Correlation between Annual Income and Amount Spent on Car - Essay Example

XXXXX Number: XXXXX XXXXX XXXXXX XXXXX of XXXXX XX – Jun - 2010 Mathematics - Assignment A. What kind of correlation do you expect to find between annual income and amount spent on car? Will it be positive or negative? Will it be a strong relationship? Base your answer on your personal guess as well as by looking through the data. The two variables, annual income and amount spent on car are expected to have a positive correlation. The main reason is that when the income is higher, people tend to spend more on their cars. Moreover, the relationship between the two variables is expected to be strong. Based on the data given, it is evident that the annual income and the amount spent on car are directly proportional to one another. In other words, they are positively correlated. However, it is important to note that some of the data indicate that at some levels of income ($ 52,000 and $ 66,000), the amount spent on cars decrease when compared to lower levels ($ 38,000 and $ 40,000). There are a few more values which differ from the rest. However, most of the data indicate that the relationship is positive. In order to identify the strength of the association, the Pearson’s correlation coefficient is computed. Correlation Coefficient R = covariance of the two variables / product of std. dev. = 0.89 The Correlation coefficient is positive confirming the positive association between the two variables. Also, the value of the coefficient is 0.89 which indicates a strong relationship between the two variables. B. What is the direction of causality in this relationship - i.e. does having a more expensive car make you earn more money, or does earning more money make you spend more on your car? In other words, define one of these variables as your dependent variable (Y) and one as your independent variable (X). In order to identify the direction of causality, the two variables are analyzed objectively. When a person spends more money on the car, it does not have any effect on his income. Hence it is evident that the amount spent on the car does not affect or have an influence on the annual income of the person. However, when a person’s annual income increases, he is more likely to spend higher on the car. In other words, annual income is the cause and the amount spent on car is the effect. Hence the annual income is the independent variable (X) and the amount spent on the car is the dependent variable (Y). The amount spent on the car (Y) depends on the annual income (X). C. What method do you think would be best for testing the relationship between your dependent and independent variable, ANOVA or regression? Explain your reasoning thoroughly with a discussion of both methods. Correlation establishes the association between two variables, however does not indicate the direction of causation or the numerical relationship that exists between the movements of the two variables. There are different methods available to test the relationship between two variables including Anova (Analysis of Variance) and Regression modeling. Anova (Analysis of Variance) as the name indicates apportions the variance of the dependent variable into different components which are caused by different sources of variation or variations in the independent variables. Shortly, Anova test mainly deals with testing several variables against each other. Regression analysis involves modelling and analysing the relationship between a dependent variable and one or more independent variables. Regression modelling and analysis is primarily conducted to observe or forecast the change in the dependent variable when one or more of the independent variables are changed. Regression differs from correlation - which establishes whether there is any association between two variables (Chatterjee and Hadi, p. 24). When there is only one independent variable the relationship of that variable with the dependent variable is said to be linear and the regression conducted to identify the numerical connection between the variables is based on linear regression model. In this case, there are only two variables (a dependent and independent variable) and it has already been established that they are strongly correlated. The correlation coefficient is found to be 0.89 which indicates a strong positive correlation. From this value, the coefficient of determination (R2) is 0.7921, indicating that about 80 % of the movement of the two variables can be explained with the help of a linear regression equation. It is evident from the above discussion that the best suited method to test the relationship between the two variables is linear regression. D. Go to this calculation page and enter in your data in the X and Y columns (dont use commas, enter 8,000 as 8000). Then click on the button "Y=MX B". Then click on the "graph" button. Write out your equation as calculated, along with your coefficients. Discuss the significance and interpretation of this result, and discuss your graph. The given data is fed into the regression calculator and the results of the linear regression (Y = MX + B) is obtained as follows: Y = 0.33 X – 3800 R = 0.89 The correlation coefficient 0.89, as discussed earlier, indicates that the two variables are positively correlated and their relationship is strong. The M value is found to be 0.33 which is the slope of the linear equation. This indicates that for every unit change in X value, the Y value will change by a proportion of 0.33 in the same direction. The B value or the Y – intercept is found to be (- 3800) which indicates that when the annual income is zero, the amount spent on the car is a negative 3,800. The graph generated by the regression calculator is given below: The scatter plot is made from the given data and the regression line is derived by fitting al ine which is of least distance from all the points on the graph. The line of best fit (Y = 0.33X – 3800) computed using the least squares method is exhibited in the graph. Works Cited Chatterjee, Samprit and Ali S. Hadi. Regression Analysis by Example. Wiley-Interscience, 2006, 21- 115. Read More