature difference instead of the temperature after heating. To minimize the variation of microwave oven temperature between each run, we cooled down the environment inside the oven by fanning after heating, and started next run until the temperature dropped to normal value. For the experiment with milk, we did exactly the same routine as with water. The result of experiment is shown in table 3. Table 3: Measured Data Result Run Block Material Shape Color Cover Before After Difference 1 Water Glass Cylinder Opaque Closed 26.4 85.2 58.8 2 Water Glass Rectangle Clear Closed 24.4 75.7 51.3 3 Water Plastic Rectangle Clear Open 24.1 83 58.9 4 Water Plastic Cylinder Opaque Open 23.6 85.2 61.6 5 Water Plastic Rectangle Opaque Closed 26.7 81 54.3 6 Water Plastic Cylinder Opaque Closed 23.5 85.5 62.0 32 Milk Plastic Rectangle Clear Closed 18.5 82.7 64.2 7. Statistical analysis of the data (Using both JMP and Design-Expert) 7.1 Temperature Response Table 4: Summary of Fit RSquare 0.846565 RSquare Adj 0.773501 Root Mean Square Error 3.194174 Mean of Response 61.69375 Observations (or Sum Wgts) 32 The R-squared value indicates how much of the total variation is explained by the regression model. It is possible sometimes this value to be inflated due to large number of factors included in the model, therefore a R-squared adjusted value is also calculated which takes into consideration the number of factors included in the model. In this case, 77.3% of the variation can be explained by the constructed model of the chosen factors, which is relatively good and gives confidence to the models ability in capturing the source of variation in the factors that we have chosen. 7.2 Factor Evaluation Table 5: Parameter Estimates Term Estimate Std Error DFDen t Ratio Prob>|t| Intercept 61.69375 4.23125 1 14.58 0.0436* Materials[Plastic] 2.075 0.564656 20 3.67 0.0015* Shape[Cylinder] 2.3375 0.564656 20 4.14 0.0005* Colors[Clear] 0.4875 0.564656 20 0.86 0.3982 Cover[Open] 0.4875 0.564656 20 0.86 0.3982 Materials[Plastic]*Shape[Cylinder] 1.49375 0.564656 20 2.65 0.0155* Materials[Plastic]*Colors[Clear] 2.04375 0.564656 20 3.62 0.0017* Materials[Plastic]*Cover[Open] -0.39375 0.564656 20 -0.70 0.4936 Shape[Cylinder]*Colors[Clear] 0.26875 0.564656 20 0.48 0.6393 Shape[Cylinder]*Cover[Open] 0.23125 0.564656 20 0.41 0.6865 Colors[Clear]*Cover[Open] 0.61875 0.564656 20 1.10 0.2862 From the p-values generated by JMP using the custom factorial model, it is evident that the significant factors in this experiment are Materials, Shape, Materials & Shape interaction, and Materials & Color interaction. Once the p-value goes below a certain value determined from the t-table and degrees of freedom, the factors can be declared as being significant. ANOVA for selected factorial model Table 6: Analysis of variance table [Partial sum of squares - Type III] Source Sum of Squares df Mean Square F Value p-value Prob > F Block 575.17 1 575.17 Model 527.03 5 105.41 11.36< 0.0001 significant a-materials 138.33 1 138.33 14.91 0.0007 b-shape 174.53 1 174.53 18.82 0.0002 c-colors 7.74 1 7.74 0.83 0.3698 ab 71.40 1 71.40 7.70 0.0103 ac 135.03 1 135.03 14.56 0.00 08 residual 231.87 25 9.27 cor total 1334.07 31 the results are further corrobarated by running the analysis in design expert and from the f-values, the same factors can be seen as significant. the f-statistic is calculated by taking the proportion of the mean square of the factor by the mean square of the residual 0.0001="" significant="" a-materials="" 138.33="" 1="" 138.33="" 14.91="" 0.0007="" b-shape="" 174.53="" 1="" 174.53="" 18.82="" 0.0002="" c-colors="" 7.74="" 1="" 7.74="" 0.83="" 0.3698="" ab="" 71.40="" 1="" 71.40="" 7.70="" 0.0103="" ac="" 135.03="" 1="" 135.03="" 14.56="" 0.00="" 08="" residual="" 231.87="" 25="" 9.27="" cor="" total="" 1334.07="" 31="" the="" results="" are="" further="" corrobarated="" by="" running="" the="" analysis="" in="" design="" expert="" and="" from="" the="" f-values,="" the="" same="" factors="" can="" be="" seen="" as="" significant.="" the="" f-statistic="" is="" calculated="" by="" taking="" the="" proportion="" of="" the="" mean="" square="" of="" the="" factor="" by="" the="" mean="" square="" of="" the="">