Quantitative Reasoning - Report Example

Quantitative Reasoning Table of Contents Descriptive Analysis 3 Central Tendency 3 Measures of Dispersion 3 Bivariate Correlation Coefficient 4 Two-sample Student’s t-test 5 References 8 Cohen, J. (2003) Applied multiple regression/correlation analysis for the behavioral sciences (3rd Edition). London, UK: Routledge. 8 Gravetter, F. J. & Wallnau, L. B. (2009) Statistics for the Behavioral Sciences. USA: Cengage. 8 Healey, J. F. (2009) The Essentials of Statistics: A Tool for Social Research. USA: Cengage. 8 Healey, J. F. & Prus, S. G. (2009) Statistics: A Tool for Social Research. USA: Cengage. 8 Reinard, J. C. (2006) Communication research statistics. London, UK: Sage. 8 Woodbury, G. (2001) An Introduction to Statistics. USA: Cengage. 8 Bibliography 8 Crawley, M. J. (2005) Statistics: an introduction using R. USA: Wiley. 8 Appendix 9 Descriptive Analysis Descriptive statistics help a researcher to analyse the basic characteristics or features of a set of observations underlying a variable (Healey & Prus, 2009). Central Tendency and Measures of Dispersion could be considered as two sections of the subject primarily targeted towards yielding descriptive details of a given set of observations. In the present paper, descriptive analysis for percentage of smokers in the population for the year 2006, have been conducted. The core target here will be to compare between the distribution of the male and female smokers in the given year. The following paragraphs will separate the results obtained after employing descriptive statistic tools to the pair of observations. Central Tendency These tools figure out the mean and the median percentage of smokers in the population, distinguished on the basis of gender. Mode has been ignored in this respect since it is irrelevant to figure out the maximum percentage to assess the average characteristics. The mean as well as the median percentage of smokers is lower for females than for males. An obvious implication from the same is that males are more inclined towards smoking than their female counterparts. Here mean is the weighted average implying approximately 19% and 27% of the female and male populations in any nation to be regular smokers, respectively. On the other hand, the median value indicates that among all nations taken collectively, in half of the cases, more than 18.9% of the females are found to smoke, in contrast to 24% among the males (Gravetter & Wallnau, 2008). Measures of Dispersion Measures of dispersion indicate the degree to which the observations are scattered around the mean value. The higher the value of the measure, greater will be the dispersion about the mean and thus, the applicability of the mean value as a core feature of the population will be disturbed. Significance of standard deviation is almost equivalent to that of variance given that the former is the positive square root of the latter. The percentage of female smokers distributed among various groups in the entire population differs from the mean value so calculated, though it is lower than the degree of dispersion of the male smokers. The variance and standard deviation statistics yield that percentage of male smokers in some nations is much higher as well as lower than the mean percentage of male smokers so calculated. However, similar statistics for females rule out such extremeness. Thus, the applicability of the mean percentage of female smokers in the population is more appropriate than it is for the percentage of male smokers in the population. The adjoining diagram compares the percentage of male and female smokers in the entire population for each of the nations being considered in the test. The graph points out the truthfulness of the above central tendency measures. The mean percentage of women and men smokers is quite evident over here, and so are the medians of the same. Bivariate Correlation Coefficient The purpose of bivariate correlation coefficient is to figure out the level of association between two variables. This level of association implies the degree to which the specific variables move in line with one another, i.e., the scale to which the trends in their movements are related to one another. However, there are common instances when the trends in two completely unrelated variables are found to be immensely attached with each other, and this is when the outcome is considered as a spurious one. In other words, prior to finding out the correlation coefficient between two variables, it is important to seek out the presence of any economic theory in support of their empirical relation (Cohen, 2003). The present section will attempt to figure out whether there exists any association between the percentage of male and female smokers in 2006. Calculating the bivariate correlation coefficient will be the ideal step in this regard where the prime target will be to figure out any compliance in the trends that the two variables follow. If the estimated figure is found to be highly positive or negative, the trends in any one of the variables could be predicted from the other. However, it does not necessarily indicate the prevalence of a unidirectional or a bidirectional causal relation between the variables (Healey, 2009).   Percentage of Female Smokers in 2006 Percentage of Male Smokers in 2006 Percentage of Female Smokers in 2006 1 Percentage of Male Smokers in 2006 -0.03250293 1 The estimated correlation coefficient statistic in this respect yields the value -0.0325 approximately, which is rather considered to be a quite low value. The absolute value together with the negative sign indicates that trends in the percentage of male smokers could be predicted by upto 3.25% from the trends in the percentage of female smokers and vice versa. Moreover, the two variables move in a negative compliance with one another as far as the present sample data is concerned. Thus, in the present case it could be said that there exists a poor and negative association between the percentage of male and female smokers in 2006. In fact, the poor association is also relevant from the scatter plot depicted underneath. It shows a clear negative association between the male and female percentage of smokers in the sample nations. Though in some of the nations, the percentage is almost equal and hence, predictable for one when the other is known, in most of the others, there exist a wide gap between the two variable values, which is reflected in the low correlation coefficient. Two-sample Student’s t-test The purpose of Two-sample Student’s t-test is to compare between the mean values of two pairs of observations. It is aimed at figuring out whether there exists any difference between the given two pairs of data each of which belong to the same parent set of observations. In most of the cases, this test is used to assess any potential changes which might have occurred due to any difference in the time of sampling or to evaluate the effect that any external factor might have on the variable (Reinard, 2006; Woodbury, 2001). Here the purpose will be to assess whether there exists any difference in the mean percentage of male smokers between the years 1990 and 2006. To be exact, the idea is to fathom whether the mean percentage in 1990 is significantly greater than that in 2006 or not. The appropriate test statistic in this regard will be two-sample Student’s t-statistic for which the null hypothesis will be, H0: Mean percentage of male smokers in 1990, M1990 = Mean percentage of male smokers in 2006, M2006, against the alternative hypothesis, H1: M1990 > M2006. The estimated result of two-sample Student’s t-statistic has been presented in the following table and is found to be equal to 2.3204. In order to figure out the fate of the null hypothesis, the predicted value has to be compared with the tabulated value at the given degrees of freedom and assumed level of significance. Level of significance implies the area under the cumulative probability distribution curve which falls under the rejection region. Given these conditions if the tabulated value is found to be greater than the estimated one, the conclusion would be that the predicted value falls within the acceptance region of the cumulative frequency distribution curve and so, the relevant null hypothesis cannot be rejected. The reverse conclusion is drawn in case that the tabulated value falls short of the estimated one. An alternative analysis could be based on p-value comparisons, where p-value implies the level of significance of the estimated statistic. If the estimated value is found to be greater than the assumed level of significance, the conclusion would be that the predicted value falls within the acceptance region of the cumulative probability distribution curve leading to a rejection of the null hypothesis. Similarly, if the estimated p-value is lower, it implies rejection of the null hypothesis at the given level of significance. As the following table will reveal, the critical Student’s t-statistic value is approximately 2.32039743. Given 15 degrees of freedom and assumed level of significance, α = 0.05, the tabulated Student’s t-statistic value is 1.75305033, which clearly is lower than the tabulated value. Moreover, the p-value of the estimated statistic is 0.0174 approximately, which is lower than the assumed level of significance, α = 0.05. Hence, the null hypothesis in this case has to be rejected at 5% level of significance. Thus, there exists significant difference in the mean percentage of male smokers between the years 1990 and 2006. % of smoker male in 1990 in certain countries % of smoker male in 2006 in certain countries Mean 37.0555556 27.3555556 Variance 102.532778 54.7427778 Observations 9 9 Hypothesized Mean Difference 0 df 15 t Stat 2.32039743 P(T Read More