Chapter 2 The CRD with a Numerical Response Continued This chapter continues the theme of Chapter 1 I begin with another example of a student project 2 1 Kymn the Rower Kymn was a member of the women s varsity crew at the University of Wisconsin Madison When she could not practice on a lake she would work out on a rowing simulation device called an ergometer One does not simply sit down at an ergometer and begin to row It is necessary to choose the setting for the machine There are four possible settings obtained by combining two dichotomies One can opt for the small gear setting or the large gear setting One can choose to have the vent open or closed Kymn decided that she was not interested in two of these settings the large gear with the vent closed would be too easy and the small gear with the vent open would too difficult for a useful workout As a result Kymn wanted to compare the following two settings Treatment 1 The small gear with the vent closed and Treatment 2 The large gear with the vent open For her response Kymn chose the time measured to the nearest second she required to row the equivalent of 2000 meters In the above I have implicitly defined Kymn s trial as sitting on the erg and rowing the equivalent of 2000 meters Kymn decided to perform a total of 10 trials in her study Kymn s data are in Table 2 1 with dot plots in Figure 2 1 Look at these data for a few minutes What do you see Below are some features that I will note 1 Every response on treatment 2 is smaller than every response on treatment 1 27 Table 2 1 Kymn s times in seconds to row 2000 meters on an ergometer Treatment 1 is the small gear with the vent closed and treatment 2 is the large gear with the vent open 1 2 3 4 5 6 Trial Treatment 2 1 1 1 2 2 Response 485 493 489 492 483 488 7 8 9 10 1 2 2 1 490 479 486 493 Figure 2 1 The dot plots for Kymn s rowing study Small Gear Vent Open 479 481 483 485 483 485 487 489 491 493 489 491 493 Large Gear Vent Closed 479 481 487 2 The variation in treatment 2 is larger than the variation in treatment 1 Having noted this fact in both treatments there is very little within treatment variation It is impressive yet perhaps unsurprising for a well conditioned athlete that in response times of slightly more than 8 minutes there is so little variation in trial to trial performance If one looks at the dot plot and remembers the center of gravity interpretation of the mean one can see that the mean on treatment 1 is a bit larger than 491 seconds and that the mean on treatment 2 is a bit smaller than 485 seconds these visual conclusions are supported by computation In particular for future reference note that the means medians and standard deviations of these data are x 491 4 x 492 s1 1 817 y 484 2 y 485 and s2 3 420 2 2 Sara s Golf Study Histograms Sara performed a balanced CRD with 80 trials Her response was the distance in yards that she hit a golf ball at a driving range She hit the ball into a net which displayed how far the ball would have traveled in real life I have no idea how accurate these devices are Sara had two treatments hitting the ball with a 3 Wood treatment 1 and hitting the ball with a 3 Iron treatment 2 If you don t know much about golf don t worry all that matters is that Sara wanted to compare two clubs with particular interest in learning which would lead to a larger response 28 Table 2 2 The distance Sara hit a golf ball in yards sorted by treatment 22 101 113 128 3 Wood 32 38 56 58 77 81 93 99 101 104 107 107 108 109 109 110 114 115 116 118 122 122 127 127 128 129 131 131 137 139 139 140 27 84 100 118 3 Iron 52 53 57 58 59 68 68 68 82 88 92 92 92 92 97 97 98 99 101 105 107 107 107 108 109 110 116 127 132 132 136 136 137 138 139 139 101 111 128 147 Figure 2 2 The dot plots for Sara s golf study 3 Wood o 20 o o 30 40 oo 50 60 3 Iron o 20 30 oo ooo 40 50 60 o o 70 o o o 70 o 80 90 o o o oo o oo o o oo o oooooooooo o oooo ooo 100 110 120 130 140 150 o o o o o o oo o o ooooo ooooo oo 90 80 o o oo o o oooo 100 110 120 130 140 150 Sara s data sorted by treatment are presented in Table 2 2 Even a cursory examination of this table reveals that within each treatment there is a huge amount of variation in Sara s responses Dot plots of Sara s data are presented in Figure 2 2 I don t like these dot plots very much but let me begin by mentioning their good features As with all dot plots each plot is a valid presentation of its observations If you want to see the exact values of all of the observations and how they relate spatially the dot plot is great In addition a dot plot is good at revealing outliers we can see the three very small response values with the 3 Wood and the one very small value with the 3 Iron Now I will discuss briefly what I don t like about these dot plots The 3 Wood data range from a minimum of 22 yards to a maximum of 147 yards This distance 125 yards towers over the number of observations 40 As a result there must be and are a large number of gaps in our picture and usually there are weird exceptions with so little data spread out so far the peaks are very short and hence likely have no scientific meaning There is another way to view the above comments the dot plot is very bumpy i e it is not very smooth As I will 29 Table 2 3 Frequency tables of the distances Sara hit a golf ball by treatment Class Width Freq Interval w f 0 25 25 1 25 50 25 2 50 75 25 2 75 100 25 4 100 125 25 18 125 150 25 13 Total 40 n1 3 Wood 3 Iron Rel Freq Density Freq Rel Freq Density rf f n1 d rf w f rf f n2 d rf w 0 025 0 001 0 0 000 0 000 0 050 0 002 1 0 025 0 001 0 050 0 002 8 0 200 0 008 0 100 0 004 …
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