BUGEN1502 Business Statistics

The Australian Football League (AFL) currently has 18 teams based in the various Australian States, Victoria, NSW, South Australia, Western Australia and Queensland. These teams play a unique Australian football code, Australian Rules football, and the season consists of 24 rounds. Football is an increasingly competitive and professional business in Australia and the budgets for the clubs reflect this. There has long been a feeling that better resourced and more popular clubs have an advantage with their larger expenditure leading to greater success on the field. The data provided give the amount spent by the Football Departments of the AFL clubs from 2003 to 2007 as well as  their results (Source: The Australian newspaper, 13 March 2008). There are three variables:

(a) Which is the independent (or predictor) variable and which is the dependent (or response) variable in this case? Give reasons for your choice and answer in Textbox (a) 
(b) Produce a scatter plot of the data (Graph (b)). The chart must have appropriate title and labels. Also you should rescale the graph axes to most clearly display the data. Based on the scatter plot, describe the type and direction of relationship between the variables. Is the relationship deterministic or probabilistic Answer in Textbox (b).
(c) Using the regression function in Excel, generate regression output for the data and place it where indicated in the Results worksheet.
i. Write down the line of best fit for the given data from the regression output obtained. Explain the meaning of any symbols used. (Textbox (c))
ii. State the value of the slope of the equation correct to three decimal places and give an economic interpretation of this value in Textbox (c).
iii. State the value of the intercept of the equation correct to three decimal places and give an interpretation of this value in Textbox (c). Does the intercept value represent a practical value in this case Explain briefly.
(d) Do the data provide statistical evidence of a linear relationship between the amount spent by the football department and the number of wins for the team? Perform an appropriate hypothesis test using the p-value method. 

i. State null and alternative hypotheses. (You are not required to type Greek letters with subscripts. You may use words or symbols instead of Greek letters. For example, Ho can be written as H_0, and β1 as beta_1, etc.)
ii. State the appropriate p-value for this test from the regression output. If the significance level were 5%, would the null hypothesis be rejected At what levels of significance will the null hypothesis be rejected.
iii. Is there are statistically significant relationship between amount spent by the football department and the number of wins for the team State your conclusion with reference to your decision about the hypotheses.

(e) State the coefficient of determination from the regression output to three decimal places.How good a fit is this linear model to the data Answer in Textbox (e)
(f) Predict the sales the number of wins (as a whole number) for a club whose football department spent:
i. 62 million dollars
ii. 70 million dollars
Comment on the reliability of each prediction. Answer in Textbox (f). 

Textbox (c)  (i) Regression line  Number of wins = -13.574 + (1.168* Money spent) Value -13.574 indicates intercept of the regression line.  Value 1.168 indicates the slope coefficient of the variable total money spent on the football team.  (ii)   Slope coefficient comes out to be 1.168.  This value is the measure of the change in the dependent variable which would incurred due to change in the independent variable.  In the present case, when the total expenditure on football team by the club has increased by 1 million Australian dollars then on an average the total number of wins of the football match would also be increased by  1.168.  (iii) Intercept comes out to be -13.574.  This value is the measure of the dependent variable when there is no contribution of the independent variable. This means,  the number of wins of  football match would be -13.574, when the club does not spend any amount on the football team. However, it is a hypothetical case, because the number of wins cannot be negative or in fractions.  

The predictor (dependent variable) variable in the present case is the total number of wins of football match from 2003 to 2007 and the response variable (independent variable) is total money spent i.e. total football expenditure (million AUD) on the football team.  It is apparent that when the total football expenditure on the team is increased then the overall performance of the team would also be enhanced.   Another reason is derived from the given information that when the team members have received high resources (total spent is high) from their club, then the probability of registering more number of wins in football match is also high. Further, the number of wins of football match is a function of performance of the team member. Hence, the selection of dependent and independent variables as total number of wins and total spent respectively is correct.  

Regression line  Number of wins = -13.574 + (1.168* Money spent) Value -13.574 indicates intercept of the regression line.  Value 1.168 indicates the slope coefficient of the variable total money spent on the football team.  (ii)   Slope coefficient comes out to be 1.168.  This value is the measure of the change in the dependent variable which would incurred due to change in the independent variable.  In the present case, when the total expenditure on football team by the club has increased by 1 million Australian dollars then on an average the total number of wins of the football match would also be increased by  1.168.  (iii) Intercept comes out to be -13.574.  This value is the measure of the dependent variable when there is no contribution of the independent variable. This means,  the number of wins of  football match would be -13.574, when the club does not spend any amount on the football team. However, it is a hypothetical case, because the number of wins cannot be negative or in fractions.  

(i) Hypotheses  H_0 (Null Hypothesis): Beta_1 = 0  H_1 (Alternative Hypothesis): Beta _1 ≠ 0 (ii) Significance level (alpha) = 5%  The p value from the regression model output (for slope coefficient) = 0.091 Conclusion: It is apparent that p value is higher than significance level and hence, null hypothesis would not be rejected. For the acceptance of alternative hypothesis and rejection of null hypothesis, it is essential that value of significance level must be higher than 9.10%.  (iii) The claim furnished in the null hypothesis is right because alternative hypothesis has not been accepted. Therefore, it can be cited that variable number of wins and total expenditure are not significantly related with each other.  

Coefficient of determination ????^????=????.????????????   Value of coefficient of determination indicates the good fit of the regression model. In the present case, 19.0% of the variation in the dependent variable has been explained by the variation in the independent variable. This means, only 19% of the variation in total number of wins has been describes by the variation in the total expenditure by the club on the football member. A regression model is termed as good fit only when significantly large percentage of variation in dependent variable would be describes by independent variable. 19% is not significantly high percentage and thus, this regression model is not termed as good fit model.  

extbox (f)  Regression line  Number of wins = -13.574 + (1.168* Money spent) (i) Money spent: 62 million dollars  Number of wins = -13.574 + (1.168* 62)  Number of wins = 58.82 ~  59   (ii) Money spent: 70 million dollars  Number of wins = -13.574 + (1.168* 70)  Number of wins =68.16 ~  68 THe model reliability is not high as is apparent from the raw data has been provided. There are instances where despite low spending, more wins are being registered and also cases where high spending does not lead to expected wins. Hence, other factors such as underlying skills of the team, coordiantion, morale, experience also matter in team peformance.