1. In Performance under Pressure in the NBA (Journal of Sports Economics, 2011),

Question

1. In Performance under Pressure in the NBA (Journal of Sports Economics, 2011), Zheng Cao and co-authors investigate how well NBA players shoot free throws at the end of close games. They examine the free throw shooting percent and also the last shot shooting percent in situations where the score is close.

They gathered play-by=play data from ESPN.com. They regressed a dummy variable (=1 for a make; = 0 for a miss) on dummies indicating the game situation. So the coefficient is the difference in the probability of making the shot between the given situation and the season average.

Here is a table from their paper.

Regression (1) is for foul shots in the last 60 seconds; regression (2) for foul shots in the last 30 seconds; and regression (3) for foul shots in the last 15 seconds. The independent variables are the game score.

So, the upper left coefficient says that, on average, the probability of making a free throw is .037 less in the last 60 seconds of a game when the shooter’s team is down 2 points. The standard error for this coefficient is .017.

a. Compute the t statistic for this coefficient. Is the estimate statistically significant? What is the null hypothesis?

b. Do NBA players choke on their free throws in pressure situations? Explain.

c. As the time remaining falls, what happens to the shooting percentage? Explain.

d. Do NBA players choke when their team is ahead by 1 point? Do they perform better than when they trail by a point?   Explain

e. Is there choking when the game is tied? Explain.

Regressions (4) through (6) analyze shots taken from the floor in the last minute.

f. Would you say that there is much evidence that NBA players choke on shots from the floor? Explain.

g. Why would a test for free throw shooting be more convincing than a test for shooting from the floor? Explain.

Table 3. Analysis of Last Minute Pressure Effects, Baseline -0.02 1 (0024) -0.062** (0.024) 0.011 (0.025) -0.009 (0.015) 0.00 (0014) 0.029 (0032) 0.043 (0031) -0.006 (0.032) 0.036 (0.023) -0.03 (0.019) 0.037 (0.017 -0.045 (0.023) -0.041(0.016) -0.066"a* (0.022) 0.015 (0.017) -0.026(0.012) -0.029(0.015) -0.018* (0.011) -0.01 9 (0.013) (0.032) (0.030) 0.034 (0.031) 0.097m* (0.032) -0.010 (0.032) 0.015 (0.017) 0.002 (0.017) -0.020 (0.042) 0.060 (0.041) 0.005 (0041) 0.031 (0026) -0.060 (0.042) -Q108 0.023 (0.040) 0.016 (0.023) 0.008 (0.022) 0.001 (0.057) 0.042 (0.053) -0.009 (0.050) 0.063* (0.032) 0.046**(0.023) 0.063 _0088 (0.043) 0.008 (0.022) 0.018 (0.029) 0.042(0.020) FinalShot x LosX x Downl FinalShot x LasX x Tied -0.039 (0.020) 414,069 Note. All models include Up5_10, Up4, Up3, Up2, Upl, Tied, Downl, Down2, Down3, Down4, Down5 10, Last60130/ 15, PrevMade, PrevMiss, Home, Playoff, inter- action of Last60130/15 and score dummies, dummies for quarter. Models 1-3 also include OneShot. Models 4-6 also include FinalShot and its separate interactions with Last60130/15 and score dummies. Observations with FinalShotI and less than 6 seconds remaining (in fourth quarter or overtime) are dropped, as are observations with Lastx-0 with less than 5 minutes remaining (this is why N varies from model to model). All models include player-season fixed effect and use robust standard errors that are given in parentheses. * ** *.* denote 10%, 5%, and 1% significance ! Up Up

Explanation / Answer

a. Compute the t statistic for this coefficient. Is the estimate statistically significant? What is the null hypothesis?
t statistic = Coefficient/ Std error = -0.037/0.017 = -2.176
Degree of freedom = N - 1 = 436898 - 1 = 436897
P-value for t = -2.176 and degree of freedom = 436897 is 0.0148
As p-value is less than the significance level (0.05), we reject the null hypothesis and conclude that the estimate is statistically significant.
Null Hypothesis H0 : The probability of making a free throw in the last 60 seconds of a game when the shooter’s team is down 2 points is same as the average probability. That is the difference between the coefficent and average coefficient is zero.

b. Do NBA players choke on their free throws in pressure situations? Explain.

Let us consider the pressure situations as the last 15 seconds of a game when the shooter’s team is down 1 or 2 points.

The probability of making a free throw is .063 less in the last 15 seconds of a game when the shooter’s team is down 2 points. The standard error of the coefficient is 0.032 at 5% level of significance.

t = -0.063/0.032 = -1.969

Degree of freedom = N-1 = 414069 - 1 = 414068

P-value for t = -1.969 and degree of freedom = 414068 is 0.0244
As p-value is less than the significance level (0.05), we reject the null hypothesis and conclude that the estimate is statistically significant.

The probability of making a free throw is .088 less in the last 15 seconds of a game when the shooter’s team is down 1 points. The standard error of the coefficient is 0.030 at 1% level of significance.

t = -0.088/0.03 = -2.933

Degree of freedom = N-1 = 414069 - 1 = 414068

P-value for t = -2.933 and degree of freedom = 414068 is 0.00168
As p-value is less than the significance level (0.01), we reject the null hypothesis and conclude that the estimate is statistically significant.

As these estimates are statistically significant, NBA players choke on their free throws in pressure situations.

c. As the time remaining falls, what happens to the shooting percentage? Explain.

When the shooter’s team is down 2 points, the shooting percentages decreases as the time remaining falls because the coefficient decreases as the time value falls.

When the shooter’s team is down 1 points, the shooting percentages decreases as the time remaining falls because the coefficient decreases as the time value falls.

When the shooter’s team score is tied, the shooting percentages increases as the time remaining falls because the coefficient increases as the time value falls.

When the shooter’s team is up 1 points, the shooting percentages decreases as the time remaining falls because the coefficient decreases as the time value falls.

When the shooter’s team is up 2 points, the shooting percentages remains constant as the time remaining falls because the coefficient remains same as the time value falls.

d. Do NBA players choke when their team is ahead by 1 point? Do they perform better than when they trail by a point?   Explain

When the shooter’s team is down 1 points, the probability of making a free throw is .041 less in the last 60 seconds of a game; the probability of making a free throw is .066 less in the last 30 seconds of a game; the probability of making a free throw is .088 less in the last 15 seconds of a game.

When the shooter’s team is up 1 points, the probability of making a free throw is .026 less in the last 60 seconds of a game; the probability of making a free throw is .029 less in the last 30 seconds of a game; the probability of making a free throw is .042 less in the last 15 seconds of a game.

Comparing these probabilities and the probability of making a free throw when the shooter’s team is up 1 points is always higher than the the probability of making a free throw when the shooter’s team is down 1 points , we conclude that NBA players choke when their team is ahead by 1 point and they perform better than when they trail by a point.

e. Is there choking when the game is tied? Explain.

The probability of making a free throw is .018 higher in the last 15 seconds of a game when the game is tied. Also, the probability of making a free throw increases as the time value falls when the game is tied.

So, there is no choking when the game is tied.

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