Statistics 512 Exam! September 25,2007

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1 Statistics 512 Exam! September 25,2007 You may assume that all random variables have normal distributions. Also assume. that other conditions of analysis of variance are met, unless you are asked to test them. Use a = 0.05 if it is not specified. 1. An experiment was conducted to test growth of four different varieties of com. Twenty experimental plots were used, and five were randomly assigned to each variety. Measurements, in centimeters of growth in a certain time interval, are as follows: (1Opts) A B C D LLYij = Total Mean (J{ ~ A. Complete the analysis of variance table, test the null hypothesis that growth is the same for all four varieties, and state your conclusions (Critical F = 3.24). 4' Vttnhe? Ire '.",.. "S'"2 ~\- -:::9 [11./ ~ i j ~ IS Y'JYZ- :; S[r}O L~ ~~ri~t =- - - ~ frf) re!ful'\$ t ~ t~h ~)9Z0 v."",~ -- $1jV'r u. fr H writ' +vtj ~I /rn~va-!col, f oj &Sb r C. 4'1-J 4 Kf) 04 I""-Q.o. Y1 HI ~ ext <2- --L] ~ 0(4 ';-0 -:!!) A r~ 4-'3'2.- 7/ 3 'J-4- :::;) rv~cj Ho. 11- Vd-n-h'~I d \ H~ r;'1t\ I" f- GM'-f-7 ~o...vt ff (c-01-~,

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3 2. An experiment was conducted to investigate the effect on yield of sugar from beets with increasing amounts of fertilizer. Eighteen plots were available for the study. Six were assigned at random to each of three levels of fertilizer (4,8, and 12 cwt./acre). The means and partial analysis of variance are given below. Linear Coefficients: (-1 0 1); Quadratic Coefficients: (-1 2-1); Critical F (Among) = 3.68; Critical F (Contrasts) = (lopts) Fertilizer 1 Mean Yield 120.s Source df SS MS F - ~!Y1s ' /z ~11-00 Among Fertilizer ~ ~ 0 ~ '}-o Linear 1 en'of) n-oo I:5S ;:. 2'hd/ ZO ) F -14" V20 -- Quadratic 1 I ()0 to i) --.L :::- ((Jljlz- 0 Within Fertilizer M5e.. ;:. 30 ij)f:;' 20

4 C. Place a 95% confidence interval on the difference between mean yield of sugar from plots receiving 8 cwt.lacre of fertilizer and the mean from those plots receiving12 cwt./acre of fertilizer. 0<.. '=- () 'OJ ~i}-\f3) t +1-1 a(y1-i) PIV1SYn = r rij;.;v ) I. '.:L ru t ( 14-0 ~/JD) i -to,ou, /) y- o 1- (1/\ c- II ou... 'v e (,./rj) i J- '3>< <9.'J IJ I doer 110..f-ev-r-~-1t( ~) f'aj.-d. I I J S',9i I/' U ~ (-/O):t $",4-, 10- rv\~tt Y\ \jltl~ ~t'\-h~tll,q"~f [ -I{'f~, -.-- t f I J ~~CJ- ~ cv.t!facf., ~~ j~ n~.-f I ~\.J\~~ +- ~~ ~ k'fv v.t\ki- ll--lljjrylct ' 3. An analysis of variance is used to study the effect of seam differences on variability in the sulfur content of coal. Ten samples were taken from each of 5 seams. The analysis of variance is given below. (lopts) 0. -:;.-)' (I~/tJ,.-- ('Il)(A -:- [hz,s N1 ) t ;:;-f3' 3 '3 A. Estimate the variance components ((12 and (12A)' I~-3 /Ins", - /IISe.. ::: v" C. Should a producer in search of a low sulfur coal be looking for seams with low sulfur content or should consider other factors that affect sulfur content ~ari:;7 S='"I f"j-0-'" Ia -.J Jd 1j'J ( (I!>t /-!V1 f k UJ.-I} U IU~ 'Z~7- o~ tt~akl:~,'" f-jlr eu>t~f 1C~~/$,} ".{tu /'0 otr~ c..w CA!'- ~ CVfVtJ I tnjj ~ r~liij\.-~ ~ tl-d f (..~ k ct*b~ I-eJ 1-0 ~ fa- c/znd, I

5 4. An experiment was conducted to investigate the effectiveness of various types of analgesics. Five (5) subjects are randomly assigned to each group (placebo, aspirin, bufferin). The variable of interest is the amount oftime until relief from pain is felt. The means and partial analysis of variance are given below. (Note: placebo is a control treatment.) (IOpts) Treatment Placebo Aspirin Bufferin Mean Time Source df SS MS F ---- Treatments (/4 ':-5'~ L, Control vs Others 1 1') 1-'J /, 88, D ( Ll- Aspirin vs Bufferin 1 :':2 r JJ,f S'b3 HQ: Error /2-4<t 4 ~I ;: Pl-t/h.2 fa?: }J3 If yu'jen ~ a )~) fov ~.~ 1nu. dl}r~~ce/ ~ S.e(ift1(3)~ J 1'0 - ~/'Jb B. Estimate the variance of the difference between the mean time to relief for aspirin and bufferin. V I ~~ - ~ \ 0- V ( i) ~V (lis ) L 5) 1-1- ::L -t- ~/Y\ ~ A % - J X Mj;fn -:::- cj 'f- 4/5 ::fl-=-

6 A. To compare all pairs of means after a significant overall F-test. f/faerl) letir+ )/lrl,'p-c-qa! clff~c..e B. To compare all pairs of means regardless of the results of the overall F-test. Dut'lC'O'1 ( 5/'J Y-, TvK E y ( S'c.k t t-te IS- taajltl~ds. C. To test a comparison suggested by the results ofthe experiment. SC,",~tf~'s- ~~~) E. To examine independence of treatment effects from the random effects (r-'j~) 1. f{~ f'f-{n J ftu ~(H\A.fu.- 1J~cw. u (Sf-) tlfa::fi ~ S~U- M.1.aM l- S~l':.:-c<~ A E~ l~<hl' i 0<: i\ :yf/' ov {r~~ -0 GtJ ~o.a /<;/ ~\ ( ~ 0<-1 ~!j_. L. Bon~s Question: ~ l. In question number 2 how would the following changes affect (increase, decrease, no change) the probability of a Type I error and power? ----~-=--~;;_:7, -= :,~'\~l ~~(J.~~~/L-~\ ~e.o.l): 11\ 6)...L~VJo...'1 Mov'A erlr-l C!<p#! N/5e -:; _L-_L----J-_ -@ ae.c~6.f:v\1 :# ~ t(f1(j (rt) -1 rv\c.~um.j ~'L~ 1'V1J'e 0. (n -/) F::. M~f)..IN1~t k(ov\fl~s S'w.-Q\\-tr ~ p(t1t.dr~tth""1" 110 clqlrt.o-.4-v ~?ovje..r 0 '\lu-~j+- ~. L.:---~ A {erf aj.:~ 9 ~ ~~ rijj-lt ~ ~~.Qrp~?J t~n1j~ ~ [0)1 b po vv ~ '\. -9 ~'i1 o...-=-o ot" i O(-=.-()rO/ (V\~ ~ \AoJ.lA~ CnhL~ F ~1 kss [1~ rv ~.Q.e...-+ tic ~ lot r 0 PDvJ«-(. ~) 6-0 ol,j,. ~ ~ r =7 povj ~ t

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A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N