b. (6 pts) State the simple linear regression models for these two regressions: Y regressed on X, and Z regressed on X.

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1 Mat 46 Exam Sprg 9 Mara Frazer Name SOLUTIONS Solve all problems, ad be careful ot to sped too muc tme o a partcular problem. All ecessary SAS fles are our usual folder (P:\data\mat\Frazer\Regresso). You may oly use te SAS fles metoed below (embryos.sas ad beetle.sas). You may ot alter tese SAS fles ay way. You may also use a calculator (or a calculator o te computer, lke Maple). You may ot use ay oter computer applcato, cludg Mtab ad Excel. Te exam s wort a total of 5 pots; pot values for eac part are pareteses. To receve maxmum credt, sow all of your work. Good luck!. (46 pts) Te fle P:\data\mat\Frazer\Regresso\embryos.dat gves te dry wegts (Y) of cck embryos ragg age from 6 to 6 days (). Use te fle P:\data\mat\Frazer\Regresso\embryos.sas to aswer te questos below. a. (6 pts) Te values of te commo logartm of te wegts (Z) are created ad two scatterplots are provded. Descrbe te relatosps betwee age () ad dry wegt (Y), log dry wegt. ad betwee age ad ( ) Wegt (Y) vs. age () as a curvlear patter; a smple lear model would probably ot be approprate ere. Usg te trasformed wegt (Z) greatly mproves te lear form; fact, te relatosp looks almost perfectly lear. b. (6 pts) State te smple lear regresso models for tese two regressos: Y regressed o, ad Z regressed o. Yˆ = Zˆ = c. (6 pts) Wc of te two regresso les part (b) as a better ft? Tat s, s t more approprate to ru a lear regresso of Y o, or of Z o? Expla your coce torougly. Z o s clearly te better ft. It as a ger R^ (99.83% vs. 74%). Plus, te stadardzed resdual plot s muc better t looks lke a radom cloud of pots, wle te Y-o- resdual plot sows a bad curved patter; ot a surprse, cosderg wat we saw (a).

2 For te ext tree parts, use te regresso tat you cose as beg more approprate part (c). d. ( pts) Fd 95% cofdece tervals for te true slope ad tercept. Iterpret eac terval wt regard to te ull ypotess tat te true parameter s. 95% CI for β : (-.7583, -.69) We are 95% cofdet tat te true value of te tercept s betwee tese two values. Ts leads us to coclude tat te tercept s sgfcatly dfferet from. 95% CI for β : (.8984,.9) We are 95% cofdet tat te true value of te slope s betwee tese two values. Ts leads us to coclude tat te slope s sgfcatly dfferet from. Tus, tere s a sgfcat lear relatosp betwee age ad log(wegt). e. ( pts) Fd ad terpret jot cofdece tervals for bot te slope ad tercept parameters wt a overall (famly) cofdece coeffcet of.98. Jot (Boferro) 98% CIs: b ± B s{ b}, b ± B s{ b } were B = t( ; ) = t( ;9) α. 4 4 = t(.995;9) = 3.5 CI for β : b ± B s{ b} =.689 ± = (.7883,.5897) CI for β : b ± B s{ b} =.9588 ± = (.87,.456) We are 98% cofdet tat te true tercept les betwee -.79 ad -.59 ad tat te true slope les betwee.87 ad.5. f. (6 pts) Fd ad terpret a approxmate 95% cofdece terval o te mea respose for a 8-day-old cck. = 8 Zˆ =. Ts correspods to observato 3 te data set. As ca be see te SAS output below, te 95% CI for E{ Z } s (-.485, -.958). Tus, we are 95% cofdet tat for ccks 8 days old, te true mea value of log(wegt) s betwee ad (Or, te true mea value of wegt s.7 ad.8.)

3 . (48 pts) I a study o geograpc varato a certa speces of beetle, te mea tba legt (U) ad te mea tarsus legt (V) were obtaed for samples of sze 5 from eac of dfferet regos spag fve souter states. Te results are provded P:\data\mat\Frazer\Regresso\beetle.dat. Use te SAS program P:\data\mat\Frazer\Regresso\beetle.sas to aswer te questos below. a. (4 pts) Fd te estmated least squares equato for predctg tba legt ( mm) from tarsus legt ( mm). We wat to predct U from V, wc meas V s te predctor varable ad U te respose. Ftted regresso le: U ˆ = V b. (6 pts) Evaluate te ft of te model by lookg at te resdual plots. Lookg at te scatterplot of U vs. V, a lear model seems to be approprate. Te stadardzed resdual plots (sresdu vs. yatu ad V) sow o outlers; owever, tere may be some o-costacy of error varace (megapoe sape?). c. (6 pts) Do you tk te ormalty assumpto s reasoable ts stuato? Justfy your respose. Te ormal probably plot of resdu does t look very stragt, altoug t s ard to tell wt so few data pots. Te stem-ad-leaf ad boxplots ( Proc Uvarate) sow a lttle leftskewess, but probably ot eoug to rule out ormalty. Te tests for ormalty (Aderso- Darlg, Cramer-vo-Mses, etc) ave very large p-values, dcatg tat tere s o evdece to reject te assumpto of ormalty of te error terms. d. (6 pts) Fd ad terpret a approxmate 98% predcto terval o te respose for a beetle wt tarsus legt of.776 mm. V =.776 Uˆ = Ts correspods to observato 7 te data set. As ca be see te SAS output below, te 98% PI for U ( ew) s (7.5999, 8.75). Tus, we are 98% cofdet tat a gve beetle wt tarsus legt of.776 mm wll ave a tba legt of betwee mm ad 8.75 mm. Ts questo sould actually read Fd PI o te mea tba legt for a group of beetles, all wt tarsus legts of.776 mm. 3

4 e. ( pts) Usg α =.5, coduct a formal test for lack of ft. H : E{ Y} = β + β H a : E{ Y} β + β From Proc Reg output, SSE = From Proc GLM, SSPE =.7568 wt d.f. So SSLF = =.58 wt 6 d.f. Tus te test statstc s.58 MSLF F* = = = =.999 MSPE Comparg ts to a F(6,) dstrbuto, we would reject H for F* > 9.3. Put aoter way, p-value = P( F6, > F*) >.5. So obvously, we would ot reject H; tus, we coclude tat te lear model seems to be approprate for ts data. f. (8 pts) Report te approprate sample correlato coeffcet betwee tarsus legt ad tba legt. Expla wy you coose tat correlato coeffcet. Pearso correlato coeffcet: r =.976 Spearma correlato coeffcet: r S =.95 Usg te Pearso measure s oly vald f U ad V are jotly bvarate ormal. I order to test f ts s true, we ca look at te ormalty of eac varable dvdually. Stem-ad-leaf ad boxplots for bot U ad V sow tat bot varables are a bt rgt-skewed. However, tests for ormalty for bot varables are ot sgfcat at ay acceptable level, dcatg tat tere s o reaso to reject te assumpto of ormalty. Tere s so lttle data tat t s dffcult to make a frm cocluso about ormalty. I tk a argumet could be made for eter measure of correlato ere. g. (8 pts) Usg te statstc foud part (f), test te ypoteses H : ρ = versus H : ρ. Make sure to report te test statstc, te p-value, ad a toroug cocluso. A Usg Pearso: Usg Spearma: Test statstc: t* = =.88 Test statstc: t* = = p-value = P( t 8 >.88) <. p-value = P( t 8 > 8.7) <. So regardless of wc statstc s used, we reject te ull ypotess ad coclude tat tere s a sgfcat correlato betwee mea tarsus legt ad mea tba legt. 4

5 3. (3 pts) Suppose tat te model Y = β + β + ε, te errors ave mea zero ad are depedet, but Var ( ε ) = κ σ, were te κ are kow costats, so te errors do ot ave equal varace. Ts stuato arses we te Y are averages of several observatos at ; ts case, f Y s a average of depedet observatos, κ =. Y = + + a. ( pts) Te model may be trasformed as follows: κ κ β κ β κ ε Z = U + V + were U = κ, V = κ, γ = κ ε. or β β γ Sow tat te ew model satsfes te assumptos of te stadard lear regresso model. Assumptos:. Expected value of te error terms s. E{ γ } = E{ κ ε } = κ E{ ε } = κ () =. Varace of te error terms s costat over te predctor (). Var{ γ } = Var{ κ ε } = κ Var{ ε } = κ κ σ = σ 3. Error terms are depedet of eac oter. Cov{ γ, γ } = Cov{ κ ε, κ ε } = Cov{ ε, ε }, sce te κ are costats j j j j =, sce te ε are depedet 4. Values of predctor () ca be tougt of as costats. Here, we ave two predctors: U ad V. Sce U = κ ad te κ are kow costats, t follows tat U are costats, as well. Ad f we assume tat te values of te orgal predctor varable are costats, t must be tat V = are costats. κ 5

6 b. ( pts) Usg te model for Z part (a), detfy te ormal equatos for fdg te least squares estmators of β ad β. DO NOT solve tese smultaeous equatos. Te least squares crtero for ts model s = ( β β ) Q = Z U V We ws to fd b ad b tat mmze Q. Take te partal dervatve of Q wt respect to β ad β : Q = U Z βu βv = = ( ) Q = V Z U V ( β β ) We we set tese partals equal to, tat wll gve us b ad b : ( ) ( ) U Z b U bv = V Z b U bv = = = () U Z b U b U V = V Z b VU b V = () Equatos () ad () are te ormal equatos for ts model. Solvg ts system wll result formulas for te least squares estmators b ad b. 6

7 c. (9 pts) Sow tat performg a least squares aalyss o te ew model, as was doe part (b), s equvalet to mmzg ( ) Y β β κ. = If we wat to mmze ( ) Q = Y β β κ, we must take te partal dervatves = wt respect to β ad β, ad set tose partals equal to order to fd b ad b, just as we dd part (b). Q Q = κ ( Y β β ) = κ ( Y β β ) = = κ = = ( Y b b ) κ ( Y b b ) (3) = = (4) We wat to sow tat Equatos () ad () are equvalet to Equatos (3) ad (4). Cosder Equato () above; plug te deftos of Z, U, ad V : () U Z b U b U V = Y b b κ κ κ κ κ = (5) κ ( Y b b ) = = Smlarly for Equato (): () V Z b VU b V = Y b b κ κ κ κ κ κ = (6) κ ( Y b b ) = = Notce tat Equatos (3) ad (5) are te same; so are Equatos (4) ad (6). Tus, te aalyss we dd part (b) s equvalet to mmzg Q. 7