Business Statistics and Use of Probability for Inference - Term Paper Example

 Abstract In making inferences on the population based on sample characteristics, one may use parametric or nonparametric statistics. Parametric statistics makes certain assumptions on the distribution of variable values across a population while nonparametric statistics do not. Nevertheless, the study conducted by this writer indicates that there are several disadvantages in using nonparametric statistics. In contrast, the use of parametric statistics can be justified if we have a large sample size (a large sample size is usually considered to be only 30 and above) or there is no strong reason to believe that the population that we are studying is non-normal. This fact is very useful in business research because profitability of operations is of special importance: business research must be inexpensive and yet the research findings must be valid and reliable. Business Statistics and the Use of Probability for Inference In the language of statistics, inference is the use of samples or the characteristics of a subset of a larger population to describe the characteristics of a population. Through the method of making inferences on the population based on the characteristics of a sample, man is able to make a diagnosis of a problem and formulate a possible solution to a problem based on the diagnosis. It is not always possible to conduct a characterization of a population through a one-by-one characterization of all members of a population to characterize a population. The reasons for this are several. Firstly, doing so is usually expensive. Secondly, for large population, it is usually not possible to describe the characteristics of all the members of a population. Lastly and thirdly, getting the data for all the members of the population can be a long process. Unfortunately, man, society, or an organization does not always have the time to wait because immediate action on a problem can be required: the time loss for a few days or years of delay in making an action is just as great as the cost of inaction itself. Thus, a decision may have to be made based on the characterization of the population based on inferences on population characteristics based on sample characteristics. To make certain inferences, man uses probability distribution functions that express a set of assumptions on how variable values are probably or possibly distributed across a population. The assumptions help man to make inferences on the population based on sample characteristics. The assumptions or the probability distribution functions allow man to associate variable values with probabilities. In other words, inferences on population characteristics associate or attach probabilities with regard to variable values in the population using certain assumptions (or which amount to the same thing: the use of a specific probability distribution function). At the same time, there is a class of statistics known as nonparametric statistics. Nonparametric statistics makes no assumption on the distribution of variable values across a population (Walpole et al. 671). 1. Non-parametric Methods Some of the non-parametric method include the Sign Test, Signed-Rank Test, Rank Sum Test, Rank-Sum Test, Kruskal-Wallist Test, Runs Tests, Tolerance Limits Analysis, and the Rank Correlation Coefficient (Walpole et al. 671-696). As suggested by the names of the tests, many of the nonparametric methods involve an analysis of ranks (Walpole et al. 671). Given several statistics under the category “nonparametric statistics”, we focus on one nonparametric statistics just to give us additional ideas on how nonparametric techniques are used. According to Walpole et al., the Sign Test may be appropriate to use if the sample size is less than 30 and there is good reason to believe that the population from which the sample was taken is not normally distributed (672). The null hypothesis for the Sign Test is that the population median is equal to a specific value while the alternate hypothesis depends on the tail of the test. If the test is two-tail then the alternate hypothesis is that the median is not equal to the specific value. In contrast, there are two possibilities for the one-tail test. One possibility is the population median is greater than the specific value and the other possible one-tail test is the population median is less than the specific value. The Sign Test makes use of a binomial probability sums. A binomial probability sums table is found in Table A.1 of Walpole et al. (742-47). Table A.1 in Walpole et al. (742-47) associates a probability value with a computed binomial probability sums. If the computed binomial probability sums is less than the associated probability or p-value associated based on Table A.1 of Walpole et al. (742-747) then the null hypothesis can be rejected in favor of the alternate hypotheses. Otherwise, the null hypotheses cannot be rejected. The alternate hypothesis may be one –tail or two-tail. Should we use nonparametric statistics? Answer: probably not unless there are very strong reasons to do so. According to Walpole et al. (671), there are even several disadvantages associated with the use of parametric statistics. One of such disadvantages is that nonparametric statistics do not use all the available information from the sample and, thus, because of this, nonparametric statistics are less inefficient (Walpole et al. 671). Walpole et al. even prescribe that when both parametric and nonparametric tests are applicable to the same set of data, a researcher must carry out the more efficient parametric tests instead of the nonparametric tests. On the other hand, Kemp & Kemp (300) pointed out that in considering whether to use nonparametric tests, one of the things that we should remember is that the nonparametric tests often use the median rather than the mean as the measure of central tendency. It is also useful to bear in mind the statement made by Moore & McCabe that the tests using assumptions of normality “are not very sensitive to moderate lack of normality, especially when the samples are reasonably large” (see 2nd page of electronic file of Chapter 14 accompanying Moore & McCabe). 2. Chi-Square Distribution The chi-square distribution is represented by the function in Walpole et al. (200). While the population itself may not be parametric, the chi-square distribution assumes that expected values are distributed according to a regular pattern, i.e., distributed according to pattern of the chi-square distribution function is captured by the formula below (taken from Soong 219). Nevertheless, the chi-square distribution is used in non-parametric statistics or where there is no assumption on the distribution of population values (Walpole et al. 201). The figures in Figure 1 are the critical values in a chi-square distribution. The same critical values of the chi-square are in Urdan (178). Comparing the figures in Walpole et al. (755-56) and Urdan (178), it appears that Urdan’s table (178) is the simpler version of the table of Walpole et. al. In the table of Urdan (178), only the figures associated with the key or the usual critical alphas are reflected. Figure 1. Critical Values of Chi-Square Distribution from Walpole et al. (755-56) Figure 2. Table of the Critical of the Chi-Square Statistics in Urdan (178) The chi-square (2) distribution is typically employed for analyzing whether two or more variables are correlated. The variables involved may be discrete or continuous or a combination of the two. The null hypothesis is that two variables are not correlated while the applicable alternative hypothesis is that the variable values are correlated. If the computed statistic is in the critical alphas, then a researcher may choose to reject a null hypothesis and accept the alternative hypothesis. 3. ANOVA Figure 3. Critical Values of F-Distribution from Walpole et al. (757) As mentioned, the chi-square test discussed earlier can be used to test the differences among several population proportions, however, the analysis of variance can be used to test hypotheses among several population means (Kaznier 114). ANOVA is a parametric test because it assumes the sample means were obtained from normally distributed population (Kaznier 114). Nevertheless, it is important to point out that the test is relatively unaffected by violations of the normality assumption when populations are unimodal and when sample sizes are approximately equal (Kaznier 14). ANOVA makes use of parametric statistics because it has certain assumptions on how the population of possible means is distributed. ANOVA makes use of the F-statistics (see Figure 3). The distribution function for the F-statistics is given by Walpole et al. (262) as: The null hypothesis is that all of the means are equal versus the alternative hypotheses that at least two of the means are not equal. There are actually several and not merely two models of ANOVA and the one-way and two-way ANOVAs are only the simplest of the models (Roussas 397). 4. Conclusion All three statistics are valuable in business statistics. Business people may use parametric statistics conveniently because there is a good foundation in theory for its use. Instead of non-parametric statistics for sample sizes less than 30, one may use the t-statistics when there is no strong reason to believe that the population we are studying is not normally distributed. For further studies, a webpage that can be useful is http://www.cob.sjsu.edu/mease_d/bus90.html. Based on research conducted by this work, it appears more useful to use parametric statistics rather than nonparametric statistics unless there are very extremely strong reasons to use the latter rather than the former. Work Cited Kaznier, Leonardo. Business Statistics. London: McGraw-Hill, 2003. Kemp, Steven & Sid Kemp. Business Statistics Demystified. New York: McGraw-Hill, 2004. Mease, David. Lectures on Business Statistics. Accessed 6 May 2010 . Moore, David and George McCabe. Introduction to the Practice of Statistics. 3rd ed. New York: W.H. Freeman and Company, 1999. Roussas, George. Introduction to Probability and Statistical Inference. Amsterdam: Academic Press, 2003. Soong, T.T. Fundamentals of Probability and Statistics for Engineers. West Sussex John Wiley & Sons, Ltd., 2004. Urdan, Timothy. Statistics in Plain English. 2nd ed. London: Lawrence Erlbaum Associates, 2005. Walpole, Ronald, Raymond Myers, Sharon Myers, and Keying Ye. Probability and Statistics for Engineers and Scientists. 8th ed. London: Pearson Education Ltd., 2007. Read More