(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] ROCKRAMA Data analysis through mathematical model of data collected from a "rock" experiment :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] An experiment in data collection and analysis is described in which collective groups of subjects are analyzed as to their abilities to differentiate masses of rocks. A reasonable sigmoidal function is needed to sit the data and comparisons between groups of subjects leads to fun discussion as to which group is better. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: ROCKRAMA Full title: Data analysis through mathematical model of data collected from a "rock" experiment. Developer: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Equipment Needed: :[font = subsubsection; inactive; preserveAspect] 5-10 small rocks. Use the same kind (i.e. same density), try to get them smooth without jagged edges, and make them in the 50 - 150 gram range. :[font = subsubsection; inactive; preserveAspect] 100 gram mass from an old science lab scale set. This will be called our known or reference mass. In lieu of this you can use a rock of "known" mass as the standard with which you compare the other rocks. :[font = subsubsection; inactive; preserveAspect] Shoebox or small container to hold the rocks and the known mass. :[font = subsubsection; inactive; preserveAspect] Forms (described below) which you use to anonymously collect data and a few pencils - attached by string to the table or box. :[font = subsubsection; inactive; preserveAspect; endGroup] A table on which to set out this equipment. The table should be in a public area where students, faculty, and staff can see it, but in a "protected" area where it would not be subject to "vandalism." :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] The Experiment: :[font = subsubsection; inactive; preserveAspect] We seek to determine how good folks are at distinguishing between different masses. Accordingly, we set out 5 - 10 masses in the 50 - 150 g range and a known or reference mass of 100 grams. We label the rocks in random order, 1, 2, 3, . . . , 9 say. (Do not label them in increasing mass order!) And we identify the known mass and its actual mass for the participants. :[font = subsubsection; inactive; preserveAspect] We ask participants to indicate whether the sample rocks are of greater mass than the known mass. We provide a slip of paper in which we could ask them some information for later references, e.g., their major, their position (student - undergraduate or graduate, faculty and discipline, administrative, staff, etc.) Do not ask for names unless, say, you wish to have a contest for "best" guesses. The real information we want is their opinion on whether the mass of the rocks, numbered 1, 2, . . . , 9, are more than the sample/known mass. So we provide a form such as the following: Sample Rock Number 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 Check if mass is more | | | | | | | | than 100 g :[font = subsubsection; inactive; preserveAspect] We set the collection table out for whatever time we believe reasonable to get sample data - one week will be sufficient especially if there is a sign, say with words like "Rock-A-Rama," "Let's Rock," "Science Experiment," or "Which Major is More Perceptive," and some catchy graphics. :[font = subsubsection; inactive; preserveAspect] Once we have collected the data, we compute the percent of respondents who say that a given rock has mass greater than the 100 g standard mass. We plot this percent vs. the actual mass of the given rock. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] If participants had perfect knowledge, e.g. a scale - which we do not provide(!), the plot of the returned data would look like this: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0.014715 0.588604 [ [(50)] .2619 .01472 0 2 Msboxa [(100)] .5 .01472 0 2 Msboxa [(150)] .7381 .01472 0 2 Msboxa [(200)] .97619 .01472 0 2 Msboxa [(Mass of Rock)] 1.025 .01472 -1 0 Msboxa [(0.2)] .01131 .13244 1 0 Msboxa [(0.4)] .01131 .25016 1 0 Msboxa [(0.6)] .01131 .36788 1 0 Msboxa [(0.8)] .01131 .4856 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [(% saying mass > 100 g)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(50)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(100)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(150)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(200)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P [(Mass of Rock)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .13244 m .03006 .13244 L s P [(0.2)] .01131 .13244 1 0 Mshowa p .002 w .02381 .25016 m .03006 .25016 L s P [(0.4)] .01131 .25016 1 0 Mshowa p .002 w .02381 .36788 m .03006 .36788 L s P [(0.6)] .01131 .36788 1 0 Mshowa p .002 w .02381 .4856 m .03006 .4856 L s P [(0.8)] .01131 .4856 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .03826 m .02756 .03826 L s P p .001 w .02381 .0618 m .02756 .0618 L s P p .001 w .02381 .08535 m .02756 .08535 L s P p .001 w .02381 .10889 m .02756 .10889 L s P p .001 w .02381 .15598 m .02756 .15598 L s P p .001 w .02381 .17952 m .02756 .17952 L s P p .001 w .02381 .20307 m .02756 .20307 L s P p .001 w .02381 .22661 m .02756 .22661 L s P p .001 w .02381 .2737 m .02756 .2737 L s P p .001 w .02381 .29724 m .02756 .29724 L s P p .001 w .02381 .32079 m .02756 .32079 L s P p .001 w .02381 .34433 m .02756 .34433 L s P p .001 w .02381 .39142 m .02756 .39142 L s P p .001 w .02381 .41497 m .02756 .41497 L s P p .001 w .02381 .43851 m .02756 .43851 L s P p .001 w .02381 .46205 m .02756 .46205 L s P p .001 w .02381 .50914 m .02756 .50914 L s P p .001 w .02381 .53269 m .02756 .53269 L s P p .001 w .02381 .55623 m .02756 .55623 L s P p .001 w .02381 .57977 m .02756 .57977 L s P [(% saying mass > 100 g)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .015 w .02381 .01472 m .06349 .01472 L .10317 .01472 L .14286 .01472 L .18254 .01472 L .22222 .01472 L .2619 .01472 L .30159 .01472 L .34127 .01472 L .38095 .01472 L .42063 .01472 L .44048 .01472 L .46032 .01472 L .47024 .01472 L .48016 .01472 L .48512 .01472 L .49008 .01472 L .49256 .01472 L .49504 .01472 L .49628 .01472 L .49752 .01472 L .5 .30902 L .50124 .60332 L .50248 .60332 L .50496 .60332 L .50992 .60332 L .51984 .60332 L .53968 .60332 L .57937 .60332 L .61905 .60332 L .65873 .60332 L .69841 .60332 L .7381 .60332 L .77778 .60332 L .81746 .60332 L .85714 .60332 L .89683 .60332 L .93651 .60332 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] If all or most participants just guessed then the plot of the returned data could look something like this: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0 0.618034 [ [(0)] .02381 0 0 2 Msboxa [(50)] .2619 0 0 2 Msboxa [(100)] .5 0 0 2 Msboxa [(150)] .7381 0 0 2 Msboxa [(200)] .97619 0 0 2 Msboxa [(Mass of Rock)] 1.025 0 -1 0 Msboxa [(0.2)] .01131 .12361 1 0 Msboxa [(0.4)] .01131 .24721 1 0 Msboxa [(0.6)] .01131 .37082 1 0 Msboxa [(0.8)] .01131 .49443 1 0 Msboxa [(1)] .01131 .61803 1 0 Msboxa [(% saying mass > 100 g)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 0 m .02381 .00625 L s P [(0)] .02381 0 0 2 Mshowa p .002 w .2619 0 m .2619 .00625 L s P [(50)] .2619 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(100)] .5 0 0 2 Mshowa p .002 w .7381 0 m .7381 .00625 L s P [(150)] .7381 0 0 2 Mshowa p .002 w .97619 0 m .97619 .00625 L s P [(200)] .97619 0 0 2 Mshowa p .001 w .07143 0 m .07143 .00375 L s P p .001 w .11905 0 m .11905 .00375 L s P p .001 w .16667 0 m .16667 .00375 L s P p .001 w .21429 0 m .21429 .00375 L s P p .001 w .30952 0 m .30952 .00375 L s P p .001 w .35714 0 m .35714 .00375 L s P p .001 w .40476 0 m .40476 .00375 L s P p .001 w .45238 0 m .45238 .00375 L s P p .001 w .54762 0 m .54762 .00375 L s P p .001 w .59524 0 m .59524 .00375 L s P p .001 w .64286 0 m .64286 .00375 L s P p .001 w .69048 0 m .69048 .00375 L s P p .001 w .78571 0 m .78571 .00375 L s P p .001 w .83333 0 m .83333 .00375 L s P p .001 w .88095 0 m .88095 .00375 L s P p .001 w .92857 0 m .92857 .00375 L s P [(Mass of Rock)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w .02381 .12361 m .03006 .12361 L s P [(0.2)] .01131 .12361 1 0 Mshowa p .002 w .02381 .24721 m .03006 .24721 L s P [(0.4)] .01131 .24721 1 0 Mshowa p .002 w .02381 .37082 m .03006 .37082 L s P [(0.6)] .01131 .37082 1 0 Mshowa p .002 w .02381 .49443 m .03006 .49443 L s P [(0.8)] .01131 .49443 1 0 Mshowa p .002 w .02381 .61803 m .03006 .61803 L s P [(1)] .01131 .61803 1 0 Mshowa p .001 w .02381 .02472 m .02756 .02472 L s P p .001 w .02381 .04944 m .02756 .04944 L s P p .001 w .02381 .07416 m .02756 .07416 L s P p .001 w .02381 .09889 m .02756 .09889 L s P p .001 w .02381 .14833 m .02756 .14833 L s P p .001 w .02381 .17305 m .02756 .17305 L s P p .001 w .02381 .19777 m .02756 .19777 L s P p .001 w .02381 .22249 m .02756 .22249 L s P p .001 w .02381 .27193 m .02756 .27193 L s P p .001 w .02381 .29666 m .02756 .29666 L s P p .001 w .02381 .32138 m .02756 .32138 L s P p .001 w .02381 .3461 m .02756 .3461 L s P p .001 w .02381 .39554 m .02756 .39554 L s P p .001 w .02381 .42026 m .02756 .42026 L s P p .001 w .02381 .44498 m .02756 .44498 L s P p .001 w .02381 .46971 m .02756 .46971 L s P p .001 w .02381 .51915 m .02756 .51915 L s P p .001 w .02381 .54387 m .02756 .54387 L s P p .001 w .02381 .56859 m .02756 .56859 L s P p .001 w .02381 .59331 m .02756 .59331 L s P [(% saying mass > 100 g)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .015 w .02381 .30902 m .06349 .30902 L .10317 .30902 L .14286 .30902 L .18254 .30902 L .22222 .30902 L .2619 .30902 L .30159 .30902 L .34127 .30902 L .38095 .30902 L .42063 .30902 L .46032 .30902 L .5 .30902 L .53968 .30902 L .57937 .30902 L .61905 .30902 L .65873 .30902 L .69841 .30902 L .7381 .30902 L .77778 .30902 L .81746 .30902 L .85714 .30902 L .89683 .30902 L .93651 .30902 L .97619 .30902 L s P P % End of Graphics MathPictureEnd :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] A reasonable population with 9 sample rocks 10, 40, 70, 90, 100, 120, 125, 150, 175 g might yield the following plot. :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -1.04083e-17 0.005 0 0.618034 [ [(25)] .125 0 0 2 Msboxa [(50)] .25 0 0 2 Msboxa [(75)] .375 0 0 2 Msboxa [(100)] .5 0 0 2 Msboxa [(125)] .625 0 0 2 Msboxa [(150)] .75 0 0 2 Msboxa [(175)] .875 0 0 2 Msboxa [(200)] 1 0 0 2 Msboxa [(Mass of Rock)] 1.025 0 -1 0 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [(% saying mass > 100 g)] 0 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .125 0 m .125 .00625 L s P [(25)] .125 0 0 2 Mshowa p .002 w .25 0 m .25 .00625 L s P [(50)] .25 0 0 2 Mshowa p .002 w .375 0 m .375 .00625 L s P [(75)] .375 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(100)] .5 0 0 2 Mshowa p .002 w .625 0 m .625 .00625 L s P [(125)] .625 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(150)] .75 0 0 2 Mshowa p .002 w .875 0 m .875 .00625 L s P [(175)] .875 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(200)] 1 0 0 2 Mshowa p .001 w .025 0 m .025 .00375 L s P p .001 w .05 0 m .05 .00375 L s P p .001 w .075 0 m .075 .00375 L s P p .001 w .1 0 m .1 .00375 L s P p .001 w .15 0 m .15 .00375 L s P p .001 w .175 0 m .175 .00375 L s P p .001 w .2 0 m .2 .00375 L s P p .001 w .225 0 m .225 .00375 L s P p .001 w .275 0 m .275 .00375 L s P p .001 w .3 0 m .3 .00375 L s P p .001 w .325 0 m .325 .00375 L s P p .001 w .35 0 m .35 .00375 L s P p .001 w .4 0 m .4 .00375 L s P p .001 w .425 0 m .425 .00375 L s P p .001 w .45 0 m .45 .00375 L s P p .001 w .475 0 m .475 .00375 L s P p .001 w .525 0 m .525 .00375 L s P p .001 w .55 0 m .55 .00375 L s P p .001 w .575 0 m .575 .00375 L s P p .001 w .6 0 m .6 .00375 L s P p .001 w .65 0 m .65 .00375 L s P p .001 w .675 0 m .675 .00375 L s P p .001 w .7 0 m .7 .00375 L s P p .001 w .725 0 m .725 .00375 L s P p .001 w .775 0 m .775 .00375 L s P p .001 w .8 0 m .8 .00375 L s P p .001 w .825 0 m .825 .00375 L s P p .001 w .85 0 m .85 .00375 L s P p .001 w .9 0 m .9 .00375 L s P p .001 w .925 0 m .925 .00375 L s P p .001 w .95 0 m .95 .00375 L s P p .001 w .975 0 m .975 .00375 L s P [(Mass of Rock)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P [(% saying mass > 100 g)] 0 .61803 0 -4 Mshowa p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .02 w .05 .01236 Mdot .2 .0618 Mdot .35 .12979 Mdot .45 .27812 Mdot .5 .30902 Mdot .6 .40172 Mdot .625 .46353 Mdot .75 .55623 Mdot .875 .60567 Mdot P P % End of Graphics MathPictureEnd :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] The Statement of the Problem - finally! :[font = subsubsection; inactive; preserveAspect; endGroup] Fit a curve (or curves) through the response data, perhaps attempting to differentiate groups. We might even wish to rank the groups based on this criteria, e.g., can geologists differentiate rock mass better than political science majors? :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Commands to generate graphics for Statement of Problem. :[font = input; Cclosed; preserveAspect; startGroup] Plot[(1+Sign[x-100])/2,{x,0,200},PlotStyle->Thickness[.015], AxesLabel->{"Mass of Rock", "% saying mass > 100 g"}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0.014715 0.588604 [ [(50)] .2619 .01472 0 2 Msboxa [(100)] .5 .01472 0 2 Msboxa [(150)] .7381 .01472 0 2 Msboxa [(200)] .97619 .01472 0 2 Msboxa [(Mass of Rock)] 1.025 .01472 -1 0 Msboxa [(0.2)] .01131 .13244 1 0 Msboxa [(0.4)] .01131 .25016 1 0 Msboxa [(0.6)] .01131 .36788 1 0 Msboxa [(0.8)] .01131 .4856 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [(% saying mass > 100 g)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(50)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(100)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(150)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(200)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P [(Mass of Rock)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .13244 m .03006 .13244 L s P [(0.2)] .01131 .13244 1 0 Mshowa p .002 w .02381 .25016 m .03006 .25016 L s P [(0.4)] .01131 .25016 1 0 Mshowa p .002 w .02381 .36788 m .03006 .36788 L s P [(0.6)] .01131 .36788 1 0 Mshowa p .002 w .02381 .4856 m .03006 .4856 L s P [(0.8)] .01131 .4856 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .03826 m .02756 .03826 L s P p .001 w .02381 .0618 m .02756 .0618 L s P p .001 w .02381 .08535 m .02756 .08535 L s P p .001 w .02381 .10889 m .02756 .10889 L s P p .001 w .02381 .15598 m .02756 .15598 L s P p .001 w .02381 .17952 m .02756 .17952 L s P p .001 w .02381 .20307 m .02756 .20307 L s P p .001 w .02381 .22661 m .02756 .22661 L s P p .001 w .02381 .2737 m .02756 .2737 L s P p .001 w .02381 .29724 m .02756 .29724 L s P p .001 w .02381 .32079 m .02756 .32079 L s P p .001 w .02381 .34433 m .02756 .34433 L s P p .001 w .02381 .39142 m .02756 .39142 L s P p .001 w .02381 .41497 m .02756 .41497 L s P p .001 w .02381 .43851 m .02756 .43851 L s P p .001 w .02381 .46205 m .02756 .46205 L s P p .001 w .02381 .50914 m .02756 .50914 L s P p .001 w .02381 .53269 m .02756 .53269 L s P p .001 w .02381 .55623 m .02756 .55623 L s P p .001 w .02381 .57977 m .02756 .57977 L s P [(% saying mass > 100 g)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .015 w .02381 .01472 m .06349 .01472 L .10317 .01472 L .14286 .01472 L .18254 .01472 L .22222 .01472 L .2619 .01472 L .30159 .01472 L .34127 .01472 L .38095 .01472 L .42063 .01472 L .44048 .01472 L .46032 .01472 L .47024 .01472 L .48016 .01472 L .48512 .01472 L .49008 .01472 L .49256 .01472 L .49504 .01472 L .49628 .01472 L .49752 .01472 L .5 .30902 L .50124 .60332 L .50248 .60332 L .50496 .60332 L .50992 .60332 L .51984 .60332 L .53968 .60332 L .57937 .60332 L .61905 .60332 L .65873 .60332 L .69841 .60332 L .7381 .60332 L .77778 .60332 L .81746 .60332 L .85714 .60332 L .89683 .60332 L .93651 .60332 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Plot[.5,{x,0,200},PlotStyle->Thickness[.015],PlotRange->{0,1}, AxesLabel->{"Mass of Rock", "% saying mass > 100 g"}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0 0.618034 [ [(0)] .02381 0 0 2 Msboxa [(50)] .2619 0 0 2 Msboxa [(100)] .5 0 0 2 Msboxa [(150)] .7381 0 0 2 Msboxa [(200)] .97619 0 0 2 Msboxa [(Mass of Rock)] 1.025 0 -1 0 Msboxa [(0.2)] .01131 .12361 1 0 Msboxa [(0.4)] .01131 .24721 1 0 Msboxa [(0.6)] .01131 .37082 1 0 Msboxa [(0.8)] .01131 .49443 1 0 Msboxa [(1)] .01131 .61803 1 0 Msboxa [(% saying mass > 100 g)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 0 m .02381 .00625 L s P [(0)] .02381 0 0 2 Mshowa p .002 w .2619 0 m .2619 .00625 L s P [(50)] .2619 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(100)] .5 0 0 2 Mshowa p .002 w .7381 0 m .7381 .00625 L s P [(150)] .7381 0 0 2 Mshowa p .002 w .97619 0 m .97619 .00625 L s P [(200)] .97619 0 0 2 Mshowa p .001 w .07143 0 m .07143 .00375 L s P p .001 w .11905 0 m .11905 .00375 L s P p .001 w .16667 0 m .16667 .00375 L s P p .001 w .21429 0 m .21429 .00375 L s P p .001 w .30952 0 m .30952 .00375 L s P p .001 w .35714 0 m .35714 .00375 L s P p .001 w .40476 0 m .40476 .00375 L s P p .001 w .45238 0 m .45238 .00375 L s P p .001 w .54762 0 m .54762 .00375 L s P p .001 w .59524 0 m .59524 .00375 L s P p .001 w .64286 0 m .64286 .00375 L s P p .001 w .69048 0 m .69048 .00375 L s P p .001 w .78571 0 m .78571 .00375 L s P p .001 w .83333 0 m .83333 .00375 L s P p .001 w .88095 0 m .88095 .00375 L s P p .001 w .92857 0 m .92857 .00375 L s P [(Mass of Rock)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w .02381 .12361 m .03006 .12361 L s P [(0.2)] .01131 .12361 1 0 Mshowa p .002 w .02381 .24721 m .03006 .24721 L s P [(0.4)] .01131 .24721 1 0 Mshowa p .002 w .02381 .37082 m .03006 .37082 L s P [(0.6)] .01131 .37082 1 0 Mshowa p .002 w .02381 .49443 m .03006 .49443 L s P [(0.8)] .01131 .49443 1 0 Mshowa p .002 w .02381 .61803 m .03006 .61803 L s P [(1)] .01131 .61803 1 0 Mshowa p .001 w .02381 .02472 m .02756 .02472 L s P p .001 w .02381 .04944 m .02756 .04944 L s P p .001 w .02381 .07416 m .02756 .07416 L s P p .001 w .02381 .09889 m .02756 .09889 L s P p .001 w .02381 .14833 m .02756 .14833 L s P p .001 w .02381 .17305 m .02756 .17305 L s P p .001 w .02381 .19777 m .02756 .19777 L s P p .001 w .02381 .22249 m .02756 .22249 L s P p .001 w .02381 .27193 m .02756 .27193 L s P p .001 w .02381 .29666 m .02756 .29666 L s P p .001 w .02381 .32138 m .02756 .32138 L s P p .001 w .02381 .3461 m .02756 .3461 L s P p .001 w .02381 .39554 m .02756 .39554 L s P p .001 w .02381 .42026 m .02756 .42026 L s P p .001 w .02381 .44498 m .02756 .44498 L s P p .001 w .02381 .46971 m .02756 .46971 L s P p .001 w .02381 .51915 m .02756 .51915 L s P p .001 w .02381 .54387 m .02756 .54387 L s P p .001 w .02381 .56859 m .02756 .56859 L s P p .001 w .02381 .59331 m .02756 .59331 L s P [(% saying mass > 100 g)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .015 w .02381 .30902 m .06349 .30902 L .10317 .30902 L .14286 .30902 L .18254 .30902 L .22222 .30902 L .2619 .30902 L .30159 .30902 L .34127 .30902 L .38095 .30902 L .42063 .30902 L .46032 .30902 L .5 .30902 L .53968 .30902 L .57937 .30902 L .61905 .30902 L .65873 .30902 L .69841 .30902 L .7381 .30902 L .77778 .30902 L .81746 .30902 L .85714 .30902 L .89683 .30902 L .93651 .30902 L .97619 .30902 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect] data = {{10,.02},{40,.1},{70,.21},{90,.45},{100,.5}, {120,.65},{125,.75},{150,.9},{175,.98}}; :[font = input; Cclosed; preserveAspect; startGroup] ListPlot[data,PlotStyle->{PointSize[.02]}, PlotRange->{{0,200},{0,1}}, AxesLabel->{"Mass of Rock", "% saying mass > 100 g"}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -1.04083e-17 0.005 0 0.618034 [ [(25)] .125 0 0 2 Msboxa [(50)] .25 0 0 2 Msboxa [(75)] .375 0 0 2 Msboxa [(100)] .5 0 0 2 Msboxa [(125)] .625 0 0 2 Msboxa [(150)] .75 0 0 2 Msboxa [(175)] .875 0 0 2 Msboxa [(200)] 1 0 0 2 Msboxa [(Mass of Rock)] 1.025 0 -1 0 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [(% saying mass > 100 g)] 0 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .125 0 m .125 .00625 L s P [(25)] .125 0 0 2 Mshowa p .002 w .25 0 m .25 .00625 L s P [(50)] .25 0 0 2 Mshowa p .002 w .375 0 m .375 .00625 L s P [(75)] .375 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(100)] .5 0 0 2 Mshowa p .002 w .625 0 m .625 .00625 L s P [(125)] .625 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(150)] .75 0 0 2 Mshowa p .002 w .875 0 m .875 .00625 L s P [(175)] .875 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(200)] 1 0 0 2 Mshowa p .001 w .025 0 m .025 .00375 L s P p .001 w .05 0 m .05 .00375 L s P p .001 w .075 0 m .075 .00375 L s P p .001 w .1 0 m .1 .00375 L s P p .001 w .15 0 m .15 .00375 L s P p .001 w .175 0 m .175 .00375 L s P p .001 w .2 0 m .2 .00375 L s P p .001 w .225 0 m .225 .00375 L s P p .001 w .275 0 m .275 .00375 L s P p .001 w .3 0 m .3 .00375 L s P p .001 w .325 0 m .325 .00375 L s P p .001 w .35 0 m .35 .00375 L s P p .001 w .4 0 m .4 .00375 L s P p .001 w .425 0 m .425 .00375 L s P p .001 w .45 0 m .45 .00375 L s P p .001 w .475 0 m .475 .00375 L s P p .001 w .525 0 m .525 .00375 L s P p .001 w .55 0 m .55 .00375 L s P p .001 w .575 0 m .575 .00375 L s P p .001 w .6 0 m .6 .00375 L s P p .001 w .65 0 m .65 .00375 L s P p .001 w .675 0 m .675 .00375 L s P p .001 w .7 0 m .7 .00375 L s P p .001 w .725 0 m .725 .00375 L s P p .001 w .775 0 m .775 .00375 L s P p .001 w .8 0 m .8 .00375 L s P p .001 w .825 0 m .825 .00375 L s P p .001 w .85 0 m .85 .00375 L s P p .001 w .9 0 m .9 .00375 L s P p .001 w .925 0 m .925 .00375 L s P p .001 w .95 0 m .95 .00375 L s P p .001 w .975 0 m .975 .00375 L s P [(Mass of Rock)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P [(% saying mass > 100 g)] 0 .61803 0 -4 Mshowa p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .02 w .05 .01236 Mdot .2 .0618 Mdot .35 .12979 Mdot .45 .27812 Mdot .5 .30902 Mdot .6 .40172 Mdot .625 .46353 Mdot .75 .55623 Mdot .875 .60567 Mdot P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Data, data collection, curve fitting, comparison curve fitting, parameter estimation, comparison criteria. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect] Students need to be willing to accept other students' opinions on what is best fit and perhaps, on what is the best type of curve to use to fit data, on what is a good parameter to use to compare groups, etc. :[font = subsubsection; inactive; preserveAspect; endGroup] Faculty need to be accepting of what comes down the pike, e.g., students will try shifts of ArcTan function, polynomials, exponentials, combinations, etc. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect] Students need to be doing some sort of curve fitting, perhaps in several variables and certainly non-linear - but software may make this easier. :[font = subsubsection; inactive; preserveAspect; endGroup] Students need to be familiar with shapes of curves, certainly a sigmoidal, leveling off curve which would model this best. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] Allow one half of a class period for groups to form, activities to be assigned, and logistics to be addressed. Spend perhaps an additional 1/2 period at a pre set-up to check that everything is ready. Then spend perhaps another half class period coming up with ideas on fitting and comparison. This latter should fit in with course activities at hand. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect] Students will get enthusiastic about setting up a data collection site, about what to ask for in addition to rock responses, about tallying the data, about picking the model, about fitting the model, and about selecting comparison criteria between groups. :[font = subsubsection; inactive; preserveAspect; endGroup] It is appropriate to have individuals or subgroups write up their own report on what they did with the data, e.g., fitting and comparison. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students will have a willingness to operate in vague areas in which their opinions and observations are valued. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect] Students could compare results with two populations, say some doing "sharp" rocks - where perhaps the sharpness in the hand may lead to the perception the rock is "heavier" and others doing "smooth" rocks. :[font = subsubsection; inactive; preserveAspect] Comparisons might be done with heavier masses, e.g., use 1 kilogram as the standard mass. :[font = subsubsection; inactive; preserveAspect; endGroup] Refinements could be done on other variables, e.g., class - soph, jr, sr., etc., major, "dorm," home state, etc. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect] Any book with a variety of curves and curve shapes from which students can choose possible types of curves. :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Any material on nonlinear curve fitting. :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We enter our data and plot it. :[font = input; preserveAspect] data = {{10,.02},{40,.1},{70,.21},{90,.45}, {100,.5},{120,.65},{125,.75},{150,.9},{175,.98}}; :[font = input; Cclosed; preserveAspect; startGroup] dataPlot = ListPlot[data,PlotStyle->{PointSize[.02]}, PlotRange->{{0,200},{0,1}}, AxesLabel->{"Mass of Rock", "% saying mass > 100 g"}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -1.04083e-17 0.005 0 0.618034 [ [(25)] .125 0 0 2 Msboxa [(50)] .25 0 0 2 Msboxa [(75)] .375 0 0 2 Msboxa [(100)] .5 0 0 2 Msboxa [(125)] .625 0 0 2 Msboxa [(150)] .75 0 0 2 Msboxa [(175)] .875 0 0 2 Msboxa [(200)] 1 0 0 2 Msboxa [(Mass of Rock)] 1.025 0 -1 0 Msboxa [(0)] -0.0125 0 1 0 Msboxa [(0.2)] -0.0125 .12361 1 0 Msboxa [(0.4)] -0.0125 .24721 1 0 Msboxa [(0.6)] -0.0125 .37082 1 0 Msboxa [(0.8)] -0.0125 .49443 1 0 Msboxa [(1)] -0.0125 .61803 1 0 Msboxa [(% saying mass > 100 g)] 0 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .125 0 m .125 .00625 L s P [(25)] .125 0 0 2 Mshowa p .002 w .25 0 m .25 .00625 L s P [(50)] .25 0 0 2 Mshowa p .002 w .375 0 m .375 .00625 L s P [(75)] .375 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(100)] .5 0 0 2 Mshowa p .002 w .625 0 m .625 .00625 L s P [(125)] .625 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(150)] .75 0 0 2 Mshowa p .002 w .875 0 m .875 .00625 L s P [(175)] .875 0 0 2 Mshowa p .002 w 1 0 m 1 .00625 L s P [(200)] 1 0 0 2 Mshowa p .001 w .025 0 m .025 .00375 L s P p .001 w .05 0 m .05 .00375 L s P p .001 w .075 0 m .075 .00375 L s P p .001 w .1 0 m .1 .00375 L s P p .001 w .15 0 m .15 .00375 L s P p .001 w .175 0 m .175 .00375 L s P p .001 w .2 0 m .2 .00375 L s P p .001 w .225 0 m .225 .00375 L s P p .001 w .275 0 m .275 .00375 L s P p .001 w .3 0 m .3 .00375 L s P p .001 w .325 0 m .325 .00375 L s P p .001 w .35 0 m .35 .00375 L s P p .001 w .4 0 m .4 .00375 L s P p .001 w .425 0 m .425 .00375 L s P p .001 w .45 0 m .45 .00375 L s P p .001 w .475 0 m .475 .00375 L s P p .001 w .525 0 m .525 .00375 L s P p .001 w .55 0 m .55 .00375 L s P p .001 w .575 0 m .575 .00375 L s P p .001 w .6 0 m .6 .00375 L s P p .001 w .65 0 m .65 .00375 L s P p .001 w .675 0 m .675 .00375 L s P p .001 w .7 0 m .7 .00375 L s P p .001 w .725 0 m .725 .00375 L s P p .001 w .775 0 m .775 .00375 L s P p .001 w .8 0 m .8 .00375 L s P p .001 w .825 0 m .825 .00375 L s P p .001 w .85 0 m .85 .00375 L s P p .001 w .9 0 m .9 .00375 L s P p .001 w .925 0 m .925 .00375 L s P p .001 w .95 0 m .95 .00375 L s P p .001 w .975 0 m .975 .00375 L s P [(Mass of Rock)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 0 m .00625 0 L s P [(0)] -0.0125 0 1 0 Mshowa p .002 w 0 .12361 m .00625 .12361 L s P [(0.2)] -0.0125 .12361 1 0 Mshowa p .002 w 0 .24721 m .00625 .24721 L s P [(0.4)] -0.0125 .24721 1 0 Mshowa p .002 w 0 .37082 m .00625 .37082 L s P [(0.6)] -0.0125 .37082 1 0 Mshowa p .002 w 0 .49443 m .00625 .49443 L s P [(0.8)] -0.0125 .49443 1 0 Mshowa p .002 w 0 .61803 m .00625 .61803 L s P [(1)] -0.0125 .61803 1 0 Mshowa p .001 w 0 .02472 m .00375 .02472 L s P p .001 w 0 .04944 m .00375 .04944 L s P p .001 w 0 .07416 m .00375 .07416 L s P p .001 w 0 .09889 m .00375 .09889 L s P p .001 w 0 .14833 m .00375 .14833 L s P p .001 w 0 .17305 m .00375 .17305 L s P p .001 w 0 .19777 m .00375 .19777 L s P p .001 w 0 .22249 m .00375 .22249 L s P p .001 w 0 .27193 m .00375 .27193 L s P p .001 w 0 .29666 m .00375 .29666 L s P p .001 w 0 .32138 m .00375 .32138 L s P p .001 w 0 .3461 m .00375 .3461 L s P p .001 w 0 .39554 m .00375 .39554 L s P p .001 w 0 .42026 m .00375 .42026 L s P p .001 w 0 .44498 m .00375 .44498 L s P p .001 w 0 .46971 m .00375 .46971 L s P p .001 w 0 .51915 m .00375 .51915 L s P p .001 w 0 .54387 m .00375 .54387 L s P p .001 w 0 .56859 m .00375 .56859 L s P p .001 w 0 .59331 m .00375 .59331 L s P [(% saying mass > 100 g)] 0 .61803 0 -4 Mshowa p .002 w 0 0 m 0 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .02 w .05 .01236 Mdot .2 .0618 Mdot .35 .12979 Mdot .45 .27812 Mdot .5 .30902 Mdot .6 .40172 Mdot .625 .46353 Mdot .75 .55623 Mdot .875 .60567 Mdot P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] The conjectured model!!! :[font = input; Cclosed; preserveAspect; startGroup] f[t_] = 1/(1 + a Exp[-b t]) :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (1 + a/E^(b*t))^(-1) ;[o] 1 -------- a 1 + ---- b t E :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We formulate a Least Squares function. :[font = input; preserveAspect] h[a_,b_] = Sum[(f[data[[i]][[1]]] - data[[i]][[2]])^2, {i,1, Length[data]}]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We can look at several contour plots to get an idea of where minimum values of least square function could be. :[font = input; Cclosed; preserveAspect; startGroup] ContourPlot[h[a,b],{a,20,80},{b,0,.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 255; pictureHeight = 255] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% ContourGraphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.301282 0.016026 0.019231 9.615385 [ [(20)] .01923 0 0 2 Msboxa [(30)] .17949 0 0 2 Msboxa [(40)] .33974 0 0 2 Msboxa [(50)] .5 0 0 2 Msboxa [(60)] .66026 0 0 2 Msboxa [(70)] .82051 0 0 2 Msboxa [(80)] .98077 0 0 2 Msboxa [(0)] -0.0125 .01923 1 0 Msboxa [(0.02)] -0.0125 .21154 1 0 Msboxa [(0.04)] -0.0125 .40385 1 0 Msboxa [(0.06)] -0.0125 .59615 1 0 Msboxa [(0.08)] -0.0125 .78846 1 0 Msboxa [(0.1)] -0.0125 .98077 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .01923 0 m .01923 .00625 L s P [(20)] .01923 0 0 2 Mshowa p .002 w .17949 0 m .17949 .00625 L s P [(30)] .17949 0 0 2 Mshowa p .002 w .33974 0 m .33974 .00625 L s P [(40)] .33974 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(50)] .5 0 0 2 Mshowa p .002 w .66026 0 m .66026 .00625 L s P [(60)] .66026 0 0 2 Mshowa p .002 w .82051 0 m .82051 .00625 L s P [(70)] .82051 0 0 2 Mshowa p .002 w .98077 0 m .98077 .00625 L s P [(80)] .98077 0 0 2 Mshowa p .001 w .05128 0 m .05128 .00375 L s P p .001 w .08333 0 m .08333 .00375 L s P p .001 w .11538 0 m .11538 .00375 L s P p .001 w .14744 0 m .14744 .00375 L s P p .001 w .21154 0 m .21154 .00375 L s P p .001 w .24359 0 m .24359 .00375 L s P p .001 w .27564 0 m .27564 .00375 L s P p .001 w .30769 0 m .30769 .00375 L s P p .001 w .37179 0 m .37179 .00375 L s P p .001 w .40385 0 m .40385 .00375 L s P p .001 w .4359 0 m .4359 .00375 L s P p .001 w .46795 0 m .46795 .00375 L s P p .001 w .53205 0 m .53205 .00375 L s P p .001 w .5641 0 m .5641 .00375 L s P p .001 w .59615 0 m .59615 .00375 L s P p .001 w .62821 0 m .62821 .00375 L s P p .001 w .69231 0 m .69231 .00375 L s P p .001 w .72436 0 m .72436 .00375 L s P p .001 w .75641 0 m .75641 .00375 L s P p .001 w .78846 0 m .78846 .00375 L s P p .001 w .85256 0 m .85256 .00375 L s P p .001 w .88462 0 m .88462 .00375 L s P p .001 w .91667 0 m .91667 .00375 L s P p .001 w .94872 0 m .94872 .00375 L s P p .002 w 0 0 m 1 0 L s P p .002 w 0 .01923 m .00625 .01923 L s P [(0)] -0.0125 .01923 1 0 Mshowa p .002 w 0 .21154 m .00625 .21154 L s P [(0.02)] -0.0125 .21154 1 0 Mshowa p .002 w 0 .40385 m .00625 .40385 L s P [(0.04)] -0.0125 .40385 1 0 Mshowa p .002 w 0 .59615 m .00625 .59615 L s P [(0.06)] -0.0125 .59615 1 0 Mshowa p .002 w 0 .78846 m .00625 .78846 L s P [(0.08)] -0.0125 .78846 1 0 Mshowa p .002 w 0 .98077 m .00625 .98077 L s P [(0.1)] -0.0125 .98077 1 0 Mshowa p .001 w 0 .05769 m .00375 .05769 L s P p .001 w 0 .09615 m .00375 .09615 L s P p .001 w 0 .13462 m .00375 .13462 L s P p .001 w 0 .17308 m .00375 .17308 L s P p .001 w 0 .25 m .00375 .25 L s P p .001 w 0 .28846 m .00375 .28846 L s P p .001 w 0 .32692 m .00375 .32692 L s P p .001 w 0 .36538 m .00375 .36538 L s P p .001 w 0 .44231 m .00375 .44231 L s P p .001 w 0 .48077 m .00375 .48077 L s P p .001 w 0 .51923 m .00375 .51923 L s P p .001 w 0 .55769 m .00375 .55769 L s P p .001 w 0 .63462 m .00375 .63462 L s P p .001 w 0 .67308 m .00375 .67308 L s P p .001 w 0 .71154 m .00375 .71154 L s P p .001 w 0 .75 m .00375 .75 L s P p .001 w 0 .82692 m .00375 .82692 L s P p .001 w 0 .86538 m .00375 .86538 L s P p .001 w 0 .90385 m .00375 .90385 L s P p .001 w 0 .94231 m .00375 .94231 L s P p .002 w 0 0 m 0 1 L s P P p p .002 w .01923 .99375 m .01923 1 L s P p .002 w .17949 .99375 m .17949 1 L s P p .002 w .33974 .99375 m .33974 1 L s P p .002 w .5 .99375 m .5 1 L s P p .002 w .66026 .99375 m .66026 1 L s P p .002 w .82051 .99375 m .82051 1 L s P p .002 w .98077 .99375 m .98077 1 L s P p .001 w .05128 .99625 m .05128 1 L s P p .001 w .08333 .99625 m .08333 1 L s P p .001 w .11538 .99625 m .11538 1 L s P p .001 w .14744 .99625 m .14744 1 L s P p .001 w .21154 .99625 m .21154 1 L s P p .001 w .24359 .99625 m .24359 1 L s P p .001 w .27564 .99625 m .27564 1 L s P p .001 w .30769 .99625 m .30769 1 L s P p .001 w .37179 .99625 m .37179 1 L s P p .001 w .40385 .99625 m .40385 1 L s P p .001 w .4359 .99625 m .4359 1 L s P p .001 w .46795 .99625 m .46795 1 L s P p .001 w .53205 .99625 m .53205 1 L s P p .001 w .5641 .99625 m .5641 1 L s P p .001 w .59615 .99625 m .59615 1 L s P p .001 w .62821 .99625 m .62821 1 L s P p .001 w .69231 .99625 m .69231 1 L s P p .001 w .72436 .99625 m .72436 1 L s P p .001 w .75641 .99625 m .75641 1 L s P p .001 w .78846 .99625 m .78846 1 L s P p .001 w .85256 .99625 m .85256 1 L s P p .001 w .88462 .99625 m .88462 1 L s P p .001 w .91667 .99625 m .91667 1 L s P p .001 w .94872 .99625 m .94872 1 L s P p .002 w 0 1 m 1 1 L s P p .002 w .99375 .01923 m 1 .01923 L s P p .002 w .99375 .21154 m 1 .21154 L s P p .002 w .99375 .40385 m 1 .40385 L s P p .002 w .99375 .59615 m 1 .59615 L s P p .002 w .99375 .78846 m 1 .78846 L s P p .002 w .99375 .98077 m 1 .98077 L s P p .001 w .99625 .05769 m 1 .05769 L s P p .001 w .99625 .09615 m 1 .09615 L s P p .001 w .99625 .13462 m 1 .13462 L s P p .001 w .99625 .17308 m 1 .17308 L s P p .001 w .99625 .25 m 1 .25 L s P p .001 w .99625 .28846 m 1 .28846 L s P p .001 w .99625 .32692 m 1 .32692 L s P p .001 w .99625 .36538 m 1 .36538 L s P p .001 w .99625 .44231 m 1 .44231 L s P p .001 w .99625 .48077 m 1 .48077 L s P p .001 w .99625 .51923 m 1 .51923 L s P p .001 w .99625 .55769 m 1 .55769 L s P p .001 w .99625 .63462 m 1 .63462 L s P p .001 w .99625 .67308 m 1 .67308 L s P p .001 w .99625 .71154 m 1 .71154 L s P p .001 w .99625 .75 m 1 .75 L s P p .001 w .99625 .82692 m 1 .82692 L s P p .001 w .99625 .86538 m 1 .86538 L s P p .001 w .99625 .90385 m 1 .90385 L s P p .001 w .99625 .94231 m 1 .94231 L s P p .002 w 1 0 m 1 1 L s P P p P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p .9 g .01923 .98077 m .98077 .98077 L .98077 .01923 L .01923 .01923 L F p 0 g .001 w s P .8 g .01923 .05144 m .08791 .06553 L .15659 .07796 L .21797 .08791 L .22527 .08903 L .29396 .09894 L .36264 .10782 L .43132 .11577 L .5 .12285 L .56868 .12915 L .63736 .13471 L .70604 .13959 L .77473 .14382 L .84341 .14747 L .91209 .15055 L .98077 .15313 L .98077 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .05144 m .08791 .06553 L .15659 .07796 L .21797 .08791 L .22527 .08903 L .29396 .09894 L .36264 .10782 L .43132 .11577 L .5 .12285 L .56868 .12915 L .63736 .13471 L .70604 .13959 L .77473 .14382 L .84341 .14747 L .91209 .15055 L .98077 .15313 L s P .7 g .01923 .08653 m .02522 .08791 L .08791 .10151 L .15659 .11416 L .22527 .12482 L .29396 .13376 L .36264 .14119 L .43132 .14728 L .5 .15219 L .56868 .15606 L .57977 .15659 L .63736 .16132 L .70604 .16664 L .77473 .17164 L .84341 .17633 L .91209 .18074 L .98077 .18488 L .98077 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .08653 m .02522 .08791 L .08791 .10151 L .15659 .11416 L .22527 .12482 L .29396 .13376 L .36264 .14119 L .43132 .14728 L .5 .15219 L .56868 .15606 L .57977 .15659 L .63736 .16132 L .70604 .16664 L .77473 .17164 L .84341 .17633 L .91209 .18074 L .98077 .18488 L s P .6 g .01923 .11365 m .08791 .12804 L .15659 .13944 L .22527 .14837 L .29396 .15527 L .30951 .15659 L .36264 .16258 L .43132 .16975 L .5 .17635 L .56868 .18241 L .63736 .18798 L .70604 .19311 L .77473 .19781 L .84341 .20214 L .91209 .20612 L .98077 .20976 L .98077 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .11365 m .08791 .12804 L .15659 .13944 L .22527 .14837 L .29396 .15527 L .30951 .15659 L .36264 .16258 L .43132 .16975 L .5 .17635 L .56868 .18241 L .63736 .18798 L .70604 .19311 L .77473 .19781 L .84341 .20214 L .91209 .20612 L .98077 .20976 L s P .5 g .01923 .13536 m .08791 .14843 L .14397 .15659 L .15659 .15845 L .22527 .16811 L .29396 .17692 L .36264 .18487 L .43132 .19203 L .5 .19846 L .56868 .20426 L .63736 .20947 L .70604 .21417 L .77473 .2184 L .84341 .22221 L .90437 .22527 L .91209 .2257 L .98077 .22938 L .98077 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .13536 m .08791 .14843 L .14397 .15659 L .15659 .15845 L .22527 .16811 L .29396 .17692 L .36264 .18487 L .43132 .19203 L .5 .19846 L .56868 .20426 L .63736 .20947 L .70604 .21417 L .77473 .2184 L .84341 .22221 L .90437 .22527 L .91209 .2257 L .98077 .22938 L s P .4 g .01923 .15422 m .03143 .15659 L .08791 .16629 L .15659 .17779 L .22527 .1881 L .29396 .19716 L .36264 .20511 L .43132 .21211 L .5 .21827 L .56868 .22374 L .58959 .22527 L .63736 .22843 L .70604 .23298 L .77473 .23745 L .84341 .24177 L .91209 .24592 L .98077 .24989 L .98077 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .15422 m .03143 .15659 L .08791 .16629 L .15659 .17779 L .22527 .1881 L .29396 .19716 L .36264 .20511 L .43132 .21211 L .5 .21827 L .56868 .22374 L .58959 .22527 L .63736 .22843 L .70604 .23298 L .77473 .23745 L .84341 .24177 L .91209 .24592 L .98077 .24989 L s P .3 g .01923 .16983 m .08791 .18476 L .15659 .19757 L .22527 .20836 L .29396 .21751 L .36189 .22527 L .36264 .22532 L .43132 .23015 L .5 .23582 L .56868 .24162 L .63736 .24726 L .70604 .25263 L .77473 .25771 L .84341 .26249 L .91209 .26698 L .98077 .2712 L .98077 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .16983 m .08791 .18476 L .15659 .19757 L .22527 .20836 L .29396 .21751 L .36189 .22527 L .36264 .22532 L .43132 .23015 L .5 .23582 L .56868 .24162 L .63736 .24726 L .70604 .25263 L .77473 .25771 L .84341 .26249 L .91209 .26698 L .98077 .2712 L s P .2 g .01923 .18929 m .08791 .20671 L .15659 .22014 L .18746 .22527 L .22527 .23096 L .29396 .2312 L .36264 .23983 L 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.95655 L .29396 .97311 L .32642 .98077 L s P .7 g .01923 .96229 m .08723 .98077 L .01923 .98077 L F p 0 g .001 w .01923 .96229 m .08723 .98077 L s P .6 g .98077 .14984 m .91209 .14017 L .84341 .13029 L .77473 .1202 L .70604 .1099 L .63736 .09939 L .56868 .08868 L .56384 .08791 L .5 .07774 L .43132 .06656 L .36264 .05508 L .29396 .04317 L .22527 .03057 L .16832 .01923 L .98077 .01923 L F p 0 g .001 w .98077 .14984 m .91209 .14017 L .84341 .13029 L .77473 .1202 L .70604 .1099 L .63736 .09939 L .56868 .08868 L .56384 .08791 L .5 .07774 L .43132 .06656 L .36264 .05508 L .29396 .04317 L .22527 .03057 L .16832 .01923 L s P .7 g .98077 .11815 m .91209 .10866 L .84341 .09897 L .77473 .08909 L .76664 .08791 L .70604 .07901 L .63736 .0687 L .56868 .05814 L .5 .04726 L .43132 .03594 L .36264 .02396 L .33693 .01923 L .98077 .01923 L F p 0 g .001 w .98077 .11815 m .91209 .10866 L .84341 .09897 L .77473 .08909 L .76664 .08791 L .70604 .07901 L .63736 .0687 L .56868 .05814 L .5 .04726 L .43132 .03594 L .36264 .02396 L .33693 .01923 L s P .8 g .98077 .08928 m .97061 .08791 L .91209 .07995 L .84341 .07042 L .77473 .06067 L .70604 .05066 L .63736 .04032 L .56868 .02956 L .50606 .01923 L .98077 .01923 L F p 0 g .001 w .98077 .08928 m .97061 .08791 L .91209 .07995 L .84341 .07042 L .77473 .06067 L .70604 .05066 L .63736 .04032 L .56868 .02956 L .50606 .01923 L s P .9 g .98077 .06256 m .91209 .0533 L .84341 .04378 L .77473 .03395 L .70604 .02372 L .67695 .01923 L .98077 .01923 L F p 0 g .001 w .98077 .06256 m .91209 .0533 L .84341 .04378 L .77473 .03395 L .70604 .02372 L .67695 .01923 L s P 1 g .98077 .03731 m .91209 .02794 L .85051 .01923 L .98077 .01923 L F p 0 g .001 w .98077 .03731 m .91209 .02794 L .85051 .01923 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] ContourGraphics["<<>>"] ;[o] -ContourGraphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We find the best fitting (in least squares sense) parameters a and b. :[font = input; Cclosed; preserveAspect; startGroup] sol = FindMinimum[h[a,b],{a,40},{b,.02}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {0.006014256661279145, {a -> 53.91258094950539, b -> 0.0401632800109223}} ;[o] {0.00601426, {a -> 53.9126, b -> 0.0401633}} :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We build our best fitting model with parameters a and b. :[font = input; Cclosed; preserveAspect; startGroup] u[t_] = f[t]/.sol[[2]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (1 + 53.91258094950539/E^(0.0401632800109223*t))^(-1) ;[o] 1 ---------------- 53.9126 1 + ------------ 0.0401633 t E :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] And plot this best fitting function :[font = input; Cclosed; preserveAspect; startGroup] fitPlot = Plot[u[t],{t,0,200}, AxesLabel->{"Mass of Rock", "% saying mass > 100 g"}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0.014715 0.598907 [ [(50)] .2619 .01472 0 2 Msboxa [(100)] .5 .01472 0 2 Msboxa [(150)] .7381 .01472 0 2 Msboxa [(200)] .97619 .01472 0 2 Msboxa [(Mass of Rock)] 1.025 .01472 -1 0 Msboxa [(0.2)] .01131 .1345 1 0 Msboxa [(0.4)] .01131 .25428 1 0 Msboxa [(0.6)] .01131 .37406 1 0 Msboxa [(0.8)] .01131 .49384 1 0 Msboxa [(1)] .01131 .61362 1 0 Msboxa [(% saying mass > 100 g)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(50)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(100)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(150)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(200)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P [(Mass of Rock)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .1345 m .03006 .1345 L s P [(0.2)] .01131 .1345 1 0 Mshowa p .002 w .02381 .25428 m .03006 .25428 L s P [(0.4)] .01131 .25428 1 0 Mshowa p .002 w .02381 .37406 m .03006 .37406 L s P [(0.6)] .01131 .37406 1 0 Mshowa p .002 w .02381 .49384 m .03006 .49384 L s P [(0.8)] .01131 .49384 1 0 Mshowa p .002 w .02381 .61362 m .03006 .61362 L s P [(1)] .01131 .61362 1 0 Mshowa p .001 w .02381 .03867 m .02756 .03867 L s P p .001 w .02381 .06263 m .02756 .06263 L s P p .001 w .02381 .08658 m .02756 .08658 L s P p .001 w .02381 .11054 m .02756 .11054 L s P p .001 w .02381 .15845 m .02756 .15845 L s P p .001 w .02381 .18241 m .02756 .18241 L s P p .001 w .02381 .20637 m .02756 .20637 L s P p .001 w .02381 .23032 m .02756 .23032 L s P p .001 w .02381 .27823 m .02756 .27823 L s P p .001 w .02381 .30219 m .02756 .30219 L s P p .001 w .02381 .32615 m .02756 .32615 L s P p .001 w .02381 .3501 m .02756 .3501 L s P p .001 w .02381 .39802 m .02756 .39802 L s P p .001 w .02381 .42197 m .02756 .42197 L s P p .001 w .02381 .44593 m .02756 .44593 L s P p .001 w .02381 .46988 m .02756 .46988 L s P p .001 w .02381 .5178 m .02756 .5178 L s P p .001 w .02381 .54175 m .02756 .54175 L s P p .001 w .02381 .56571 m .02756 .56571 L s P p .001 w .02381 .58967 m .02756 .58967 L s P [(% saying mass > 100 g)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .02562 m .06349 .02985 L .10317 .03565 L .14286 .04357 L .18254 .05429 L .22222 .0686 L .2619 .08742 L .30159 .11165 L .34127 .14199 L .38095 .17873 L .42063 .22143 L .46032 .26875 L .5 .31851 L .53968 .36803 L .57937 .41468 L .61905 .45641 L .65873 .49205 L .69841 .5213 L .7381 .54453 L .77778 .56251 L .81746 .57614 L .85714 .58631 L .89683 .59382 L .93651 .59932 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] over the data to see how good the fit looks to us. :[font = input; Cclosed; preserveAspect; startGroup] Show[fitPlot,dataPlot] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0.014715 0.598907 [ [(50)] .2619 .01472 0 2 Msboxa [(100)] .5 .01472 0 2 Msboxa [(150)] .7381 .01472 0 2 Msboxa [(200)] .97619 .01472 0 2 Msboxa [(Mass of Rock)] 1.025 .01472 -1 0 Msboxa [(0.2)] .01131 .1345 1 0 Msboxa [(0.4)] .01131 .25428 1 0 Msboxa [(0.6)] .01131 .37406 1 0 Msboxa [(0.8)] .01131 .49384 1 0 Msboxa [(1)] .01131 .61362 1 0 Msboxa [(% saying mass > 100 g)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(50)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(100)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(150)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(200)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P [(Mass of Rock)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .1345 m .03006 .1345 L s P [(0.2)] .01131 .1345 1 0 Mshowa p .002 w .02381 .25428 m .03006 .25428 L s P [(0.4)] .01131 .25428 1 0 Mshowa p .002 w .02381 .37406 m .03006 .37406 L s P [(0.6)] .01131 .37406 1 0 Mshowa p .002 w .02381 .49384 m .03006 .49384 L s P [(0.8)] .01131 .49384 1 0 Mshowa p .002 w .02381 .61362 m .03006 .61362 L s P [(1)] .01131 .61362 1 0 Mshowa p .001 w .02381 .03867 m .02756 .03867 L s P p .001 w .02381 .06263 m .02756 .06263 L s P p .001 w .02381 .08658 m .02756 .08658 L s P p .001 w .02381 .11054 m .02756 .11054 L s P p .001 w .02381 .15845 m .02756 .15845 L s P p .001 w .02381 .18241 m .02756 .18241 L s P p .001 w .02381 .20637 m .02756 .20637 L s P p .001 w .02381 .23032 m .02756 .23032 L s P p .001 w .02381 .27823 m .02756 .27823 L s P p .001 w .02381 .30219 m .02756 .30219 L s P p .001 w .02381 .32615 m .02756 .32615 L s P p .001 w .02381 .3501 m .02756 .3501 L s P p .001 w .02381 .39802 m .02756 .39802 L s P p .001 w .02381 .42197 m .02756 .42197 L s P p .001 w .02381 .44593 m .02756 .44593 L s P p .001 w .02381 .46988 m .02756 .46988 L s P p .001 w .02381 .5178 m .02756 .5178 L s P p .001 w .02381 .54175 m .02756 .54175 L s P p .001 w .02381 .56571 m .02756 .56571 L s P p .001 w .02381 .58967 m .02756 .58967 L s P [(% saying mass > 100 g)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .004 w .02381 .02562 m .06349 .02985 L .10317 .03565 L .14286 .04357 L .18254 .05429 L .22222 .0686 L .2619 .08742 L .30159 .11165 L .34127 .14199 L .38095 .17873 L .42063 .22143 L .46032 .26875 L .5 .31851 L .53968 .36803 L .57937 .41468 L .61905 .45641 L .65873 .49205 L .69841 .5213 L .7381 .54453 L .77778 .56251 L .81746 .57614 L .85714 .58631 L .89683 .59382 L .93651 .59932 L .97619 .60332 L s P P p p .02 w .07143 .02669 Mdot .21429 .07461 Mdot .35714 .14049 Mdot .45238 .28422 Mdot .5 .31417 Mdot .59524 .404 Mdot .61905 .4639 Mdot .7381 .55373 Mdot .85714 .60164 Mdot P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; endGroup] This looks pretty good to us. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Now as to parameters for comparison, one could use the "steepness" in the center of the fitted plot, i.e. find where the first derivative is maximum and compare maximum first derivative values. This idea comes from the fact that the perfect knowledge model has "infinite" steepness at the 100 g mass. :[font = input; Cclosed; preserveAspect; startGroup] Plot[u'[t],{t,0, 200}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0.014715 58.621174 [ [(50)] .2619 .01472 0 2 Msboxa [(100)] .5 .01472 0 2 Msboxa [(150)] .7381 .01472 0 2 Msboxa [(200)] .97619 .01472 0 2 Msboxa [(0.002)] .01131 .13196 1 0 Msboxa [(0.004)] .01131 .2492 1 0 Msboxa [(0.006)] .01131 .36644 1 0 Msboxa [(0.008)] .01131 .48368 1 0 Msboxa [(0.01)] .01131 .60093 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(50)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(100)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(150)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(200)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .13196 m .03006 .13196 L s P [(0.002)] .01131 .13196 1 0 Mshowa p .002 w .02381 .2492 m .03006 .2492 L s P [(0.004)] .01131 .2492 1 0 Mshowa p .002 w .02381 .36644 m .03006 .36644 L s P [(0.006)] .01131 .36644 1 0 Mshowa p .002 w .02381 .48368 m .03006 .48368 L s P [(0.008)] .01131 .48368 1 0 Mshowa p .002 w .02381 .60093 m .03006 .60093 L s P [(0.01)] .01131 .60093 1 0 Mshowa p .001 w .02381 .03816 m .02756 .03816 L s P p .001 w .02381 .06161 m .02756 .06161 L s P p .001 w .02381 .08506 m .02756 .08506 L s P p .001 w .02381 .10851 m .02756 .10851 L s P p .001 w .02381 .15541 m .02756 .15541 L s P p .001 w .02381 .17885 m .02756 .17885 L s P p .001 w .02381 .2023 m .02756 .2023 L s P p .001 w .02381 .22575 m .02756 .22575 L s P p .001 w .02381 .27265 m .02756 .27265 L s P p .001 w .02381 .2961 m .02756 .2961 L s P p .001 w .02381 .31955 m .02756 .31955 L s P p .001 w .02381 .34299 m .02756 .34299 L s P p .001 w .02381 .38989 m .02756 .38989 L s P p .001 w .02381 .41334 m .02756 .41334 L s P p .001 w .02381 .43679 m .02756 .43679 L s P p .001 w .02381 .46024 m .02756 .46024 L s P p .001 w .02381 .50713 m .02756 .50713 L s P p .001 w .02381 .53058 m .02756 .53058 L s P p .001 w .02381 .55403 m .02756 .55403 L s P p .001 w .02381 .57748 m .02756 .57748 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .05681 m .06349 .0727 L .10317 .09415 L .14286 .1227 L .18254 .16001 L .22222 .2075 L .2619 .26585 L .30159 .33411 L .34127 .40874 L .38095 .48292 L .40079 .51686 L .42063 .54686 L .44048 .57158 L .4504 .58156 L .46032 .58978 L .47024 .59612 L .4752 .59857 L .48016 .60051 L .48512 .60195 L .4876 .60248 L .48884 .6027 L .49008 .60288 L .49132 .60303 L .49256 .60315 L .4938 .60324 L .49504 .6033 L .49628 .60332 L .49752 .60331 L .49876 .60327 L .5 .6032 L .50124 .60309 L .50248 .60295 L .50496 .60258 L .50744 .60208 L .50992 .60146 L .51488 .59982 L .51984 .59769 L .52976 .59193 L .53968 .58428 L .55952 .5637 L .57937 .53701 L .61905 .47051 L .65873 .39562 L .69841 .3217 L .7381 .25499 L .77778 .19851 L .81746 .15285 L .85714 .11718 L .89683 .08997 L .93651 .06959 L Mistroke .97619 .05452 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] sol = FindRoot[u''[t]==0,{t,100}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 99.2788403334185} ;[o] {t -> 99.2788} :[font = input; Cclosed; preserveAspect; startGroup] u'[t]/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.01004082000273057 ;[o] 0.0100408 :[font = subsection; inactive; preserveAspect] Now compare this maximum value of u'[t] to other comparable values for other populations. :[font = subsection; inactive; preserveAspect] Of course, one could use the sum of the squares obtained as the first output in the FindMinimum command to compare groups, but perhaps it would be better to compare the sum of squares of the differences between the group's response and the perfect response. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] For that purpose we need the "perfect response" function. :[font = input; preserveAspect] g[x_] = If[x<100,0,1]; :[font = input; Cclosed; preserveAspect; startGroup] Plot[g[x],{x,0,200}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.004762 0.014715 0.588604 [ [(50)] .2619 .01472 0 2 Msboxa [(100)] .5 .01472 0 2 Msboxa [(150)] .7381 .01472 0 2 Msboxa [(200)] .97619 .01472 0 2 Msboxa [(0.2)] .01131 .13244 1 0 Msboxa [(0.4)] .01131 .25016 1 0 Msboxa [(0.6)] .01131 .36788 1 0 Msboxa [(0.8)] .01131 .4856 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(50)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(100)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(150)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(200)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .13244 m .03006 .13244 L s P [(0.2)] .01131 .13244 1 0 Mshowa p .002 w .02381 .25016 m .03006 .25016 L s P [(0.4)] .01131 .25016 1 0 Mshowa p .002 w .02381 .36788 m .03006 .36788 L s P [(0.6)] .01131 .36788 1 0 Mshowa p .002 w .02381 .4856 m .03006 .4856 L s P [(0.8)] .01131 .4856 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .03826 m .02756 .03826 L s P p .001 w .02381 .0618 m .02756 .0618 L s P p .001 w .02381 .08535 m .02756 .08535 L s P p .001 w .02381 .10889 m .02756 .10889 L s P p .001 w .02381 .15598 m .02756 .15598 L s P p .001 w .02381 .17952 m .02756 .17952 L s P p .001 w .02381 .20307 m .02756 .20307 L s P p .001 w .02381 .22661 m .02756 .22661 L s P p .001 w .02381 .2737 m .02756 .2737 L s P p .001 w .02381 .29724 m .02756 .29724 L s P p .001 w .02381 .32079 m .02756 .32079 L s P p .001 w .02381 .34433 m .02756 .34433 L s P p .001 w .02381 .39142 m .02756 .39142 L s P p .001 w .02381 .41497 m .02756 .41497 L s P p .001 w .02381 .43851 m .02756 .43851 L s P p .001 w .02381 .46205 m .02756 .46205 L s P p .001 w .02381 .50914 m .02756 .50914 L s P p .001 w .02381 .53269 m .02756 .53269 L s P p .001 w .02381 .55623 m .02756 .55623 L s P p .001 w .02381 .57977 m .02756 .57977 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .01472 L .10317 .01472 L .14286 .01472 L .18254 .01472 L .22222 .01472 L .2619 .01472 L .30159 .01472 L .34127 .01472 L .38095 .01472 L .42063 .01472 L .44048 .01472 L .46032 .01472 L .47024 .01472 L .48016 .01472 L .48512 .01472 L .49008 .01472 L .49256 .01472 L .49504 .01472 L .49628 .01472 L .49752 .01472 L .49876 .01472 L .5 .60332 L .50124 .60332 L .50248 .60332 L .50496 .60332 L .50992 .60332 L .51984 .60332 L .53968 .60332 L .57937 .60332 L .61905 .60332 L .65873 .60332 L .69841 .60332 L .7381 .60332 L .77778 .60332 L .81746 .60332 L .85714 .60332 L .89683 .60332 L .93651 .60332 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] SumOfSquares = Sum[(data[[i]][[2]] - g[data[[i]][[1]] ])^2, {i,1,Length[data]}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.7024 ;[o] 0.7024 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We note that the SumOfSquares between the perfect response and the guessing response would be :[font = input; Cclosed; preserveAspect; startGroup] 9 .5^2 :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2.25 ;[o] 2.25 :[font = subsection; inactive; preserveAspect; endGroup] Every group should do better (less) than this!!! :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect] Students need to be given lots of freedom in this experiment to design the display table set up, to decide what they want to compare (or even if they want to compare), the time duration for the data collection, variations in the experiment, and the curve fitting routines and curves themselves. :[font = subsection; inactive; preserveAspect; endGroup; endGroup] We have used this several times in a liberal arts college setting and students really get into the collection and comparison of real data. Something not often done in a mathematics class. ^*)