{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT 256 13 "ACCESS - MATH" }}{PARA 257 "" 0 "" {TEXT -1 9 "July 2008" }}{PARA 256 "" 0 "" {TEXT -1 49 "Notes on Body Mass Index and Actual National \+ Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 28 "What is the Body Mass Index?" }}{PARA 0 "" 0 "" {TEXT -1 383 " If you read newspapers and magazines it i s likely that once or twice a year you run across an article about the body mass index (B.M.I.) , and its use in determining health risk fac tors for overweight and underweight people. If you search the interne t for \"body mass index\" you will find many sites which let you compu te your B.M.I., and which tell you a little bit about it. " }}{PARA 0 "" 0 "" {TEXT -1 931 " A person's B.M.I. is computed by dividin g their weight by the square of their height, and then multiplying by \+ a universal constant. If you measure weight in Newtons, i.e. mass in \+ kilograms, and height in meters, this constant is the fine number one. If you use pounds and inches instead, and are good at conversions you will work out that the constant is about 703. Or you might just find a calculator on the internet which confirms this. Since your B.M.I. \+ is supposed to indicate health risks, proponants of the B.M.I index a re claiming that if for two different people, their weights are the s ame multiple (i.e. their common B.M.I.) of their heights-squared, then these people have comparable health risks. Thus, unless height itsel f is a health risk, the B.M.I. hypothesis can be interpreted as a biom etric scaling law that says that on average, weight should scale propo rtionately to the square of height in people." }}{PARA 0 "" 0 "" {TEXT -1 876 " If people were to scale equally in all directions ( \"self-similar\") when they grew, volume and hence weight would scale \+ as the cube of height. That particular power law seems a little high , since adults don't look like uniformly expanded versions of babies; \+ we seem to get relatively stretched out when we grow taller. One migh t expect the best predictive power for weight as a function of height \+ to be somewhere between 2 and 3, if one expected a power law at all. \+ If there is a predictive power, and if it is much larger than 2, then \+ one could argue that the body mass index might need to be modified to \+ reflect this fact. (In fact, when you find body mass index tables, the y often explain how and why you should modifiy the acceptable BMI valu es for children. Is it possible that if a different power law had been used, no such modification would have been needed?)" }}{PARA 0 "" 0 " " {TEXT -1 172 " Each group will use our common data, see if it i s consistent with a power law relating weights to heights, and decide \+ whether the B.M.I. power of 2 is a good choice. " }}{PARA 0 "" 0 "" {TEXT -1 982 " Several years ago I found a national data base at t he U.S. Center for Disease Control web site. It contained a wide vari ety of body measurements collected between 1976 and 1980 (when America ns were skinnier), including national median heights and weights for b oys and girls, age 2-19. By using only the national medians, a lot of the variance has been taken out of the data, compared to what yours w ill look like. The national data is very consistent with a power law, \+ with power = 2.6. When you do a literature and internet search into \+ the history of the B.M.I. index and height-weight scaling laws, as par t of your project work for this week, you might discover that (by cent uries) we're not the first people to stumble upon a power close to tha t value, based on empirical data. As far as I know there is no mathem atical model to explain any power law for height-weight scaling in peo ple, despite the empirical evidence which shows that the data does see m to follow one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 31 "How do you test for power laws?" }}{PARA 0 "" 0 "" {TEXT -1 157 " Yesterday Erin explained how to look for power law fits to data points: you look for a best line fit to the correspondi ng ln-ln data points. Recall: " }}{PARA 0 "" 0 "" {TEXT -1 148 " \+ Suppose we have a set of \"n\" data points, which you can think of as your height-weight data, but which could really be any set of paired \+ data:" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "[[x[1], y[1]], [x[2], y[2]], [x[3], y[3]], `...`, [x[n], y[n]]];" "6#7'7$&%\"xG6#\"\"\"&%\"yG6#F(7 $&F&6#\"\"#&F*6#F/7$&F&6#\"\"$&F*6#F5%$...G7$&F&6#%\"nG&F*6#F<" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "We want to see if there \+ is a power m and a proportionality constant " }{XPPEDIT 18 0 "b;" "6#% \"bG" }{TEXT -1 21 " so that the formula" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "y = b*x^m;" "6#/%\"yG*&%\"bG\"\"\")%\"xG%\"mGF'" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "effectively mirrors the r eal data. " }{TEXT 259 60 "Taking (natural) logarithms of the propose d power law yields" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "ln(y) = ln(b)+m *ln(x);" "6#/-%#lnG6#%\"yG,&-F%6#%\"bG\"\"\"*&%\"mGF,-F%6#%\"xGF,F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "So, if we write " } {XPPEDIT 18 0 "Y = ln(y);" "6#/%\"YG-%#lnG6#%\"yG" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "X = ln(x);" "6#/%\"XG-%#lnG6#%\"xG" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "B = ln(b);" "6#/%\"BG-%#lnG6#%\"bG" }{TEXT -1 59 ", this becomes the equation of a line in the new variables " }{XPPEDIT 18 0 "X;" "6#%\"XG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y;" "6#%\"YG" }{TEXT -1 1 ":" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "Y = mX+B;" "6#/%\"Y G,&%#mXG\"\"\"%\"BGF'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Th us," }{TEXT 262 1 " " }{TEXT 263 207 "in order for there to be a power law for the original data, the ln-ln data should (approximately) sati sfy the equation of a line, and vise verse. If we get a good line fit to the ln-ln data, then the slope " }{XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT 269 75 " of this line is the power relating the original data, a nd the exponential " }{XPPEDIT 18 0 "exp(B);" "6#-%$expG6#%\"BG" } {TEXT 264 8 " of the " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 270 43 "-in tercept is the proportionality constant " }{XPPEDIT 18 0 "b" "6#%\"bG " }{TEXT 267 26 " in the original relation " }{XPPEDIT 18 0 "y = b*x^m " "6#/%\"yG*&%\"bG\"\"\")%\"xG%\"mGF'" }{TEXT 268 3 ". " }{TEXT -1 161 "With real data it is not too hard to see if the ln-ln data is wel l approximated by a line, in which case the original data is well-appr oximated by a power law. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 24 "National data example: " }} {PARA 0 "" 0 "" {TEXT -1 31 " Here is the national data." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "boyhw:=[[35.9,29.8],[38.9,34.1],[41.9,38.8],[44.3,42 .8],\n [47.2,48.6],[49.6,54.8],[51.4,60.8],[53.6,66.5],\n \+ [55.7,76.8],[57.3,82.3],[59.8,93.8],[62.8,106.8],\n [66.0,124 .3],[67.3,132.6],[68.4,142.4],[68.9,145.1],\n [69.6,155.3],[69. 6,153.2]]:\n#boy heights (inches) weights (pounds): Ntl medians for ag es 2-19" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "girlhw:=[[35.4, 28.0],[38.4,32.6],[41.1,36.8],[43.9,41.8],\n [46.6,47.0],[48.9, 52.5],[51.4,60.8],[53.1,65.5],\n [55.7,76.1],[58.2,89.0],[61.0, 100.1],[62.6,108.1],\n [63.3,117.1],[64.2,117.6],[64.3,122.6],[ 64.2,128.8],\n [64.1,124.5],[64.5,126]]:\n#girl heights(inches) weights (pounds): Ntl medians for ages 2-19 " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 133 "boys:=pointplot(boyhw,color=blue):\ngirls:=po intplot(girlhw,color=pink): #plots of the points for girl\n #and b oy height-weights.\n\n" }}}{PARA 0 "" 0 "" {TEXT -1 124 " I used co mmands like those in Erin's file to fit the X=ln(ht) to Y=ln(wt) data \+ to a line. The line I got was Y=mX+B with" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "m:=2.593828004; #the power I found\nB:=-6.038703303; \+ #intercept\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"+/!GQf#!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG$!+.LqQg!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 274 "Here's the graph of the best-fit line to the ln-ln \+ data points (girls are pink, boys are blue). Notice that the boy and \+ girl data seem to be close to the same line, which may justify why I d idn't ask you to keep track of whether your data points came from male s or females." }}{PARA 269 "" 0 "" {GLPLOT2D 340 339 339 {PLOTDATA 2 " 6(-%'POINTSG657$$\"+&HP2e$!\"*$\"+%R3XR$F)7$$\"+^U*4m$F)$\"+%Q(HHNF)7$ $\"+FeGNPF)$\"+Z-UeOF)7$$\"+xY)4z$F)$\"+.\"Qlv$F)7$$\"+$*QRaQF)$\"+JNi $)QF)7$$\"+M3*R!RF)$\"+%>!p.SF)7$$\"+s\"Q'RRF)$\"+*y*e2TF)7$$\"+o!\\:) RF)$\"+[>?(>%F)7$$\"+Z,)*>SF)$\"+SY?TVF)7$$\"+C1I[SF)$\"+36P5WF)7$$\"+ hc+\"4%F)$\"+c[;TXF)7$$\"+t]&*RTF)$\"+Fz&4n%F)7$$\"+UZl*=%F)$\"+**zpA[ F)7$$\"+P-;4UF)$\"+yqL()[F)7$$\"+DGPDUF)$\"+***R'e\\F)7$$\"+yhlKUF)$\" +gJUx\\F)7$$\"+nXwUUF)$\"+I(e`/&F)7$Fbp$\"+dUuJ]F)-%'COLOURG6&%$RGBG$ \"\"!F^qF]q$\"*++++\"!\")-F$657$$\"+?=rmNF)$\"+5X?KLF)7$$\"+gu0[OF)$\" +)G7V[$F)7$$\"+A\"3gr$F)$\"+Xy\\0OF)7$$\"+?V\">y$F)$\"+Sj*Gt$F)7$$\"+T 0gTQF)$\"+-w9]QF)7$$\"+'Rx(*)QF)$\"+qJ\"3'RF)FE7$$\"+GpGO[F)-Fjp6&F\\q\"\"\"$\"*w6%HvF)$\"+9Vygz!#5-%'CU RVESG6$7S7$$\"3++++++++N!#<$\"3*4++5r%pRIFdw7$$\"35++vQuh>NFdw$\"3S0l* H(*y04$Fdw7$$\"3!)*\\7tY'oONFdw$\"3x$*HG(4`[8$Fdw7$$\"3$***\\7vB)eb$Fd w$\"3+f#H0)Rk%=$Fdw7$$\"3&***\\(e_0_d$Fdw$\"3dA,25\\wMKFdw7$$\"3;+D1Po V%f$Fdw$\"3l)*=eBwk%G$Fdw7$$\"3=+DJCnE7OFdw$\"3U5\"RWG&*3L$Fdw7$$\"3** *\\7)R&G2j$Fdw$\"3>#p([e?yyLFdw7$$\"3A+D\"ex@)\\OFdw$\"3=t9rGmIGMFdw7$ $\"3C+DczP&)oOFdw$\"3Sc/,tBnxMFdw7$$\"39++vM0V)o$Fdw$\"3)fCV)36XGNFdw7 $$\"3****\\P^Pn0PFdw$\"3o4U+ZqQFdw7$$\"30+++'RFdw7$$\"3.+DJmG(\\(QF dw$\"3S6:QK)4B,%Fdw7$$\"3I+v$*elX$*QFdw$\"3#ecH>N`-1%Fdw7$$\"32+]7`Bu7 RFdw$\"3H#[s\\Qx-6%Fdw7$$\"3F+DcKFXIRFdw$\"3SN*=J1:i:%Fdw7$$\"3I+]Ppdb \\RFdw$\"3:UY'\\0ld?%Fdw7$$\"3<+D\"eX)RpRFdw$\"3J>3sqNBdUFdw7$$\"3R+vo s:n')RFdw$\"3bgv)\\1P?I%Fdw7$$\"3(***\\77qK0SFdw$\"32suAmgU]VFdw7$$\"3 w****\\1**fCSFdw$\"3A#[v$RmT+WFdw7$$\"3))**\\itYXVSFdw$\"3h0KNiEK\\WFd w7$$\"3_+DJIwphSFdw$\"3y-]Ga3%Fdw$\"3GV4>Q:=\\X Fdw7$$\"3Z++]kR:+TFdw$\"3qA<$3(3R'f%Fdw7$$\"3C++vLqe>TFdw$\"3*[/c+\"pz YYFdw7$$\"3'**\\7GG'>PTFdw$\"3#ew.MFsCp%Fdw7$$\"3\\++DVyWcTFdw$\"3Cas) *4vSUZFdw7$$\"3%)*\\781iX<%Fdw$\"3'yKogn#R*y%Fdw7$$\"3K+DcQm\\$>%Fdw$ \"3!4G$y7d]Q[Fdw7$$\"3O++v['3?@%Fdw$\"3!4&z@$oAl)[Fdw7$$\"30+D\"y+*QJU Fdw$\"3PP=G,?zO\\Fdw7$$\"35+++(fa+D%Fdw$\"3'\\`F#Ht?&)\\Fdw7$$\"3#*** \\(y&G9pUFdw$\"33nF$[**=Z.&Fdw7$$\"3S+DJeI2)G%Fdw$\"3K*))[)y1#Q3&Fdw7$ $\"3J++]YzY0VFdw$\"3wug&p-S*G^Fdw7$$\"3H+]P^WSDVFdw$\"3?KFg\"*=l!=&Fdw 7$$\"3i+++**eBVVFdw$\"3G=V!)*f.pA&Fdw7$$\"3$**\\78%zCiVFdw$\"3)4rnvc " 0 "" {MPLTEXT 1 0 10 "b:=exp(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+x!\\YQ#!#7 " }}}{PARA 0 "" 0 "" {TEXT -1 173 "Finally, here's a graph showing the power law we derived, and the national boy/girl data points. Boys ar e blue and girls are pink. Notice in the command lines below how to" }}{PARA 0 "" 0 "" {TEXT -1 20 " (i) make a title" }}{PARA 0 "" 0 " " {TEXT -1 63 " (ii) get the axes labeled \"h\" and \"w\" for heigh t and weight" }}{PARA 0 "" 0 "" {TEXT -1 243 " (iii) get the displa y to include appropriate ranges of height and weight - to contain all \+ our data\n (iv) Create plots of the boy and girl ht-wt data points, a plot of the power function, and then display all three graphs in on e picture." }}{PARA 0 "" 0 "" {TEXT -1 134 "If I wanted to, I could co py and paste my display somewhere else in the text of this document, l ike I did for the ln-ln picture above." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "boys:=pointplot(boyhw,color=blue):\ngirls:=pointplot (girlhw,color=pink): #plots of the points for girl\n #and boy heig ht-weights." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "powerplot:=p lot(b*h^m,h=0..80,w=0..200,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "display(\{powerplot,boys,girls\},title=\n `power la w for national height-weight data`);\n#by calling the variables h and \+ w, and giving their ranges, I\n#get Maple to label the axes as I want \n" }}{PARA 13 "" 1 "" {GLPLOT2D 503 503 503 {PLOTDATA 2 "6(-%'CURVESG 6$7S7$$\"\"!F)F(7$$\"3ELLLLBxV7$$\"3ymmm;$>5 E$F-$\"319/e#z@l6&F07$$\"3MLLLLAKn\\F-$\"3#o8EUZ9U_\"!#=7$$\"3]MLLLc$ \\o'F-$\"3EqC(yGZHH$F;7$$\"3ammm;bQ%R)F-$\"3kHb'pX\">WfF;7$$\"3lLLL$Qk #z**F-$\"3Zj`k_)R#4$*F;7$$\"3#******\\YJ?;\"!#;$\"3/[5L:'3F-7$$\"32+++&[A4]\"FN$\"3asM8\"FN7$$\"3%pmmmqFc#GFN$\"3SIf=/it%Q\"FN7 $$\"32+++X,H.IFN$\"3%)ot??c-A;FN7$$\"3cmmm1:bgJFN$\"3X+&[.4=;&=FN7$$\" 3`+++X@4LLFN$\"3-bj.B/MD@FN7$$\"3K+++N;R(\\$FN$\"3lr%\\Y(>'yS#FN7$$\"3 Rnmm;4#)oOFN$\"3kKQ^*)42EFFN7$$\"3!omm;^Yi#QFN$\"3MwhK6=%*RIFN7$$\"3EL LL$G^g*RFN$\"3(y#Qwb!3YFN7$$\"33+ ++!G5Jm%FN$\"3xu9L,t(y2&FN7$$\"3C+++5#32$[FN$\"3ZSL29#>\\c&FN7$$\"3W++ +Dy'G*\\FN$\"3Zl8%)=(4D1'FN7$$\"3++++I%=H<&FN$\"3@9_IHS*fk'FN7$$\"35mm m1>qM`FN$\"3[()eFA=l)>(FN7$$\"3Q++++.W2bFN$\"3^q#\\@(o&*=yFN7$$\"3(HLL $ep'Rm&FN$\"3Q#euS<\"\\3%)FN7$$\"3%4+++%>4NeFN$\"3^44:nmFN$\"3zb>&=!ya$G\"Faw7$$\"3NMLL.a#o$oFN$\"3zc9UPF+q8Faw7$$ \"3gnmm^Q40qFN$\"3z![pJw'=f9Faw7$$\"3&3+++3:(frFN$\"3yM`cN^?W:Faw7$$\" 3%ymmmXGpL(FN$\"3g1\\*[u4`k\"Faw7$$\"3GMLL8-V&\\(FN$\"3ey$*\\+++XhUkwFN$\"3^z&>oBSE%=Faw7$$\"3)3++]\"oFa w7$$\"#!)F)$\"3yv8C@mEf?Faw-%'COLOURG6&%$RGBGF)F)F)-%'POINTSG657$$\"$f $!\"\"$\"$)HFc[l7$$\"$*QFc[l$\"$T$Fc[l7$$\"$>%Fc[l$\"$)QFc[l7$$\"$V%Fc [l$\"$G%Fc[l7$$\"$s%Fc[l$\"$'[Fc[l7$$\"$'\\Fc[l$\"$[&Fc[l7$$\"$9&Fc[l$ \"$3'Fc[l7$$\"$O&Fc[l$\"$l'Fc[l7$$\"$d&Fc[l$\"$o(Fc[l7$$\"$t&Fc[l$\"$B )Fc[l7$$\"$)fFc[l$\"$Q*Fc[l7$$\"$G'Fc[l$\"%o5Fc[l7$$\"$g'Fc[l$\"%V7Fc[ l7$$\"$t'Fc[l$\"%E8Fc[l7$$\"$%oFc[l$\"%C9Fc[l7$$\"$*oFc[l$\"%^9Fc[l7$$ \"$'pFc[l$\"%`:Fc[l7$Fb`l$\"%K:Fc[l-Fjz6&F\\[lF(F($\"*++++\"!\")-F^[l6 57$$\"$a$Fc[l$\"$!GFc[l7$$\"$%QFc[l$\"$E$Fc[l7$$\"$6%Fc[l$\"$o$Fc[l7$$ \"$R%Fc[l$\"$=%Fc[l7$$\"$m%Fc[l$\"$q%Fc[l7$$\"$*[Fc[l$\"$D&Fc[lF_]l7$$ \"$J&Fc[l$\"$b'Fc[l7$Fj]l$\"$h(Fc[l7$$\"$#eFc[l$\"$!*)Fc[l7$$\"$5'Fc[l $\"%,5Fc[l7$$\"$E'Fc[l$\"%\"3\"Fc[l7$$\"$L'Fc[l$\"%r6Fc[l7$$\"$U'Fc[l$ \"%w6Fc[l7$$\"$V'Fc[l$\"%E7Fc[l7$F[el$\"%)G\"Fc[l7$$\"$T'Fc[l$\"%X7Fc[ l7$$\"$X'Fc[l$\"$E\"F)-Fjz6&F\\[l\"\"\"$\"*w6%Hv!\"*$\"+9Vygz!#5-%&TIT LEG6#%Jpower~law~for~national~height-weight~dataG-%+AXESLABELSG6$Q\"h6 \"Q\"wFbgl-%%VIEWG6$;F(Fez;F($\"$+#F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {PARA 0 "" 0 "" {TEXT -1 283 "Notice that the power graph fits the dat a pretty well. What power will you get from the ACCESS data? Would e ither power be different if you disallowed kids younger than 4, with t he claim that their baby-fat and big heads are messing with how people may scale later on? Just asking." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "42 0" 134 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }