Chi-square Distribution - Essay Example

Chi – Square Test: Sales Software Test There are numerous statistical methods available for forecasting and deciding on the significance of various scenarios. In probability statistics we consider the three techniques of hypothesis testing, parametric analysis and multivariate. These three are most of the time used together since they are just the components of probability analysis for given statistical data. Hypothesis is the first stage in probability analysis where assumptions are made towards the outcome and set the base for testing the variables to be considered. Using the hypothesis together with the set variables, data is collected then tested always with an aim of proving, rejecting or simply inval8idating the null (Ho) hypothesis. We consequently consider the alternative hypothesis usually denoted as Ha and conclude by either accepting as true, the null hypothesis, or rejecting it (Bowerman, O'Connell, Orris & Murphree, 2010). In the case of WidgeCorp Company, we use the parametric analysis which involves categorical data. We are considering the relationship between sales volume and the use of software are not numerical consideration since the sales force is spread over the four regions of Southeast, West, Central and Northeast. Using the stated Nominal variables and Ordinal variables, we are able to do a qualitative classification of the data set. In the non- parametric test, we use the ANOVA technique of analysis. Suitable for this data therefore is the two way ANOVA technique which will allow us to consider the regions where the sample software is used as one way and the sales per region as another (CEC, 2007). Using the Chi-square Distribution By applying the chi-square distribution and analysis to the information provided for WidgeCorp Company, we are able to find whether the data and the chi test are independent and thus be able to accept or reject the null hypothesis. We realize that the sales data and the use of software will present with categorical data which ultimately allows us to use non-parametric analysis through chi squared test. The answers to possible questions from the variables could ‘yes, there is a relationship between sales and use of software’ or ‘no’ (Bowerman, O'Connell, Orris & Murphree, 2010). Considering the distribution of the sales force of WidgeCorp, it is convenient to assume that in each region, there are 500 sales persons which give the total of 500. Since we are told that only half of this number was given the software during the test period, we hypothesize that suppose this number is equally divided for the Southeast and the West region, then all of the agents in these two regions were given the software leading to the chi square analysis below. Ho: Sales depending on the use of software Ha: Sales do not depend on the software Consequently, the data can be presented in a contingency table as indicated below to facilitate the computation of chi squared for the stated hypothesis. Table 1: Contingency table for Regional Usage of software Northeast West Total Used 200 150 350 Did not use 100 100 200 Total 300 250 550 These data forms a 2x2 contingency table from where the chi tests X2 can be computed using the formula: = 2.62 From the calculations we obtain a chi – square of 2.62 with a degree of freedom for the data set 1 (obtained from rows and columns). From the chi distribution table in appendix 1 with section provided below, we obtain a chi square probability of 3.81 which is greater than the calculated x2 and we therefore reject the null hypothesis in this regard. The calculations prove an adequate dependence of sales on the use of software. Using the ANOVA test we are able to test the hypothesis that all sales agents in the four regions sell equal volumes of software leading to the alternative hypothesis stated as below. Ha: sales in NE=sales in SE= Sales in Central =- sales in West. Consequently, the null hypothesis in this case will be that the alternative hypothesis is not true where the sales volume in the three regions will not be the same but vary (Bartholomew, Steele, Moustaki, & Galbraith, 2002). We develop the contingency table below and test as in the subsequent table. Table 2: Contingency table for sales by Region Region NE C W SE Total Used 25 75 62 85 247 Did Not use 75 56 72 50 253 Total 100 131 134 135 500 Table 3: Chi Squared Analysis for Sales volume by Region Observed Expected /O-E/ (O-E)2 (O-E)2/E 25 49.4 24.4 595.36 12.0518 75 64.714 10.286 105.802 1.63491 62 66.196 4.196 17.6064 0.26597 85 66.69 18.31 335.256 5.02708 75 50.6 24.4 595.36 11.766 56 66.286 10.286 105.802 1.59614 72 67.804 4.196 17.6064 0.25967 50 68.31 18.31 335.256 4.90786 X2       37.5095 The degree of freedom for this data is obtained from the expression: DF= (columns-1) (rows-1) = (4-1) (2-1) = 3 Therefore, considering a probability level (alpha) of 0.05 and checking on the chi distribution table given as appendix 1 against the degree of freedom obtained we find a probability of 7.815 which is far less than our computed chi squared value of 37.5095. This therefore confirms the positive hypothesis of a relationship between the use of software and volume of sales and so we reject the null hypothesis. Conclusion Chi-square test rejects the null hypothesis that all the sales agents of company W must sale same volume of the software and so we realize a variation in sales for those who used the software and those who did not use the software. We find from the analysis, that those who did not use the software sold more but cannot hastily conclude based on the sales volume but through the chi squared test which confirms the alternative hypothesis while rejecting the null. The company will have a better conclusion in deciding on whether to use the software in its sales or not if this method of chi squared analysis is adopted. References Bartholomew, D.J., Steele, F., Moustaki, I., & Galbraith, J. I. (2002). The analysis and interpretation of multivariate data for social scientist. Chapman & HaLL/CRC. Bowerman, B.L., O'Connell, R. T., Orris, J. B. & Murphree, E. S. (2010) Essential of Business Statistics (3rd E.d.) McGraw Hill Irwin Boston, New York Career Education Corporation (2007) Statistical test conclusions Chi-square table of critical values. (n.d.). Statistics help. Retrieved November 2, 2012, from http://www.statisticsmentor.com/tables/table_chi.htm Appendix I: Chi alpha distribution table  α  0.995 0.99 0.975 0.95 0.9 0.1 0.05 0.025 0.01 0.005 df=1 --- --- 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 2 0.01 0.02 0.051 0.103 0.211 4.605 5.991 7.378 9.21 10.597 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.86 5 0.412 0.554 0.831 1.145 1.61 9.236 11.07 12.833 15.086 16.75 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 7 0.989 1.239 1.69 2.167 2.833 12.017 14.067 16.013 18.475 20.278 8 1.344 1.646 2.18 2.733 3.49 13.362 15.507 17.535 20.09 21.955 9 1.735 2.088 2.7 3.325 4.168 14.684 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.94 4.865 15.987 18.307 20.483 23.209 25.188 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.92 24.725 26.757 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.3 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 14 4.075 4.66 5.629 6.571 7.79 21.064 23.685 26.119 29.141 31.319 15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32 34.267 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 18 6.265 7.015 8.231 9.39 10.865 25.989 28.869 31.526 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 20 7.434 8.26 9.591 10.851 12.443 28.412 31.41 34.17 37.566 39.997 21 8.034 8.897 10.283 11.591 13.24 29.615 32.671 35.479 38.932 41.401 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796 23 9.26 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.98 45.559 25 10.52 11.524 13.12 14.611 16.473 34.382 37.652 40.646 44.314 46.928 26 11.16 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.29 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.42 76.154 79.49 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 80 51.172 53.54 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169 Source: http://www.statisticsmentor.com/tables/table_chi.htm Read More