6 How does string tension in tennis rackets affect the speed of the ball coming off

Example 11.6 How does string tension in tennis rackets affect the speed of the ball coming off

  the racket? The article “Elite Tennis Player Sensitivity to Changes in String Tension and the Effect on Resulting Ball Dynamics” (Sports Engr., 2008: 31–36) described an experiment in which four different string tensions (N) were used, and balls projected from a machine were hit by 18 different players. The rebound speed (kmh) was then determined for each tension-player combination. Consider the following data in Table 11.4 from a similar experiment involving just six play- ers (the resulting ANOVA is in good agreement with what was reported in the article).

  The ANOVA calculations are summarized in Table 11.5. The P-value for testing to see whether true average rebound speed depends on string tension is .049. Thus

  H 0 :a 1 5a 2 5a 3 5a 4 50 is barely rejected at significance level .05 in favor of

  the conclusion that true average speed does vary with tension (F .05,3,15 5 3.29) . Application of Tukey’s procedure to identify significant differences among tensions requires Q .05,4,15 5 4.08 . Then w 5 7.464 . The difference between the largest and smallest sample mean tensions is 6.87. So although the F test is significant, Tukey’s

  11.1 Two-Factor ANOVA with K ij 51 429

  Table 11.4 Rebound Speed Data for Example 11.6 Player

  Table 11.5 ANOVA Table for Example 11.6 Source

  method does not identify any significant differences. This occasionally happens when the null hypothesis is just barely rejected. The configuration of sample means in the cited article is similar to ours. The authors commented that the results were contrary to previous laboratory-based tests, where higher rebound speeds are typically associated with low string tension.

  ■ In most randomized block experiments in which subjects serve as blocks, the

  subjects actually participating in the experiment are selected from a large population. The subjects then contribute random rather than fixed effects. This does not affect

  the procedure for comparing treatments when K ij 51 (one observation per “cell,” as

  in this section), but the procedure is altered if K ij 5K.1 . We will shortly consider two-factor models in which effects are random.

  More on Blocking When I52 , either the F test or the paired differences t test can

  be used to analyze the data. The resulting conclusion will not depend on which

  procedure is used, since T 2 5F and t 2 a2,n 5F a,1,n .

  Just as with pairing, blocking entails both a potential gain and a potential loss in precision. If there is a great deal of heterogeneity in experimental units, the value

  of the variance parameter s 2 in the one-way model will be large. The effect of block- ing is to filter out the variation represented by s 2 in the two-way model appropriate

  for a randomized block experiment. Other things being equal, a smaller value of s 2

  results in a test that is more likely to detect departures from H 0 (i.e., a test with

  greater power).

  However, other things are not equal here, since the single-factor F test is based on I(J 2 1) degrees of freedom (df) for error, whereas the two-factor F test is based on (I 2 1)(J 2 1)

  df for error. Fewer error df results in a decrease in power, essentially because the denominator estimator of s 2 is not as precise. This loss in df can be

  especially serious if the experimenter can afford only a small number of observations. Nevertheless, if it appears that blocking will significantly reduce variability, the sacrifice of error df is sensible.

  CHAPTER 11 Multifactor Analysis of Variance