Introduction to Linear Regression Analysis

Question: Describe about the Introduction to Linear Regression Analysis. Answer: a. The amount spent is the independent factor and the wins is the dependent factor. This is because the success of a team is dependent on the resources it spends and hence the wins is dependent on the amount spent. Hence wins is the dependent factor and amount spent is the independent factor. (George A. F. Seber, 2012) b. The scatter plot of the data is given below From the scatter plot it can be seen that the number of wins increases with the amount spent. Thus it can be said that the amount spend and number of wins are positively correlated. This relationship is deterministic. c. The equation of the best fit for the given data from the regression output is y = 1.167* x 13.574, where y is the wins and x is the amount spent. The slope of the line is 1.167. This means that for every million dollar spent by the club, the number of wins increases by 1.167. The value of the intercept is 13.574. This means that if the club does not spends any amount then the wins will be -13.574 or we can say that the club will lose 13.574 matches. The intercept in this case cannot be negative as the wins cannot be less than zero. But if we consider the negative value as losses, the negative intercept can be practically correct.(Yan, 2009) d. Null hypothesis H_0 : There is a no linear relationship between the amount spent by the football department and the number of wins for the team. i.e. Beta_1 = 0 Alternate hypothesis H_1 : There is linear relationship between the amount spent by the football department and the number of wins for the team. i.e. Beta_1 0 Alpha for 95% confidence level is 0.05 The excel p value column gives value for 2 sided p value. The p value from the regression output is 0.091. At 95% confidence interval, p value (Beta_1) = 0.091 alpha 0.05. Hence we cannot reject the null hypothesis. The null hypothesis can be rejected when p value (Beta_1) is less than alpha. Thus if alpha = 0.1 i.e. at 90% confidence level, the null hypothesis can be rejected. iii) As the p value (Beta_1) = 0.091 alpha 0.05, we can conclude that there is no statistically significant linear relationship between the amount spent by the football department and the number of wins for the team. (Ning-Zhong Shi, 2008) e The coefficient of determination from the regression output is 0.190. This implies that 19% of the variation in the wins can be attributed to the amount spent by the club whereas the remaining 81% is unexplained. Thus the linear model is not a good fit to the data. (Douglas C. Montgomery, 2012) f. Using the regression equation, y = 1.167* x 13.574 i) The number of wins for amount spent = 62 million dollars will be y = 1.167* 62 13.574 = 58.780 ii) The number of wins for amount spent = 70 million dollars will be y = 1.167* 70 13.574 = 68.116 The reliability of the results is low as the model is able to explain only 19% of the variation in wins due to the amount spent by the club whereas the remaining 81% is unexplained and caused by other factors. (Allen, 2007) References Allen, M. P. (2007). Understanding Regression Analysis. Springer Science Business Media. Douglas C. Montgomery, E. A. (2012). Introduction to Linear Regression Analysis. John Wiley Sons. George A. F. Seber, A. J. (2012). Linear Regression Analysis . John Wiley Sons. Ning-Zhong Shi, J. T. (2008). Statistical Hypothesis Testing: Theory and Methods. World Scientific. Yan, X. (2009). Linear Regression Analysis. World Scientific.