A Study on Box Office Revenues - Research Paper Example

Understanding Information Report Part Correlation and Regression The correlation coefficient, r, measures the strength and the direction of a linear relationship between two variables. The degree to which one or more predictors (i.e., independent or X variables) are related to the dependent (Y) variable is expressed in the correlation coefficient r. In a study involving box office revenues, the principal variables influencing gross box office earnings such as the number of cinema sites, the number of cinema screens, cinema revenue per screen, revenue per admission and the total number of admissions were examined vis a vis their correlation to the gross box office earnings. From a consideration of the correlation coefficients obtained (Table 1) it was clear that r being greater than 0.8 in all the cases, the correlation was generally strong with all the predictors. However, the variable relating to the number of cinema screens yielding the highest r value of 0.985 could be considered as the best predictor for the gross box office earnings. On the other hand, the number of cinema sites was not as good a predictor since some cinema sites might house multiple screens (“multiplexes”). Similarly, the gross revenue was more influenced by the cinema revenue per admission (r = 0.972) than the total number of admissions (r = 0.956) because the value of each ticket (i.e., admission) can differ. However, other important determinants such as the actors/directors/category (comedy, tragedy, action) of the film have not been taken into consideration in this study. Scatterplots are helpful in visually identifying relationships between any two variables. In the case of gross box office earnings and the number of cinema screens, the regression line on the scatterplot (Fig. 1) being a straight line shows that the two variables have a strong positive linear correlation (r = 0.985). A straight line depicts a linear trend in the data. This means that by increasing the number of cinema screens, the gross box office earnings can be enhanced. Also, since there is significant linear correlation, the line can be used to estimate the gross box office revenue (dependent variable) for any given value of the number of cinema screens (independent variable). The equation obtained for the regression line from the scatterplot is: Gross box office revenue = 0.2674 * the number of cinema screens – 179.41 This equation signifies that for every additional screen, the gross revenue will increase by 0.2674 million pound sterling. This is because the term -179.41 in the equation which is a constant, does not have any practical meaning as such; it is just the point on the graph where the regression line, if extended, will cross the y-axis on which the gross box office earnings are plotted. In other words, -179.41 is the gross box office earning when the number of cinema screens is zero which is, of course, absurd. The value of the coefficient of determination, r2, as obtained from the scatterplot (Fig.1) is 0.9711. This value indicates how well the regression line represents the data. While the strength of the relationship between gross box office revenue and the number of cinema screens was indicated by the correlation coefficient, r, to measure the strength of the relationship, the coefficient of determination, r2, is useful since it denotes the percentage of the data that is the closest to the line of best fit. It describes the proportion of variance in common between the two variables. Thus, in this case, 97.1% of the total variation in the gross box office earnings was matched or shared by the variation in the number of screens. What influences the residual 3 per cent of the variation is unknown. We can predict fairly accurately either through intrapolation of the regression line or using the regression equation, the gross box office revenues that a particular number of cinema screens will yield, provided that number is within the data range (1919 to 3596) that has been used to draw the scatterplot. For example, gross earnings from 2541 operational cinema screens can be predicted to be around 500 million pounds sterling [(0.2674 x 2541 ) – 179.41 = 500]. However, calculation of gross revenue from say, 500 cinema screens using the regression equation yields a negative value that is, -45.71 million pounds sterling. Hence, although the gross box office revenues show a high covariance and a near-perfect relationship with the number of cinema screens, the relationship and regression equation cannot be used to make accurate predictions based on extrapolation. Table 1. Relationship between gross box office earnings and different predictors Predictor Correlation Coefficient Number of cinema sites 0.889568 Number of cinema screens 0.985465 Total number of admissions 0.956026 Cinema revenue per admission 0.971857 Cinema revenue per screen 0.931812 Fig. 1. Scatterplot showing the relationship between number of cinema screens and gross box office earnings. Part 2 Time Series Analysis and Forecasting Data collected over a period of five years and six months from January 2003 to June 2008 on consumer expenditure on transport in the U.K. shows a typical and constant annual trend (Fig. 1). The annual trend was examined by calculating the coefficient of determination, r2. With a linear trendline, the value of r2 was obtained as 0.503 and, with the logarithmic trendline, it was 0.409. It is clear from the time series chart (Fig. 1), that the linear trendline follows the data very closely whereas the logarithmic trendline does not reflect the constant annual increase in consumer expenditure that is apparent in the graph. A seasonal trend is also very evident from the graph. The expenditure is maximum during July to September every year and it shows a minimum year after year, during the late fall and early winter months, from October to December. The expenditure during the remaining 6 months (from January to June) is midway between the high and low values. The obvious conclusion is that people travel much more during fair weather and least during winter. Furthermore, the linear upward trendline suggests that the expenditure on transport by the consumers will continue to show an annual average increase of, at least if not more than,1278.5 million pounds sterling in the future, too. Fig. 1. Graph showing quarterly trend in consumer expenditure on transport from January 2003 to June 2008. Linear trendline and the corresponding coefficient of determination (r2) are depicted in red; logarithmic trendline and corresponding r2 value are shown in blue. Table 1. Average annual consumer expenditure from 2003-2008 Year Average consumer expenditure on transport (£ million) 2003 26142.25 2004 27114.00 2005 28542.00 2006 29624.25 2007 31027.75 2008 (up to June) 32534.50 Average annual increase in consumer expenditure on transport = (32534.5 – 26142.25) / 5 = 1278.5 million pounds sterling References Electronic Textbook StatSoft. Accessed July 25, 2009 http://www.statsoft.com/textbook/stmulreg.html http://www.stat.berkeley.edu/~stark/SticiGui/Text/regression.htm Accessed July 25, 2009 Appendix 1 cinema screens (number) Gross box office takings (£ millions) 1919 355.8 2003 354.2 2166 407.2 2383 486.2 2638 504.9 2825 549.7 3017 572.8 3248 645.0 3402 755.3 3433 742.0 3475 769.6 3486 770.3 3569 762.1 3596 821.0   Appendix 2 Year £million 2003 25755   25952   29056   23806 2004 26398   26711   30078   25269 2005 27537   28579   31683   26369 2006 29129   29494   32964   26910 2007 29977   30937   34386   28811 2008 32243   32826 Read More