JOBINC: Quantitative Analysis - Assignment Example

Data Analysis and Reporting Data Analysis Part Quantitative analysis refers to a statistical technique that deals with numerical data in estimation of model parameters (Day & Underwood 1986). This study will seek to perform various analysis, stating the underlying assumptions and interpreting the results obtained from the SPSS output. The variable under study will be JOBINC – Applicant’s monthly income from current job. a) The graphical representation of JOBINC was best shown using a histogram which is a pictorial representation of data, a histogram is very useful in displaying continuous and categorical data variables and it shows the number of people in each category. From the graph it is evident that the monthly income for the current job varies at different frequencies and the graph is not normally distributed. JOBINC histogram clears indicates a sharp drop below 20 and therefore this implies that most applicants earn a monthly income of between 0-1,000 while only few applicants earn a monthly income of above 1,000. It is therefore worth noting that most KDS customers earn very little from their current jobs. b) Descriptive statistics is a very crucial aspect in quantitative data analysis as it helps us to organize, summarise and present data in a convenient and informative manner. They are also used to describe the basic characteristics of data collected from a study in different ways (Holcomb 1998). Descriptive statistics includes the measures of central tendency, measures of dispersion, variance, skewness and kurtosis. Statistics Jobinc N Valid 106 Missing 7 Mean 994.8868 Std. Error of Mean 58.00976 Median 800.0000 Mode 600.00a Std. Deviation 597.24699 Variance 356703.968 Skewness 2.085 Std. Error of Skewness .235 Kurtosis 6.175 Std. Error of Kurtosis .465 Range 3700.00 Minimum 300.00 Maximum 4000.00 Sum 105458.00 a. Multiple modes exist. The smallest value is shown Measures of central tendency These refers to the single numbers that are used to summarise a given set of data in a distribution. There are three measures of central tendency namely mean, median and mode. Mean is the sum of all the values divided by the total number of values it is also referred as the average (Holcomb 1998). From the summary statistics the mean of JOBINC is 994.8868 and therefore the average monthly income of the applicants in KDS is 994.8868. The median is another measure which shows the middle value in an ordered set of data (Holcomb 1998). In determination of the median the first step is to arrange the data in a systematic manner. The middle value for JOBINC is 800, this implies that the mean is greater than the median of JOBINC. Additionally, the mode represents the value that appears most often in a given distribution that the value with the highest frequency. From the summary statistics the mode is 600 implying that the most applicants earn a monthly income of 600. Measures of dispersion These measures enable us to determine how far or near the observed values are spread from the average value. They show the extent to which these value differ from the mean (Weinberg & Abramowitz 2008), when the values are close to the mean we say that they have a low dispersion and the converse is also true. One of the measures of dispersion is the range which refers to the difference between the highest and the lowest values in a data set. The range only uses two values, maximum and minimum values. The range of JOBINC is 3700, this value of the range is slightly affected by some outliers in the data. 3700 shows the difference between the maximum income that an applicant earns monthly (4000) and the minimum income earned by an applicant monthly (300). Another measure of dispersion is the variance and standard deviation. Variance can be defined as the sum of square deviations divided by the total number of observations whereas the standard deviation is the square root of variance, the variance of JOBINC is 356703.968 and the standard deviation is 597.24699. These values represents the deviations of the monthly income in the current job from the average income. Skewness Skewness determines the level of symmetry in a distribution. A skewed distribution occurs when higher frequencies appear in one end of the distribution, the value of skewness for JOBINC is 2.085, this implies that the distribution is positively skewed in the sense that mean ˃ median ˃ mode. Kurtosis Kurtosis is a measure of normality or shape which explains the way data is distributed. It describes the degree of peakedness or steepness of a distribution (Weinberg & Abramowitz 2008), JOBINC has a kurtosis of 6.175. Between -4 and 4 is sufficient to approximate to a normal distribution which is the value for a normal distribution the monthly income is leptokurtic implying highly peaked. This situation is best shown by the histogram. c) Significance tests on two and more than two population means. T test is a statistical tool that is used in comparing the one and two population means (Weinberg & Abramowitz 2008). An independent sample t test enables us to compare two different population means. From the SPSS output, we wish to understand if the applicants monthly income is different based on the sex of the applicants, that is, the dependent variable will be represented by JOBINC while the independent variable will be represented by sex which has two groups: 1 to represent male and 0 to represent female. Independent Samples Test Levenes Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper Jobinc Equal variances assumed 10.470 .002 -3.432 104 .001 -544.00147 158.49568 -858.30435 -229.69858 Equal variances not assumed -7.783 103.263 .000 -544.00147 69.89989 -682.62722 -405.37571 Hypothesis: HO: The monthly income for male and female is different H1: The monthly income for male and female is not different The Levene’s test for equality of variances in the output is 0.02 which is less than 0.05. Therefore we fail to reject the null hypothesis and conclude that the applicant’s monthly income in the current job is significantly different in male and female. The significance 2-tailed t-test is 0.01 with 104 degrees of freedom, and since 0.01 is less than 0.05, we conclude that there is a statistically significant difference between the mean monthly income in the current job for male and female applicants. The 95% confidence interval for the difference is (-858.30435, -229.69858), this interval shows that since zero is not included in the interval, then there is a significant difference in the mean of the monthly income for both sexes. However, when the independent sample t-test is only used for comparing the means of two populations, one–way ANOVA is used to compare means for more than two populations. The variables under study are JOBINC and sex of the applicants. The hypothesis: HO: The applicant’s monthly income is different between male and female H1: The applicant’s monthly income is not different between male and female ANOVA Jobinc Sum of Squares df Mean Square F Sig. Between Groups 3810894.491 1 3810894.491 11.781 .001 Within Groups 33643022.151 104 323490.598 Total 37453916.642 105 The ANOVA analysis an F value of 11.78 and comparing this value with significant level of 0.01. The significance level is less than 0.05 therefore, we conclude that there is a statistically significant difference between the applicant’s monthly income and sex. (Appendix 4). Both the independent sample t-test and the one-way ANOVA work under some underlying assumptions and they include; i. The dependent variable should be a continuous variable and also should follow an approximately normal distribution. ii. The independent variable should contain two or more groups iii. Existence of outliers should not be allowed. Part 2: Regression analysis Regression analysis is a statistical concept that refers to the changes that occur in the dependent variable occasioned by the changes in the independent variable, it also measure the level of linear relationship between the two variables under consideration (Seber, Lee & Wiley 2003). It is one of the best model that is used in predicting/explaining one variable given another. For the purpose of our study we will have one dependent and one independent variable: JOBINC and age respectively. Regression analysis will enable us to predict the monthly incomes that a given number of applicant’s based on age will earn in a particular period of time. The general representation of regression analysis is; dependent variable = a + b (independent variable) where a represent the intercept and b represents the slope. For a regression model to hold, the following basic assumptions apply; i. There exists a linear relationship between the variables ii. The error term should be normally distributed and also independent iii. The observed values of the predictor variable should be independent iv. The expected value of the random error term is zero v. The variance of the error term is a constant value Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B B Std. Error Beta Lower Bound Upper Bound 1 (Constant) 294.087 156.171 1.883 .062 -15.605 603.779 age 21.752 4.562 .424 4.768 .000 12.706 30.799 a. Dependent Variable: jobinc Violating none of the above constraints, the analysis can be done and good results will come forth. The coefficient table gives the coefficients used for formulation the regression model; JOBINC = 294.087 + 21.752 (age) Model Summaryb Model R R Square Adjusted R Square Std. Error of the Estimate Durbin-Watson 1 .424a .179 .172 543.62463 1.961 a. Predictors: (Constant), age b. Dependent Variable: jobinc From the output, the model summary table gives the value of R which is 0.319 which is the correlation coefficient, 0.424 shows low degree of correlation between JOBINC and age. Another column gives the value of R2 which is 0.179. R2 is the percentage of total variation that is explained by the independent variable therefore 17.9% is the total variation in JOBINC that is explained by sex of the applicants while the remaining 82.1% is the total unexplained variation. Hypothesis: HO: The monthly income for the applicants can be estimated by their age (best fit) H1: The monthly income for the applicants cannot be estimated by their age ANOVAa Model Sum of Squares df Mean Square F Sig. 1 Regression 6719031.890 1 6719031.890 22.736 .000b Residual 30734884.752 104 295527.738 Total 37453916.642 105 a. Dependent Variable: jobinc b. Predictors: (Constant), age The ANOVA table on the other hand gives the significant p value of 0.001 which is less than 0.05 and therefore we conclude that the model is a best fit for the data. Thus the monthly income for the current job can be estimated by the age of the applicants. This table also gives the 95% confidence interval for the coefficients of the model and the normal p-p plot are used to test the normality of the residuals in the model. From the p-p chart the points are scattered randomly with some values deviating from the regression line, this points are known as outliers. Therefore, the residuals are not normally distributed thus breaking the assumptions of regression analysis. Once the model has been developed and analyzed it is also important to note some of the problems that might arise if one or more assumption is violated. One of the major estimation problem of violation of the basic assumption will be obtaining misleading or incorrect information from the analysis: if there exists outliers, the normality assumption is not met and this will render the goodness of fit test a failure. Similarly, the independence test when violated causes the linearity aspect to be inappropriate. The Durbin-Watson statistic is used for the test of independence and from the output, the value of this statistic is 1.961 which shows that JOBINC and age are not completely independent f each other. Some violations that are small tend to cause little problem on the model but other great violations can render the results obtained inaccurate, increase the residual variance therefore reducing the chances of rejecting the stated hypothesis thus making it difficult to interpret the results. In conclusion the ‘best’ model for estimating/ predicting JOBINC is the regression analysis as outlined above though there seems to be a problem with the presence of outliers which can be removed by rebuilding the model. This has a little impact on the model since points do not lie too far from the line implying that the model is still valid and it is a good test of fitness and also the model is useful in estimating the applicant’s monthly income for their current job based on their age. References Day, R. A., & Underwood, A. L. (1986). Quantitative analysis. Englewood Cliffs, N.J: Prentice Hall. Holcomb, Z. C. (1998). Fundamentals of descriptive statistics. Los Angeles, CA: Pyrczak Pub. Seber, G. A. F., Lee, A. J., & Wiley InterScience (Online service). (2003). Linear regression Analysis. Hoboken, N.J: Wiley-Interscience. Weinberg, S. L., & Abramowitz, S. K. (2008). Statistics using SPSS: An integrative approach. Cambridge: Cambridge University Press. Read More