Simple Linear Regression - Scalar Form

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Joan Shelton
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1 Smple Lear Regresso - Scalar Form Q.. Model Y X,..., p..a. Derve the ormal equatos that mmze Q. p..b. Solve for the ordary least squares estmators, p..c. Derve E, V, E, V, COV, p..d. Derve the mea ad varace of Y X ad ad COV Y, e e Y Y Q.. A electrcal cotractor fts a smple lear regresso model, relatg cost to wre a house (Y, dollars) to the sze of the house (X, ft ). She fts a model, based o a sample of =6 houses ad obtas the followg results. Y 5..X s 6 X X 4 X p..a. Compute the estmated stadard errors of p..b. Compute a 95% Cofdece Iterval for ad p..c. Compute a 95% Cofdece Iterval for the mea of all homes wth X = p..d. Compute a 95% Predcto Iterval for her brother--laws house wth X = Q.3. A researcher s terested the relatoshp betwee the educato level ad salares rural coutes the U.S. He obtas the percetage of adults over 5 wth a college educato each couty (X) ad the per capta come of the couty (Y, $s). He obtas the followg summary statstcs, based o a sample of = 3 coutes. X X 7.45 X X Y Y Y Y X 4.9 Y p.3.a. Compute least squares estmates of ad p.3.b. Show that Y Y Y Y X X Y Y X X p.3.c. Use p.3.b. to compute a ubased estmate of

2 Q.4. Cosder the regresso through the org model: p.4.a. Derve the least squares estmator of p.4.b. Derve the mea ad varace of the least squres estmator. Y X ~ NID,,..., Y p.4.c. Cosder the estmator ~ X. Derve ts mea ad varace. p.4.d. Whch estmator has the smallest varace? Why? Q.5. For the smple lear regresso model wth a tercept, show that Y Y Q.6. For smple regresso, we get: X X Y ad Y Y COV, Y??? SXX Q.7. For a smple lear regresso model, derve COV, completg the followg parts: p.7.a. Wrte a Y ad by statg explctly what the a ad b values (fuctos) are p.7.b. Usg rules of Covaraces of lear fuctos of radom varables to derve COV, p.7.c. Researchers the U.S. ft regressos of relatoshp betwee vscosty (Y) ad temperature (X) degrees Fahrehet, whle foreg researchers work wth temperature degrees Celsus. The temperatures for the expermetal rus are gve below. Gve the COV, (they use the same uts for Y): Ru # X(F) X(C) for each set of researchers as a fucto of COV, Fahrehet = Celsus =

3 Q.8. For the smple regresso model (scalar form): Y X,..., ~ NID, we get: X X Y, Y Y, Y X SXX p.8.a. Derve p.8.b. Derve,, E E Y E,, COV,,, COV, V V Y Y V Q.9. A Austra study cosdered Breath Alcohol Elmato Rates (X, mg/l/hr*) ad Blood Alcohol Elmato Rates (Y, g/l/hr*) a sample of = 7 adult females. The sample meas, stadard devatos ad correlatos are gve below. Complete the followg table for the smple lear regresso relatg Blood Elmato Rate (Y) to Breath Elmato Rate (X). X Y s.65 s r.8786 Regresso Statstcs R Square Resdual Std Error Observatos X Y XY ANOVA Regresso Resdual Total df SS MS F F(.5) Itercept X Coeffcetsadard Erro t Stat t(.5) Q.. For the smple regresso model (scalar form): Y X X,..., ~ NID, p..a. Derve least squares estmators of ad. p..b. Derve E, E, V, V, COV,

4 Q.. Cosder the cetered (wth respect to the depedet varable) model scalar form: Y X X,..., ~ NID, p..a. Obta the ormal equatos ad the least squares estmated for the parameters ad. p..b. Derve COV, Ht: COV ay, bjyj ab jcov Y, Yj j j Q.. A lear regresso was ru o a set of data, based o a smple lear regresso. You are gve oly the followg partal formato: ANOVA df SS MS F P-value Regresso Resdual Total Coeffcetstadard Erro t Stat P-value Itercept X p..a. Compute a 95% Cofdece Iterval for : p..b. Gve the F-statstc ad rejecto for testg H : = vs H A: at sgfcace level. (Ht: thk of coecto betwee t- ad F-tests) p..c. Compute the coeffcet of determato, R. Q.3. A smple lear regresso model s to be ft, relatg Mea Aual Temperature (Y, F) to Year 957 (X). That s, the org s 957. The data are for the years ( = 58). The sample meas ad sums of squares ad cross-products are gve below for Model. X 8.5 Y X X Y Y X X Y Y p.3.a. Compute the least squares estmates of ad, ad wrte out the predcted equato.

5 p.3.b. Y Y Use ths to obta a ubased estmate of. p.3.c. Obta a 95% Cofdece Iterval for (the amout, o average, that mea temperature creases per year). p.3.d. For the year 4, the (observed) average temperature was Obta the predcted temperature ad the resdual for that year. Q.4. Derve the two ormal equatos by mmzg Q wth respect to ad. X X Q.5. The ftted value for Yj ca be wrtte as Y j X j X Y SS XX p.5.a. Gve COV Yj, Y j p.5.b. Gve COV k, j Y Y k j Q.6. Derve EMSR ad EMSE Q.7. A smple lear regresso model s ft relatg farway accuracy (Y, percet) to average drve dstace (X, yards) for a radom sample of = professoal wome golfers from the 9 seaso. p.7.a. Complete the followg table cells.

6 Regresso Statstcs R Square Observatos ANOVA Source df SS MS F F(.5) Regresso Resdual #N/A #N/A Total 47.5 #N/A #N/A #N/A Parameter Coeffcets Stadard Error t Stat t(.5) Itercept drve p.7.b. What s your cocluso regardg the test of H : H A :? Reject H / Fal to Reject H p.7.c. Gve the sample correlato, r, betwee Y ad X. Q.8. A smple lear regresso model s ft, relatg plat growth over year (y) to amout of fertlzer provded (x). Twety fve plats are selected, 5 each assged to each of the fertlzer levels (, 5, 8,, 4). The results of the model ft are gve below: Model (Costat) x a. Depedet Varable: y Ustadardzed Coeffcets Coeffcets a B Std. Error t Sg p.8.a. Ca we coclude that there s a assocato betwee fertlzer ad plat growth at the.5 sgfcace level? Why (be very specfc). p.8.b. Gve the estmated mea growth amog plats recevg uts of fertlzer. p.8.c. The estmated stadard error of the estmated mea at uts s. ( 8) Gve a 95% CI for the mea at uts of fertlzer.

7 Q.9. A study was coducted to relate weght ga chckes (Y) to the amout of the amo acd lyse gested by the chcke (X). A smple lear regresso s ft to the data. ANOVA df SS MS F P-value Regresso Resdual Total Coeffcets Stadard Error t Stat P-value Itercept lyse(x) p.9.a. Gve the ftted equato, ad the predcted value for X=. p.9.b. Gve a 95% Cofdece Iterval for the MEAN weght ga of all chckes wth X=. (Note: the mea of X s.6 ad SXX=.) p.9.c. What proporto of the varato weght ga s explaed by lyse take? Q.. A researcher reports that the correlato betwee legth (ches) ad weght (pouds) of a sample of 6 male adults of a speces s r=.4. p..a. Test whether she ca coclude there s a POSITIVE correlato the populato of all adult males of ths speces: H : = H A: > o Test Statstc: o Rejecto Rego (=.5): o Coclude: Postve Assocato or No Postve Assocato p..b. A colleague from Europe trasforms the data from legth ches to cetmeters ( ch=.54 cm) ad weght from pouds to klograms ( poud=. kg). What s the colleague s estmate of the correlato? Q.. A smple lear regresso s to be ft, relatg fuel effcecy (Y gallos/ mles) to cars weght (X, pouds), based o a sample of =45 cars. You are gve the followg formato: ( X X ) X X ( Y Y) 3385 ( Y Y) 6.5 X 739 Y 3.4 Y Y.835 Compute the followg quattes: p..a. p..b. p..c. Resdual Std. Devato se

8 p..d. Estmate of mea effcecy for all cars of x*= pouds p..e. 95% Cofdece Iterval for all cars of x*= pouds Lower Boud = Upper Boud = p..f. Regresso Sum of Squares SSR = p..g. Proporto of Varato Effcecy Explaed by Weght Q.. A regresso model was ft, relatg reveues (Y) to total cost of producto ad dstrbuto (X) for a radom sample of =3 RKO flms from the 93s (the total cost raged from 79 to 53): Y Y X S X X Y X S xx e 467 p..a. Obta a 95% Cofdece Iterval for the mea reveues for all moves wth total costs of x * = 685. Note: SE 95%CI : y p..b. Obta a 95% Predcto Iterval for tomorrow s ew flm that had total costs of x * = y SE 95%PI : y Q.3. A researcher s terested the correlato betwee heght (X) ad weght (Y) amog year old male chldre. He selects a radom sample of = 8 male -year olds from a school dstrct, ad teds o testg H: = versus HA:, where s the populato correlato coeffcet. Hs sample correlato s r X X Y Y X X Y Y.6 Test H: = versus HA: : Test Statstc = Rejecto Rego: Q.4. A study was coducted to determe the effects of daly temperature (X, C) o Electrcty Cosumpto (Y, s of Wh) a expermetal house over a perod of = 3 days. Cosder the followg model: E Y X SSR 594. SSE 4.4 TSS SXX 58.5 X 7. Y X

9 p.4.a. What proporto of the varato Electrcty cosumpto s explaed by daly temperature (X)? p.4.b. Compute the resdual stadard devato, se p.4.c. Obta the estmated mea electrcty cosumpto whe x * = 7. degrees, ad the 95% Cofdece Iterval. Estmated Mea: 95% CI: p.4.d. Compute a 95% Cofdece Iterval for Q.5. For the smple regresso model (scalar form): Y X,..., ~ NID, we get: X X Y, Y Y, Y X SXX p.5.a. Derve p.5.b. Derve,, E E Y E,, COV,,, COV, V V Y Y V Q.6. For the smple regresso model (scalar form): Y X X,..., ~ NID, p.6.a. Derve least squares estmators of ad. p.6.b. Derve E, E, V, V, COV,

STA302/1001-Fall 2008 Midterm Test October 21, 2008

STA302/1001-Fall 2008 Midterm Test October 21, 2008 STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from