Maximizing Popping of Corns - Math Problem Example

Student Instructor Professor Course Maximizing Popping of Corns Experiment Description It is well known that not every kernel of the corns pops. During the preparation of pop corns, some un-popped corns remains at the bottom of the baking bag and as result of this, the quality of the pop corns reduces. Because of this, a number of questions remains ringing in the minds of many people in relation to the causes of the un-popped corns. For instance, if you take your time waiting for the popping of the corns and identify the number of variables that contributes the level of the popped corns, you might end up listing more than ten variables. In this case, only three variables are chosen out of the possible variables that can be identified as the causer of the popping of corns (Khan, 2013). The factors considered were; time of cooking, temperature subjected to the corns and the type of the corns used with an objective of maximizing the popping of corns. Factor Levels and Range In all these three factors, each factor was categorized into two levels; low or high. This levelling of the factors was used in determining the effect of popping with the change from the actual values. Table 1 shows the summary of the factors, their levels and range. Table 1: Summary of the Factors, Factor Level and their Ranges Factor under consideration Low level High levels Type of pop corn White (-) Yellow (-) Cooking time 20 (-) 40 (+) Base and Top Coat 140 (-) 160 (-) In addition to the factors considered, the following factors also were perceived to contribute to the popping of corns and as a consequence, they were kept to constant. Ambient temperature Type of cooking utensil Surrounding environment Amount of oil used in cooking the corns Response variable Following to the objective of the experiment of determining the factors that might contribute to the maximization of the popping of the corns and their contribution towards this, a choice on the number of popped corns out of the sample size of 300 corns was done. In this case, the number of popped corns under the different conditions was counted and recorded against the possible combinations of experiment. In order to obtain the required results with minimal interference from the environment, the experiment was carried out with carefulness by controlling the effect of the surrounding from the factors under consideration. Experimental procedure Ensure that all the environmental impacts other than the factors under considerations are controlled. Ensure that the corns are free from dust and any other foreign particles Engage the Subject the corns to the same conditions of the factors under considerations while varying one at time. Record the number of popped corns for every experiment under consideration Experimental Design By considering the two levels of each factor, the type of corn was taken as the categorical factor while the other two factors were taken as quantitative factors. As it shown in table 1, each factor is categorized into two levels. In our design, the experiment requires factorial design because there are three factor and two level for each factor. Thus, in order to increase the level of accuracy, the trials will be replicated two time making a total of sixteen runs. As a consequence, the effect of each factor on the performance of popping of corns will be measured at different levels of the factors under consideration. In this case, the number of popped corns will be used as the determining factor of the contribution of the factor varied. In addition to that, an interaction effect between and among the factors will also be conducted. Through this, the adequacy model check will be performed. Table 2 shows the summary of the design matrix used. Table 2: Summary of the Design Matrix Used Std Order Run Order Center Blocks Pop Corn Type Cooking Time Temperature 1 1 1 1 White(-1) 20 140 5 2 1 1 White(-1) 20 160 11 3 1 1 White(-1) 40 140 10 4 1 1 yellow(+) 20 140 3 5 1 1 White(-1) 40 140 16 6 1 1 yellow(+) 40 160 12 7 1 1 yellow(+) 40 140 4 8 1 1 yellow(+) 40 140 7 9 1 1 White(-1) 40 160 14 10 1 1 yellow(+) 20 160 15 11 1 1 White(-1) 40 160 2 12 1 1 yellow(+) 20 140 9 13 1 1 White(-1) 20 140 8 14 1 1 yellow(+) 40 160 6 15 1 1 yellow(+) 20 160 13 16 1 1 White(-1) 20 160 From the above design matrix, it can be observed that the two levels considered for each factor for purpose of analysis after considering the type of corn as a numerical value is as shown in table 3 below. Table 3: Summary of Coded Factors and Levels and Numerical value Factor under consideration Low level High levels Type of pop corn White (-) Yellow (-) Cooking time 20 (-) 40 (+) Base and Top Coat 140 (-) 160 (-) In this case, the blocked design was chosen in order to minimize the nuisance factor. As a result, the figure 1 shows the randomized complete block design of the 2 blocks in which each contains eight runs that represents the replicates that was generated from Minitab. Figure 1: Randomized Complete Block Design STATISTICAL ANALYSIS When the experiment was run, a total of 16 responses were obtained and were added into the design matrix according to the run order. The statistical analysis was done using Minitab in order to obtain the ANOVA tables, interaction plots, residual plots, normal probability plots and cube plots as explained in each of below: ANOVA Table The main ANOVA analysis was carried out starting from the main effects, that is, cooking time, temperature and the type of corn used. As a consequence, figure 2 shows the results obtained from Minitab softaware Figure 2: ANOVA Analysis of the Effect Factors only By considering the results of the ANOVA table above, the significant value is at p-value is 0.709. Thus, the two factors with great impact on the popped corns is the temperature and cooking time with p-values of 0.709 and 0.664 respectively. In addition that, a consideration on the effect of interaction with the two or three factors was also done and it gave the following ANOVA table as shown in figure 3 below: Figure 3: Effect of two or three factor interaction From the results of the two or three interaction factors, it was found that none of the values was significant but cooking time and temperature interaction and AC showed the effect of maximizing popping. Thus, these factors were included in our model to determine their effect to popping of corns. As a result of the above, it was observed that the factors type of corn, cooking time and AC and BC interaction contributed to the quality of popping as depicted by the standardized normal plot of figure 4 below: Figure 4: Normal plots of the Standardized effects Thus the fitted model designed from the Minitab software was as follows; For the coded factors, the actual equation was Thus For the case of the actual factors, Thus ADEQUACY CHECKING Normal probability plot Figure 5 shows the normal probability plot of the data based on the residual values. Figure 5: Normal probability plot From the plot, it is observed that the plot has a flat pencil plot. Thus, the ANOVA assumes a normality whose errors are at a mean of zero and standard deviation of zero. Residual VS Predicted Figure 6 shows the plot of residual vs. the predicted of popping. Figure 6: A plot of Residual Vs. Predicted From the graph, it can be observed that the structure is structure-less. Therefore, constant of variance holds Conclusion and Recommendation The experiment to determine the effect of cooking time, temperature and type of popcorn used in maximizing popping has been done. By basing on our analysis from ANOVA, it has been determined that factors A, B, C, AC and BC are the factors that contributes greatly to the outcome of the popping. Thus, has it was predicted, yellow corns gives better results on popping. In addition to that, popping is also affected by the time of cooking and temperature the corns are subjected to. Thus, it is recommended to vary the amount of temperature, select best corn seeds, yellow seeds and increase time of cooking in order to achieve the best results. In addition to that, future experiments need to be done in order to determine other factors that might lead to reduction of popping in popcorn seeds. Work cited Khan, R. M. Problem solving and data analysis using Minitab: A clear and easy guide to six sigma methodology (1st ed.).2013. 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