Executive summary
The aim of the report was to determine the consumption pattern as well as the magnitude and direction of the correlations of various inputs in the production process of aluminum grade. Effective control of inputs cost is a result of efficient analysis. Therefore, the report analyzed the electrical as well as chemical and plumbing to compare their respective amount and standard deviation. The study showed that the process produces more plumbing and chemical grade as compared to electrical aluminum grade. Again, the report probed on the appropriate purchase amount for the alumina against a set value. Statistical analysis confirmed that the manager decision to rely on the set amount, which was 2.69 tons of alumina, was reliable. Moreover, the study sought to determine whether there exist a difference between the amount of electrical grade produced at the plant and the amount of total chemical/plumbing and sheet metal grade (put together) produced. The paired t-test indicated that there is significant statistical difference between the two outputs. More specifically, electrical grade was less than the other two combined.
In order to establish the nature of relationship between pot temperature and power consumption, the report undertook regression for the two variables. The findings indicated that there is an inverse relationship between the two variables. Therefore, since an increase in pot temperature reduces power consumption, the manager can decide to increase the temperatures during the aluminum production to cut the power cost. Additionally, the coefficient of determination was substantially low. This results from violation of the normality assumption in the power consumption data. Consequently, factors outside the regression model accounted for significant portion of changes in the power consumption. Further, a dot graph proved that there exists a linear relationship between the regressed variables, which is important in ascertaining the suitability of the model. Finally, the estimated model forecasted the values of power consumption given several values of pot temperatures. The results proved that an increase in the value of the pot temperature decreases the power consumption.
1.0 Introduction
Customarily, an organization management ensures that different departments record and maintain sizable data that bears insignificant meaning to the management in the raw form. Therefore, statistical analysis puts to use the numerous data in such departments as the procurement and production to aid in the decision-making and drafting of the strategic plan. More particularly, for the aluminum-producing plant to optimize the resources, it must evaluate its policies and plans based on reliable analysis of the recorded data over time. As such, the report will focus on analysis and interpretation of the historical data recorded. Also, the report will fit models capable of making the forecast with high precision levels. Firstly, the study will apply descriptive statistics to compare the outputs alongside their variability using measures of central tendency such as the means as well as the standard deviation. Moreover, to aid the production manager in setting the appropriate purchase amount for the alumina, which is a significant input, the study will apply one sample t-test against the set value. The t-test will test whether the hypothesized value is different from the average consumption of the alumina in the production process. Besides, to make an informed decision in determining whether there exist a significant difference between two of the aluminum grade produced, electricity and combined chemical/plumbing and sheet metal, the analysis will apply the paired t test. Using the same test, the study will determine the nature and the magnitude of the correlation between the two variables. The aim of the report was to determine the consumption pattern as well as the magnitude and direction of the correlations of various inputs in the production process of aluminum grade. Effective control of inputs cost is a result of efficient analysis. Therefore, the report analyzed the electrical as well as chemical and plumbing to compare their respective amount and standard deviation. The study showed that the process produces more plumbing and chemical grade as compared to electrical aluminum grade
Evidently, to understand and predict a relationship between one response and one or more explanatory variable(s) it is prudent to use linear regression model. Thereby, the study will apply a linear regression model to determine the relationship between power consumption and the pot temperatures. Moreover, the report will determine the suitability of the model using the ANOVA as well as ascertain the percentage of the variable response change explained within the model. After that, the investigation will interpret the coefficient and test their statistical significance in the model. It is equally important to note that the report will investigate whether there is a linear relationship in the regression model using a dot graph with a line of best fit. At this point, the estimated model will predict the estimates of power consumption using different values of pot temperatures. Again, histograms will be used in determining the normality of the variables factored in the regression model to establish ways of improving the coefficient of determination. Finally, the report will list the recommendations based on conclusions derived from the report.
The total electrical grade aluminum produced is 37.199000 tons for all the pots while the average production amounts to .46498750 tons per pot. The standard deviation for the electrical grade is .051348029; it implies that the values of aluminum produced in each pot are close to the mean because it is relatively low. On the other hand, the total chemical and plumbing aluminum produced for all the pots amounts to 44.562072 tons while the mean output for each pot averages to .55702590 tons that are higher than the output for the electrical grade. Similarly, the standard deviation of the output in the chemical and pluming is higher. This shows that the output per pot is widely spread. The standard deviation is also consistent with the range per pot, which is higher in the chemical and plumbing than in the electrical output. Subsequently, it implies a wider deviation from the mean. This discussion basis its analysis from table no.1
The above test assumes that the sample observation was random for scientific reasons. Besides, the variance of the population is unknown while its distribution is normal and its mean known. Again, the cases should be dependent (MCAfee 2011).
Therefore, in line with the stated assumptions above, the average alumina consumption is appropriate at an average level of 2.69 tons per pot. The table no.2 above indicate that the p-value {Sig. (2-tailed)}, 0.000, is less the 0.05 which is the acceptable level of significance chosen for this test. Impliedly, the value of the average alumina consumption is not significantly different from the hypothesized value, 2.69 tons.
Likewise, the fact that both lower and upper boundary bear negative signs indicates that the test value is not significantly different from the average alumina consumption per pot. Hence, the analysis shows consistency and reliability and the manager can confidently make amount purchase decision based on the value suggested.
The paired sample test takes into account several assumptions before the conducting the test. Firstly, there is an assumption that the data is normally distributed, and only the matched pairs are fit for the test (Verma 2013). Thereby, the samples size for variables under consideration must be equal. Additionally, there is an assumption of equal variances for two samples while the cases should be independent. Thus, in consideration of the enumerated assumptions, the standard deviation for the electricity grade is lower than the total for the chemical/plumbing and metal sheet grade as indicated in the table no.3 above. This implies that the values of output for the grade with a lower standard deviation are closer to the mean in comparison with one having a higher standard deviation.
The Sig. (2-tailed) value, 0.000, recorded in the paired sample test table below, Table no.5, is less than the set level of significance. This implies that there exist a significant difference between the amount of electrical grade produced at the plant and the amount of total chemical/plumbing and sheet metal grade (put together) produced. On average, the amount of the electrical grade produced is .466225900 tons less than the combined amount of chemical/plumbing and sheet metal grade (put together).
For the linear regression model to give optimum results, it should factor some assumptions. First, the data set should contain one response variable and one or more explanatory variable(s). Also, the relationship between the explanatory and response variable should be strictly linear (Gro€ 2003). More importantly, all the variables in the model should be multivariate normal to produce a suitable regression line and strong coefficient of determination. Still, the regression analysis requires that the data exhibit little or no autocorrelation, that is, the variables should be independent.
In the regression model for this report, power consumption is the response variable since it is determined by the amount of temperature applied in the process. On the other hand, pot temperatures data represent the explanatory variable. This is so because other factors that an outside the regression model determine its values. The equation below represents the estimated model;
Y= β0+ β1X
Where;
Y = power consumption
X = pot temperature
β0= is the autonomous power consumption that is not influenced by the pot temperature which is constant
β1= is the coefficient for the power consumption that results from the pot temperature
The table below shows that the Sig. Value, 0.00, is less than the hypothesized level of significance in the model. Subsequently, this implies that the model is statistically significant from table no.7 below. As such, the model is fit to analyze the data provided as well as make the prediction for the purpose of management.
5.3 Analysis of the coefficients in the fitted regression model
The table above shows that the constant represented by β0 amounts to 16.190 (x 1000 kWh) while the value of the coefficient for the pot temperature is -.717 o C from table no.8 above. Deductively, the results imply that a decrease in temperature by .717 o C leads to an increase in the power consumption by one unit during the aluminum production process. Additionally, the Sig. Value for both the constant and the pot temperature coefficient are less than 0.05; this implies that they are statistically significant from zero. Therefore, the estimated model is as shown below;
Y= 16.190 - .717 X
Similarly, the relationship between the power consumption and the pot temperature in a line graph is as illustrated below. The line has a negative gradient implying the inverse relationship between the power consumption and the temperature. Moreover, the fact that there exists a line of best fit is an indication that the relationship between the response and explanatory variable is linear as shown in graph no.1 below.
Graph no.1
Source: Author
5.4 Forecasting using the estimated linear regression model
Where Y represent the estimated power consumption while X in the pot temperature. Thus, by substituting for X with 111, 940 and 890 you obtain the solutions as -63.397, -657.79 and -621.9 (x 1000 kWh) respectively.
Causes of low R-Squared
The first histogram indicates that in the pot temperature the level of normality is acceptable. However, the second histogram the power consumption data violates the assumptions of the linearregression model, which is the underlying cause of a low R-squared. Therefore, an increase in the normality of power consumption data would yield better R-squared as shown in graph no. 2 below.
Graph no.2
Source: Author
6.0 Conclusion
The analysis of the aluminum production data provided for this report yields a couple of conclusions. Firstly, the analysis shows there is the higher production of the chemical and plumbing aluminum alloy. Thus, the plant manager should engage most of his/her sales people to the marketing of the product that is in abundance. Similarly, if the demand for the electrical aluminum grade is higher, then the plant should increase its production levels to meet the demand. Moreover, the average production of the electric aluminum is closer to the average value; this means that there is consistency in production, and thus the manager would reliably meet the sales orders. One sample t-test confirmed that the manager could make purchase order based on the value that was initially set since the analysis provided statistical evidence that the value is not significantly different from the average input on alumina. The report probed on the appropriate purchase amount for the alumina against a set value. Statistical analysis confirmed that the manager decision to rely on the set amount, which was 2.69 tons of alumina, was reliable. Moreover, the study sought to determine whether there exist a difference between the amount of electrical grade produced at the plant and the amount of total chemical/plumbing and sheet metal grade (put together) produced. The paired t-test indicated that there is significant statistical difference between the two outputs. More specifically, electrical grade was less than the other two combined. Additionally, the coefficient of determination was substantially low. This results from violation of the normality assumption in the power consumption data. Consequently, factors outside the regression model accounted for significant portion of changes in the power consumption. Further, a dot graph proved that there exists a linear relationship between the regressed variables, which is important in ascertaining the suitability of the model. The relationship between the power consumption and the pot temperature in a line graph is as illustrated below. The line has a negative gradient implying the inverse relationship between the power consumption and the temperature. Moreover, the fact that there exists a line of best fit is an indication that the relationship between the response and explanatory variable is linear
Again, the study found that the production of the grade electricity aluminum and the total for the chemical/plumbing and metal sheet grade has a positive correlation. Therefore, if the plant receives an increase in sales order of one of the product it has to lay out marketing strategies for other products as they will also increase. On the other hand, reduction in demand for one product would adversely affect the other products since their production would reduce because of the output that is in low demand. However, despite the fact that the products have positive correlation the paired t-test indicated that production of electrical aluminum is different from that chemical/plumbing and metal sheet grade. Hence, cycles of its demand would have an insignificant effect on the demands of other products. Additionally, the analysis of regression model shows that power consumption is inversely correlated with the pot temperatures, thereby, increase in power temperature decrease the cost of power. Moreover, lack of normality in the power consumption data led to the low coefficient of determination in the linear relationship between power consumption and the pot temperatures.
7.0 Recommendations
The conclusion resulting from the analysis should be the basis of recommendations to the manager. Firstly, the plant manager should make sure that chemical and plumbing aluminum is sufficient to meet the sales demand in case the production is low since it has high standard deviation. Still, the manager should always keep the production of chemical/ plumbing and metal sheet grade high to have enough electrical aluminum grade since their production is different. To keep the carriage and storage cost low, the manager should maintain the amount of purchase order at 2.69 tons. This would also reduce opportunity cost if the alumina is too low to support of production process. It is essential to remember that the aim of the report was to determine the consumption pattern as well as the magnitude and direction of the correlations of various inputs in the production process of aluminum grade. Effective control of inputs cost is a result of efficient analysis. Therefore, the report analyzed the electrical as well as chemical and plumbing to compare their respective amount and standard deviation. The analysis from the linear regression suggests that the manager should increase the pot temperature to reduce the cost of power. Therefore, reduced cost will increase the profit margin for the plant. Also, the managers should ensure normality in the power consumption data to increase the coefficient of determination. Consequently, the prediction of the model will yield the better result.
References
Gro€, J. 2003. Linear Regression. Berlin, Heidelberg, Springer Berlin Heidelberg. Available at: http://dx.doi.org/10.1007/978-3-642-55864-1
MCAfee, G. 2011. Master math AP statistics. Boston, MA, Course Technology PTR. Available at:
http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=325093.
Verma, J. P. 2013. Data analysis in management with SPSS software. New Delhi, Springer. Available at: http://site.ebrary.com/id/10656855.
