Suppose that you've been hired to work as an analyst at a sports consultancy firm. Your task is to evaluate the effectiveness of players on the client's team. Your observe the following data on seven players at the end of the season: Player # Season Goals Total Field Time (minutes) 1 Y1 = 3 X1 = 120 2 Y2 = 5 X2 = 340 3 Y3 = 8 X3 = 560 4 Y4 = 15 X4 = 590 5 Y5 = 27 X5 = 780 6 Y6 = 45 X5 = 1120 7 Y7 = 46 X5 = 1800 Table 1: Player Effectiveness: Goals vs Field Time You decide to use these data to estimate the following regression model: Yi = 0 + 1Xi + ui (1) and obtain the following OLS estimates: 1 Estimate St. Deviation 0 -1.72 5.23 1 0.03 0.006 Table 2: Model 1 Coefficient Estimates (a) (8 points) Using the estimates of model coefficient and of the standard errors (deviations) from Table 2, determine whether field time has a statistically-significant effect on the number of field goals? Formulate the appropriate hypothesis, and test it at = 5% level of significance. (b) (8 points) Does spending more time on the field increase the number of season goals? Formulate the appropriate hypothesis, and test it at = 1% level of significance. (c) (7 points) If a player from this team spends an additional minute on the field, the number of season goals is expected to change by at most how much, with 95% level of confidence? (Hint: use the estimates from Table 2 to find the upper 95% confidence bound (one-sided confidence interval) for 2). (d) (8 points) Use the data provided in Table 3 below to find the Residual Sum of Squares (RSS), Explained Sum of Squares (ESS) and the Total Sum of Squares (TSS) for this regression (note that you can find ESS as ESS = TSS - RSS). Player # (Yi Y ) 2 (Y i Y ) 2 u 2 1 334 376 1.25 2 265 163 12 3 176 38 50 4 39 28 0.96 5 32 0.16 28 6 562 112 172 7 610 960 39 Table 3: RSS, ESS and TSS components (e) (6 points) Use the RSS, TSS and/or ESS to calculate the coefficient of determination (R2 ) for this regression. What can you conclude about the overall goodness of fit? (f) Which of the seven players in your sample spends his or her time on the field most effectively? Which player is least effective? (Hint: calculate and consider the residuals ui).