The CRD with a Numerical Response: ContinuedKymn the RowerSara's Golf Study; HistogramsKernel DensitiesInterpreting the Standard DeviationCathy's Running StudyComputingSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 2The CRD with a Numerical Response:ContinuedThis chapter continues the theme of Chapter 1. I begin with another example of a student project.2.1 Kymn the RowerKymn was a member of the women’s varsity crew at the University o f Wisconsin-Madison. Whenshe could not practice on a lake, she would work out on a rowing simulation device called anergometer. One does not simply sit down at an ergometer and begin to row. It is necessary tochoose the setting for the machine. There are four possible sett ings, obtained by combining t wodichotomies:• One can opt for the small gear setting or the large gear setting.• One can choose to have th e vent open or closed.Kymn decided that she was not interested in two of these settings: the large gear w ith the ventclosed would be too easy and the small gear with the vent open would too di fficult for a usefulworkout. 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 th e vent open.For her response, Kymn chose the time, measured to the n earest second, she required to row theequivalent of 2000 meters.In the above, I have implicitly defined Kymn’s trial as sitting on the erg and rowing the equiv-alent 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 Figu re 2.1. Look at these data for a fewminutes. Wh at do y ou s ee? Below are some features that I will note.1. Every response on treatment 2 is smaller than every response on treatment 1.27Table 2.1: Kymn’s times, in seconds, to row 2000 meters on an ergometer. Treatment 1 is the smallgear with the vent clos ed; and treatm ent 2 is the l arge gear with the vent open.Trial:1 2 3 4 5 6 7 8 9 10Treatment: 2 1 1 1 2 2 1 2 2 1Response:485 493 489 492 483 488 490 479 486 493Figure 2.1: The dot plots for Kymn’s rowing study.Small Gear, Vent Open:479 481 483 485 487 489 491 493• • • ••Large Gear, Vent Closed:479 481 483 485 487 489 491 493• • • • •2. The variation in treatment 2 is larger than the variation in treatment 1. Having noted thisfact, in both treatments t here is very little with in-treatment variation. It is impressive, yetperhaps unsurprising for a well-condition ed athlete, that in response times of slightly mo rethan 8 minutes, there is so little variation in trial-to-trial performance.If one looks at the d ot plot, and remem bers the center of gravity interpretation of the mean, one cansee that the mean on treatment 1 is a bit larger than 491 seconds and that the mean on treatment 2is a bit smaller than 485 seconds; these visual conclusion s are supported by computation. Inparticular, for future reference note that the means, medians and standard deviations of these dataare:¯x = 491.4, ˜x = 492, s1= 1.817, ¯y = 484.2, ˜y = 485 and s2= 3.420.2.2 Sara’s Golf Study; HistogramsSara performed a balanced CRD with 80 tri al s. Her respo nse was the distance—in yards—that shehit a golf ball at a driving range. (She hit the ball into a net which displayed how far the ball wouldhave 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 youdon’t know much about golf, don’t worry; all that m at ters is that Sara wanted to compare two clubswith particular interest in learning which would lead to a larger response.28Table 2.2: The distance Sara hit a golf ball, in yards, sorted by treatment.3-Wood22 32 38 56 58 77 81 93 99 101101 101 104 107 107 108 109 109 110 111113 114 115 116 118 122 122 127 127 128128 128 129 131 131 137 139 139 140 1473-Iron27 52 53 57 58 59 68 68 68 8284 88 92 92 92 92 97 97 98 99100 101 105 107 107 107 108 109 110 116118 127 132 132 136 136 137 138 139 139Figure 2.2: The dot plots for Sara’s golf study.20 30 40 50 60 70 80 90 100 110 120 130 140 1503-Wood:o o o oo o o o ooooo oooooooooooo oooooooooooooo o20 30 40 50 60 70 8090100 110 120 130 140 1503-Iron:o oo ooo ooooo o oooooooooo ooooooo oo o ooooooooSara’s data, sorted by treatment, are presented in Table 2.2. Even a cursory examination of thistable reveals that, within each treatment, there is a hug e amount of variation in Sara’s responses.Dot plots of Sara’s data are presented in Figure 2.2.I d on’t like these dot plots very much, but let m e begin by mentioning their good features. Aswith all dot plots, each pl ot is a valid presentation of its observations. If you want to see the exactvalues of all of the observations and how they relate spati ally, the dot plot is great. In addition,a dot plot is good at revealing outliers: we can s ee the t hree very small response values with the3-Wood and the one very small value w ith the 3-Iron. Now I wi ll discuss, briefly, what I don’t likeabout 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 num ber of observations , 40. As a result, there must be, and are, a largenumber of gaps in our picture and usually (there are wei rd exceptions) wi th so little data spreadout so far, the peaks are very short and, hence, likely have no scientific meaning. There is anotherway to view the above comment s: the dot plot is very bumpy; i.e., it is not very smooth. As I will29Table 2.3: Frequency tables of the distances Sara h it a golf ball, by treatment.3-Wood 3-IronClass Width Freq. Rel. Freq. Density Freq. Rel. Freq. DensityInterval (w)(f) (rf= f/n1) (d = rf/w) (f) (rf= f/n2) (d = rf/w)0–25 25 1 0.025 0.001 0 0.000 0.00025–50 25 2 0.050 0.002 1 0.025 0.00150–75 25 2 0.050 0.002 8 0.200 0.00875–100 254 0.100 0.004 11 0.275 0.011100–125 25 18 0.450 0.018 11 0.275 0.011125–150 25 13 0.325 0.013 9 0.225 0.009Total — 40 1.000 — 40 1.000 —(n1) (n2)discuss later in the subsection on kernel densities, smoothness is very important to sci entists.Here is what I mean by bumpy. Imagine the number line is a road and the dots are bumps inthe road. Driving a car (or if you prefer a greener example, riding a bike) along the road will resultin a flat road (the gaps) interrupted by numerous bumps
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