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# APS Section 4.1(3)

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### APS Section 4.1(3)

1. 1. hat about 1/y (sec/ swing X 205 vs? X (length)? This changes the direction of the relationship. (Why? ) LSRL and Residuals Residuals only Find m re Free Wallpapers n www PIcsDe5k! opc
2. 2. In Fact, many of the relationships we will look at will take one of the following two Forms: I: = O ' bx mls (’on\$*l-u CO! ‘ Exponential Relationship Power Relationship But these aren't lines! How can we make these equations look like lines? , "r i -4‘ “ ‘ « ; If s WW: ~33» ""; .;". ‘.‘". '.. L . . ». v="‘ s~ : .i_& , ._
3. 3. 4 . . ' '-_ , . ~. .- H)‘. -- :4» u . . 1' -. ---. .. IA. ..u. .. .‘. IF we take the log of both sides of the equation y = a-bx we get: 109(y) = logca-bx) logo) = 1og<a> + Iogcbo Q 1090/) = 109(0) + X-10909) g*u So there is a linear relationship between ‘ ‘ ‘ log(y) and x. The slope is log(b) and the y—intercept is log(a). mn-. l.nLu. - = i‘. ua . nnu"- ' an
4. 4. '. ,7 . ‘ "Zv‘ﬁ7V ' ‘v . v _z. . 7‘ / , 5-3 . .5" v. ’ '9 ‘ Exponential vs. Linear Growth "-A variable grows (or decays) linearly over time if it adds a Fixed increment in each equal time period. ’ *Exponential growth (or decay) occurs when a variable is multiplied by a Fixed number in each equal time period. C -6 :2 . J“ , . , (at if 1. ’» ' . . . ” "“* 4.-315’; ~.'*. *-’N .4‘. ‘,7. ~“. '—. ‘;. '*o'; ” ‘ “ zzuuum. L. .Wa-.9 99,5 °rI. ’ww. : : cs: E5e. sKLeP-cam, ‘J’ 2:7. -. .i‘: 1*‘ ¢'J§? '.‘. C3 . ‘ -
5. 5. .. _ -. .-—_4 - - -, I. .r 5 r . -3 / V , ;_ J , E3‘*: §“-(«r-,3:-” ‘*‘ Va. ‘ ‘ ‘; .,v_ . h ‘ . ~, . ‘-v, ,' < _ >- 1", K. .. ,1 . V} ‘ ‘; M « ¥‘ , ‘:'. u.a. [§Er§b. ‘v. {.. ‘. . . -. .'-A. /.’ . ' v, . . i. ‘ . . -. §i~'. _,. ?‘€*5.7"". .. "- .145. '4 ’ . a. -,. . - . *W. -;t,1‘u. ' . - u_ ‘Straightening an Exponential Relationship 3— ,5 — --. —- 1 — see we e. .. e 3 ; ? Look at your Skittles data. Does it appear lg; .5f that we're multiplying each y—value by the f: Ly" same amount as the x—value increases by 1? "E i (What is that value? ) Check the ratios of *1. each y—value to the previous one. ;; ‘; XL . £é . , , _ _ . L2m=1. 799349549.. . U : :f This is the main indicator of an exponential ~ relationship. Plot the log of the counts against the trial __ number, and comment on the suitability of a f. § linear model for this transformed data. 3: FL “"‘ y. ‘ -, ;_. ._ ,4» , _“ 2“ ', ._- -. ___ ' . ., _;. ~. <’: "- ; ;“:3“_: -«'. "3.'»f «: '‘‘«''§»? ‘‘ ‘‘; ’.‘3' ‘ . .. , ~ . A. ‘ . .&»‘1’, ': rs : _ x~ ', ,f£. n.} ‘V’ ‘1A‘ .5 . :r'». ;. . z*__' . ;.. _:_ , *- ; ; --
6. 6. vv- 3'" ‘ ‘ ' , .. . '4' ' , . “% *"~ x 5 1' '» €ZiA; /2mw/ Strighteing the Skittles Data 1 _ﬁ LinRe9 u= a+bx ‘ a=1.8B7?228 ': b= '.36158313B6 log y w~ P2=.9944@53561 . r= '.99?198?546 I 4 X . 1 ~, Regression Output| [Scatter Plot ResidUCI1 P1013 N0|"m P1019 P1013 * The transformed Skittles data looks quite -'4' «. linear, and the residuals also look random. E3 « v ‘if * The normal probability plot of the _ ' residuals also appears relatively linear.
7. 7. ;~-vs’ _ <’_, . V. .-. _,f - 1' - '. ,2, . ‘ zr. -“ix --=5. , ,. _ . .W‘, '.xR“‘ )1 are-1: v-x1.. —.~1'. _‘ , - ' re": -.-. -_ > . ‘Av-to. -_6-. -“, .'-”“": i u-1) ', ‘ , ,r, . , _-: .". '57'ij_-: a;—. .a. ' . .9" . ~>“. , . V-: " flu-. Au. -'= :24! it A- Straightening a Power Relationship If we take the log of both sides of the 9? equation y = a-xb we get: 109(0-X”) 5*? 71-‘r'J‘5. . di; ;': :~ ; .,. ~l3 log(a) + b-log(x) ¥f 109(y) 109(y) = 109(0) + 109(x”) 109(y) = y“. So there is a linear relationship between ' ‘ log(y) and log(x). The slope is b and the 3: y—intercept is log(a). E F. :4’ r- '53. ‘. =.k-gr. ,. . ' I. _ - :0. Q~’£25iA»'}'-.5". -‘. ‘;'ui. .Z; 52.91 7%. I www Phat; _o 0-. » v-
8. 8. ‘id * .97 . ._. /. :~. ;,_, 1 Straightening the heerios Data 3.. ‘‘’?3‘6' / ’‘? X '_ LinRe-El _ _ I r_‘~""' ' : fa§E§81@8B6? 1 .3-E-1y-. ... ,., ,.. .a"’ ' r b= i.?1241993e 1°9 3’ - r~2=. 9?s2943?15 _. r it “W , . r= .988B3IE15519 . , I _ . , , , . , , .. _ ' 1 log x 1-- Regression Output Scatter Plot Res1dUCI1 P101: “NON” PROP P1015 A3» 1 . The transformed Cheerios data also looks .11 “ ’ quite linear, and the residuals look random. :; .‘5| T- :15 M; .. 3 3 . -9’. -. :.£3-. .;-E. '. '&uf: * . »*-’~ The normal probability plot of the residuals , " ‘also appears relatively linear. (o3{%ab°3:057—l “ A ‘.1 ; - :1 10aC'hetrI 3'5) 7- C? -05” /0"3a"‘”” *— tom“ » . _.
9. 9. 1 Comparing Transformations} 3 Cheerios vs. r2 log(Cheerios) vs. 109CFD L1nR-39 . 33a§3§3193337 LPQEEEX 3:1 _ 712419935 .336. 395139399 r2=.9?62B43?15 P72-959959937 ', ..= _ 9333335519 l"‘: —. 9?31771E‘|51 r—. 9354973924 For the ' Cheerios data , Regression Outpu Regression Outpu either of these . transformations " - would work. Scatter Plot Scatter Plot
10. 10. Comparing Transform ions Here's the pendulum data Pendulum # of swings length gin) per ; Q_§eg 6.5 22 9 20 11.5 17 14.5 16 18 14