The study involves the two variables such as the number of transformers for the year 2007, 2008 and 2009 and the sales of the refrigerator. Both variables have the ratio scale of measurements.
Milestone Two
Southern New Hampshire University
QSO-510
The study involves the two variables such as the number of transformers for the year 2007, 2008 and 2009 and the sales of the refrigerator. Both variables have the ratio scale of measurements. We have to find the linear relationship exists between these two variables. We will use the regression analysis for the prediction purpose of the number of transformers. We would consider the dependent variable as the number of transformers and the independent variable as the sales of the refrigerator (vanElst, 2013).
The data for the number of transformers for the years 2007, 2008 and 2009 is given as below:
| Number of transformers | Year |
| 779 | 2006 |
| 845 | 2007 |
| 857 | 2008 |
| 802 | 2006 |
| 739 | 2007 |
| 881 | 2008 |
| 818 | 2006 |
| 871 | 2007 |
| 937 | 2008 |
| 888 | 2006 |
| 927 | 2007 |
| 1159 | 2008 |
| 898 | 2006 |
| 1133 | 2007 |
| 1072 | 2008 |
| 902 | 2006 |
| 1124 | 2007 |
| 1246 | 2008 |
| 916 | 2006 |
| 1056 | 2007 |
| 1198 | 2008 |
| 708 | 2006 |
| 889 | 2007 |
| 922 | 2008 |
| 695 | 2006 |
| 857 | 2007 |
| 798 | 2008 |
| 708 | 2006 |
| 772 | 2007 |
| 879 | 2008 |
| 716 | 2006 |
| 751 | 2007 |
| 945 | 2008 |
| 784 | 2006 |
| 820 | 2007 |
| 990 | 2008 |
Now, we have to find the descriptive statistics for the number of transformers for the given three years. Descriptive statistics provides us the general idea about the data under study (Reid, 2000). Below, I have included the descriptive statistics for the number of transformers.
| Descriptive Summary | |||
| Number of transformers | |||
| 2006 | 2007 | 2008 | |
| Mean | 801.1666667 | 898.6666667 | 990.3333333 |
| Median | 793 | 864 | 941 |
| Mode | 708 | #N/A | #N/A |
| Minimum | 695 | 739 | 798 |
| Maximum | 916 | 1133 | 1246 |
| Range | 221 | 394 | 448 |
| Variance | 7020.5152 | 18750.0606 | 21117.8788 |
| Standard Deviation | 83.7885 | 136.9309 | 145.3199 |
| Coeff. of Variation | 10.46% | 15.24% | 14.67% |
| Skewness | 0.1223 | 0.7720 | 0.6496 |
| Kurtosis | -1.6266 | -0.6091 | -0.8528 |
| Count | 12 | 12 | 12 |
| Standard Error | 24.1877 | 39.5285 | 41.9502 |
For the variable number of transformers, the average number of transformers for the year 2006 is given as 801.1667 with the variance of 7020.515. The average number of transformers for the year 2007 is given as 898.6667 with the variance of 18750.06 while the average number of transformers for the year 2008 is observed as the 990.3333 with the variance of 21117.88. From this information it is noted that significant differences appear in the average number of transformers within the three years. The coefficient of variation is more for the year 2007 as compare to the year 2006 and 2008. This means there is more variation for the year 2007.
We have to check these significant differences by using the one-way analysis of variance in the next topic.
This is because both deterministic model and stochastic inventory models concur to the fact that the amount of inventory to be ordered or stock emanates from the demand. Indeed the
Anova single factor has facilitated by noting that there is change. i.e The results (F = 6.871 and p = 0.003202) suggest that the mean number of transformers has changed over the period 2006–2008(Chandrakantha, L., 2015).
| 2006 Number of transformers | 2007 Number of transformers | 2008 Number of transformers |
| 779 | 845 | 857 |
| 802 | 739 | 881 |
| 818 | 871 | 937 |
| 888 | 927 | 1159 |
| 898 | 1133 | 1072 |
| 902 | 1124 | 1246 |
| 916 | 1056 | 1198 |
| 708 | 889 | 922 |
| 695 | 857 | 798 |
| 708 | 772 | 879 |
| 716 | 751 | 945 |
| 784 | 820 | 990 |
| ANOVA: Single Factor | ||||||
| SUMMARY | ||||||
| Groups | Count | Sum | Average | Variance | ||
| 2006 Number of transformers | 12 | 9614 | 801.1666667 | 7020.5152 | ||
| 2007 Number of transformers | 12 | 10784 | 898.6666667 | 18750.0606 | ||
| 2008 Number of transformers | 12 | 11884 | 990.3333333 | 21117.8788 | ||
| ANOVA | ||||||
| Source of Variation | SS | df | MS | F | P-value | F crit |
| Between Groups | 214772.2222 | 2 | 107386.1111 | 6.8707 | 0.0032 | 3.2849 |
| Within Groups | 515773.0000 | 33 | 15629.4848 | |||
| Total | 730545.2222 | 35 | ||||
| Level of significance | 0.05 |
For above ANOVA test for checking the significant differences between groups and within groups, we get the test statistic value f as 6.87 with p-value as 0.003 which is very less than the given level of significance or alpha value as 0.05 or 5%. We know as per decision rule we reject the null hypothesis if the p-value is less than the given level of significance or alpha value. Here, p-value is less than the level of significance, so we reject the null hypothesis that there is no any significant difference between the averages of given groups. This means we conclude that there is a significant difference exists between the average values of given groups(Kreinovich&Servic, 2015).
Now, we have to see the regression model for the purpose of estimation of the number of transformers based on the sale of the transformers which is given as below:
| Sales of Refrigerators | Transformer Requirements |
| 3832 | 2399 |
| 5032 | 2688 |
| 3947 | 2319 |
| 3291 | 2208 |
| 4007 | 2455 |
| 5903 | 3184 |
| 4274 | 2802 |
| 3692 | 2343 |
| 4826 | 2675 |
| 6492 | 3477 |
| 4765 | 2918 |
| 4972 | 2814 |
| 5411 | 2874 |
| 7678 | 3774 |
| 5774 | 3247 |
| 6007 | 3107 |
| 6290 | 2776 |
| 8332 | 3571 |
| 6107 | 3354 |
| 6792 | 3513 |
| SUMMARY OUTPUT | ||||||
| Regression Statistics | ||||||
| Multiple R | 0.925948991 | |||||
| R Square | 0.857381533 | |||||
| Adjusted R Square | 0.849458285 | |||||
| Standard Error | 179.467867 | |||||
| Observations | 20 | |||||
| ANOVA | ||||||
| df | SS | MS | F | Significance F | ||
| Regression | 1 | 3485332.925 | 3485333 | 108.2109 | 4.84979E-09 | |
| Residual | 18 | 579756.8751 | 32208.72 | |||
| Total | 19 | 4065089.8 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 1233.499456 | 167.475485 | 7.365254 | 7.79E-07 | 881.646519 | 1585.352393 |
| Sales of transformers | 0.314901799 | 0.030271902 | 10.40245 | 4.85E-09 | 0.251302893 | 0.378500705 |
For this regression model, we get the correlation coefficient between the number of transformers and sales of transformers as 0.9259 which means there is very strong positive correlation or linear relationship exists between the given two variables. The R square or the coefficient of determination is given as 0.8574 which means about 85.74% of the variation in the dependent variable number of transformers is explained by the independent variable sales of transformers. The p-value for this regression model is given as 0.00 which is less than the given level of significance so we conclude that there is a significant relationship exists between the dependent variable number of transformers and independent variable sales of transformers. The regression line for this model is given as below:
Number of transformer = 1233.50 + 0.3149*Sales of transformer
By using this regression equation, we can estimate the required numbers of transformers by using the given sales of transformer.
Next, we have to check the hypothesis or claim and figure out whether the average number of transformers for the year 2006 is less than 745 transformers or not. This will help explain whether the hypothesis was correct. For checking this claim we use the one sample t-test for the population mean (Kreinovich&Servic, 2015). The null and alternative hypotheses for this test are shown below:
Null hypothesis: H0: The population average for transformers is 745.
Alternative hypothesis: Ha: The population average for transformers is less than 745.
Symbolically it is written as below:
H0: µ = 745 versus Ha: µ < 745
This is a one tailed test. This is lower tailed test or left tailed test.
We assume the level of significance or alpha value for this test as 5% or 0.05.
The formula for test statistics is given as below:
Test statistic = t = (Xbar – µ) / [SD / sqrt(n)]
Now, from the given data we have
| t Test for Hypothesis of the Mean | |
| Data | |
| Null Hypothesis m= | 745 |
| Level of Significance | 0.05 |
| Sample Size | 12 |
| Sample Mean | 801.1666667 |
| Sample Standard Deviation | 83.78851444 |
| Intermediate Calculations | |
| Standard Error of the Mean | 24.1877 |
| Degrees of Freedom | 11 |
| t Test Statistic | 2.3221 |
| Lower-Tail Test | |
| Lower Critical Value | -1.7959 |
| p-Value | 0.9798 |
| Do not reject the null hypothesis |
We reject the null hypothesis if the p-value is less than the given level of significance or test statistic value is greater than the critical value (Grusho et al., 2015). So, here, we get the p-value as 0.9798 which is greater than the given level of significance or alpha value 0.05, so we do not reject the null hypothesis that the population average for transformers is 745. We conclude that the population average for transformers is 745.
References
Babbie, E. R. (2009). The Practice of Social Research. Wadsworth. Retrieved from: https://books.google.com/books?hl=en&lr=&id=bS9BBAAAQBAJ&oi=fnd&pg=PR5&ots=Pvu39RtuO2&sig=ABbVdyvTXocc7XHdv9GR9m6dG18#v=onepage&q&f=false
Chandrakantha, L. (2015). Simulating One-Way ANOVA Using Resampling. Electronic Journal of Mathematics & Technology, 9(4), 281-296.
Grusho, A.A., Grusho, N.A., &Timonina, E.E. (2015). Statistical Decision Functions Based on Bans. AIP Conference Proceedings, 1648(1), 1-4. Doi:10.1063/1.4912506
Kreinovich, V., &Servic, C. (2015). How to Test Hypotheses When Exact Values Are Replaced by Intervals to Protect Privacy: Case of t-tests. International Journal of Intelligent Technologies & Applied Statistics, 8(2), 93-102. doi: 10.6148/IJITAS.2015.0802.0
Reid, N., (2000). The University of Toronto. Theoretical Statistics and Asymptotics. Retrieved from http://www.utstat.utoronto.ca/reid/research/neuchatel.pdf
Van Elst, H. (2013). Foundations of Descriptive and Inferential Statistics. Cornell University Library.Retrieved from: http://arxiv.org/abs/1302.2525?
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