The Effect of Temperature, Time, and Humidity on the Bounce of Tennis Balls

Bryan Reimer

IME412

April 29, 1997

Table of Contents

Objective 3

Procedure 5

Conclusion 6

Data 10

Tables 11

t-test 13

Graphs 14

Presentation Slides 18

Objective

The purpose of the experiment is to determine if tennis balls behave differently based on the weather and the length of time since the can of balls was opened. Two different types of balls, Penn Championship and Non-Championship will be compared. Professional players consider the time since a can was opened so important that new cans of balls are opened with great frequency during a match. Even amateur players consider the effects important enough to open a new can every couple of days. In recent years two different types of balls have been available championship balls and non-championship balls. The data from this experiment will show their differences. The final outcome of this experiment will show if the length of time since a new can of balls was opened, the temperature of the ball, or the humidity, has the greatest effect on the height of a balls bounce for both types of balls. Two variables will be omitted in the study, how much the ball is played with and different court surfaces. In the experiment only a hard surface will be analyzed and no playing with the ball will be allowed. Analysis of the data will either conclude that the common practices of opening a new can balls often is necessary to keep a consistent bounce height, or show that it is unnecessary and balls out of cans previously opened bounce with the same height as newly opened cans. Other conclusions will try to prove what the optimum playing temperature and humidity is, and if these factors affect the speed of the game. The conclusion for the two different ball types will show which is in fact is a better ball in each of the different situations. Players will be able to use the conclusions in many ways. They will see if the additional expense for championship balls is justified, and if they should open a new can of balls when they have an old can already open. Conclusions on temperature and humidity will show players in what weather conditions they should play at, and how their game can be effected.

Procedure

Conclusion

Some interesting results can be found in an analysis of the experiment results. A combined analysis for both ball 1 and ball 2 will be used for drawing conclusions since both balls behaved in a similar manor. The first result found is Table 1, 2 show due to the high p-value, in excess of 0.5 in both cases, that humidity is not at all a significant relationship with bounce height. Time was analyzed next in Table 3, 4. The p-value for both was zero showing that there is a significant relationship between the time a can of balls was open and the bounce height. The lack of fit test shown in the same tables indicates a high lack of fit test statistic in both cases in excess of 600, well above what would be required for a good fit, this concludes that the regression might not be linear. Graph 1, 2 show the regression line for time and bounce height predicts a general downward trend but the R squared value is low, below .17 in both cases. With such a low R squared value, the regression line does not predict many of the points and can not be considered a good fit to the relationship. A quadratic regression was next tried as illustrated in Table 7, 8. The p-value remains, as it should at zero, when the regression was plotted in Graph 5, 6 the R squared value showed little improvement, few points are still predicted by the regressed line. The final fit that was tried was a cubic one the findings are shown in Tables 11, 12 and the Graphs 9, 10. The cubic fit is shown by R squared values of .396 and .331 and, the regression is able to predict between two and three times as many of the points as the linear regression. By far the cubic fit is the best out of the three regression models analyzed but the R squared values are all well below 0.9 which is what would be needed as a minimum to make the models good predictors for future trials. No conclusions can be drawn from this data because of such a high lack of fit. The temperature regression was carried out last. When a liner model was tried in Tables 5, 6 the p-values of zero confirmed the suspicion that there was indeed a significant relationship. The regression Graphs 3, 4 show R squared values considerably higher than any of the time regression models. For ball 1 a value of 0.925 was found and 0.908 was found for ball 2. These acquired values show that the line is a good predictor of future points and where they will fall. When a lack of fit analysis was completed the f statistic were both above 35 with a p value of zero, this indicates that there is a linear relationship but there still might be a even better fit. A residual analysis was next completed for the liner regression. The residual vs. Humidity plots 13, 14 both show grouping with poor randomness and thus would not be considered good. Next Residuals vs. Temperature were Plotted in 15, 16 both again show similar patterns indicating a upward trend in the data and poor randomness, both would be considered poor indications of a good fit. The residuals vs. time graphs 17, 18 and residuals vs. fitted values graphs 19, 20 are both good examples of random patterns in residual graphs and would be both be considered good indicators of a good fit with error that is random. The normal probability plot of residual graphs 21, 22 both show generally a straight line, when the extremities are neglected. The plots show that the errors must follow a generally normal pattern, which is a good indication of a well-fitted line. Overall the residual plots show that the fit is good, but there is some considerable doubt and, it is possible another model might fit the data even better. A quadratic fit was next tried it is represented in Tables 9, 10 and Graphs 7, 8. An increase in R squared to values of 0.974 and 0.957 were seen and suggests that the quadratic fit is even better than the linear one. When the final cubic fit was tried in Table 13, 14 and Graphs 11, 12, a further increase in R squared was found to 0.979 and 0.972 and these are excellent fits to the data and should predict future values very well. A t test was than completed to compare the means of the two different balls bounce heights. The value of the test statistic was found to be –0.40997 with an alpha level of 0.05 the H0 hypotheses would not be rejected. By not rejecting H0 means the two different balls would be considered equal. The sample standard deviations for ball 1 and 2 were now found. Ball 1 had a sample standard deviation of 3.312 while ball 2 was 3.177. This showed that the non-championship ball was more consistent than the championship ball in the test.

The analysis of all the data proves some of the questions posed. Time out of the can does pose a significant relationship, although that relationship can not be proven with the data from this experiment. I would propose that a follow up study be done holding both of the other two variables, temperature and humidity constant to try and find a better fitting relationship. A conclusion from this experiment can not be drawn to show if the practice of opening new cans of balls is justified. Temperature has a significant positive relationship with the bounce height of tennis balls. A game played at higher temperatures will be much faster since the ball will rebound from contact with any surface better than at low temperatures. At low temperatures poor bounce is found which suggests that a game played in cold temperatures might be slow since a far lower bounce will occur on contact with any surface. Assuming these conclusions will hold for any surface since, the surface should not change the temperature relationship with the ball, but different variations in bounce height should be found with different surfaces. Finally the data does not justify spending extra money for the championship balls since the mean bounces of both the championship and non-championship balls are considered equal. A further study should be completed to see if anything else effects the balls differently to try and determine what makes the balls different. Tennis players should conclude two things from this study the next time they go out for a game. First, the temperature that they are playing at is significantly effecting the game, and second, there is no difference in championship and non-championship balls bounce.

Tables

t-test

t test on the sample means

H0: µ1=µ2

H1: µ1<µ2

t.05,246 =-1.645

Graphs