Applied Statistics and Probability for Engineers, 5 th edition February 23, b) y ˆ = (85) =

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2 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, y x b) y ˆ (85).6836 c) y ˆ (9).744 d) ˆ.46-3 a) Regresso Aalyss: Ratg Pots versus Meters per Att The regresso equato s y x Predctor Coef SE Coef T P Costat x S 5.97 R-Sq 67.% R-Sq(adj) 65.9% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total y + x + ε (4.74) S xx (4.74)(74.) S xy ˆ Sxy S xx ˆ (.39)

3 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, yˆ 4. + x SSE 8.5 ˆ σ MSE b) y ˆ 4. + (6.9) 9. c) ˆ d).9 e) y ˆ 4. + (6.59) There are two resduals e y yˆ e e a) Regresso Aalyss - Lear model: Y a+bx Depedet varable: SalePrce Idepedet varable: Taxes Stadard T Prob. Parameter Estmate Error Value Level Itercept Slope Aalyss of Varace Source Sum of Squares Df Mea Square F-Rato Prob. Level -3

4 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Model Resdual Total (Corr.) Correlato Coeffcet R-squared percet Std. Error of Est..964 ˆ σ If the calculatos were to be doe by had, use Equatos (-7) ad (-8). yˆ x b) y ˆ (7.3) c) y ˆ (5.639) yˆ e y yˆ d) All the pots would le alog a 45 degree le. That s, the regresso model would estmate the values exactly. At ths pot, the graph of observed vs. predcted dcates that the smple lear regresso model provdes a reasoable ft to the data. Plot of Observed values versus predcted 5 45 Predcted Observed -5 a) Regresso Aalyss - Lear model: Y a+bx Depedet varable: Usage Idepedet varable: Temperature Stadard Parameter Estmate Error T Prob. -4

5 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Itercept Slope Aalyss of Varace Source DF Sum of Squares Mea Square F Prob. Level Model Resdual 34 3 Total Std. Error of Est R-Sq 99.9% Correlato Coeffcet.9999 ˆ σ 3 If the calculatos were to be doe by had, use Equatos (-7) ad (-8). yˆ x b) y ˆ (3) 8.67 c) If mothly temperature creases by.5 C, ŷ creases by 7.59 d) y ˆ (8) 9.7 y ˆ 9.7 e y yˆ a) The regresso equato s MPG Ege Dsplacemet Predctor Coef SE Coef T P Costat Ege Dsplacemet S R-Sq 59.% R-Sq(adj) 57.% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total 65.4 ˆ σ 4. yˆ 39..4x -5

6 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, b) y ˆ 39..4(5) 33.7 c) y ˆ e y yˆ a) 6 y x 95 5 Predctor Coef StDev T P Costat x S.76 R-Sq 8.% R-Sq(adj) 78.% Aalyss of Varace Source DF SS MS F P Regresso Error Total a) ˆ σ 7.3 yˆ x b) y ˆ e c) y ˆ (96) 5.7 y x 9-6

7 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Yes, a lear regresso would seem approprate, but oe or two pots mght be outlers. Predctor Coef SE Coef T P Costat x S.38 R-Sq 74.8% R-Sq(adj) 73.4% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total 9 4. b) ˆ σ.737 ad yˆ x c) y ˆ at x 9-9 a) 5 y 5 x 3 4 Yes, a lear regresso model appears to be plausble. Predctor Coef StDev T P Costat x S 9.96 R-Sq 87.9% R-Sq(adj) 86.% Aalyss of Varace Source DF SS MS F P Regresso Error Total 8 37 b) ˆ σ ad yˆ x c) y ˆ (3) 8.84 d) yˆ e

8 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, - a) 4 3 y x Yes, a smple lear regresso model seems approprate for these data. Predctor Coef StDev T P Costat x S 3.76 R-Sq 85.% R-Sq(adj) 84.3% Aalyss of Varace Source DF SS MS F P Regresso Error Total b) ˆ σ 3.8 yˆ x c) y ˆ ().38 d) yˆ e.579 for x.63 - a) -8

9 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Yes, a smple lear regresso (straght-le) model seems plausble for ths stuato. Predctor Coef SE Coef T P Costat x S R-Sq 89.7% R-Sq(adj) 89.% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total b) ˆ σ 4699 yˆ x c) y ˆ () 999. d) If there were o error, the values would all le alog the 45 le. The plot dcates age s a reasoable regressor varable. -9

10 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, - a) The regresso equato s Porosty Temperature Predctor Coef SE Coef T P Costat Temperature S R-Sq 6.% R-Sq(adj).3% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total yˆ x ˆ σ b) y ˆ (4) 7.86 c) y ˆ 4.39 e 7. -

11 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, d) The smple lear regresso model does t seem approprate because the scatter plot does t show a lear relatoshp betwee the data. -3 a) The regresso equato s BOD Tme Predctor Coef SE Coef T P Costat Tme S.878 R-Sq 94.7% R-Sq(adj) 94.% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total 4.87 yˆ x ˆ σ.83 b) y ˆ (5) 3.38 c).78(3).534 d) y ˆ (6).76 e y yˆ

12 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, e) All the pots would le alog the 45 degree le y yˆ. That s, the regresso model would estmate the values exactly. At ths pot, the graph of observed vs. predcted dcates that the smple lear regresso model provdes a reasoable ft to the data. -4 a) The regresso equato s Deflecto Stress level Predctor Coef SE Coef T P Costat Stress level S.5743 R-Sq 85.% R-Sq(adj) 83.% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total ˆ σ.8 -

13 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, b) y ˆ (64) 4.3 c) (.77)(5).385 d) e) y ˆ (75).75 e y yˆ It s possble to ft ths data wth straght-le model, but t s ot a good ft. There s a curvature show o the scatter plot. a) The regresso equato s y x Predctor Coef SE Coef T P Costat x S.5959 R-Sq 67.7% R-Sq(adj) 66.4% Aalyss of Varace -3

14 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Source DF SS MS F P Regresso Resdual Error Total yˆ. +.87x ˆ σ.53 b) y ˆ. +.87().3357 c) If the relatoshp betwee legth ad age was determstc, the pots would fall o the 45 degree le y yˆ. Because the pots ths plot vary substatally from ths le, t appears that age s ot a reasoable choce for the regressor varable ths model. 9-6 a) yˆ ( x+ 3) yˆ x yˆ x b) ˆ Let x ege dsplacemet (cm 3 ) ad x old ege dsplacemet ( 3 ) a) The old regresso equato s y 39..4x old Because cm 3, the ew regresso equato s yˆ 39..4( x/6.387) 39..5x b) ˆ.5-8 ˆ + ˆ ˆ ˆ x ( y x) + x y -4

15 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -9 a) The slopes of both regresso models wll be the same, but the tercept wll be shfted. b) yˆ x ˆ ˆ vs. ˆ 3.4 ˆ a) The least squares estmate mmzes obta ( y x ) ( x ) [ y x + x ] yx ˆ Therefore, x. ( y x ). Upo settg the dervatve equal to zero, we b) yˆ.346x. The model seems very approprate a eve better ft chlorde watershed -5

16 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Secto -4 - a) T ˆ.857 se( ) P-value [P( T 8 >.4583)] ad P-value < (.5). ˆ.3445 T.387 se( ).5 P-value [P( T 8 >.387)] ad P-value < (.5). SSE 7.55 MSE MSR 9.43 F MS.938 E P-value s ear zero b) Because the P-value of the F-test s less tha α.5, we reject the ull hypothess that at the.5 level of sgfcace. Ths s the same result obtaed from the T test. If the assumptos are vald, a useful lear relatoshp exsts. c) ˆ σ MS E a) T ˆ se( ).373 P-value [P( T 4 >.739)] ad P-value < (.5). ˆ.4756 T se( ).63 P-value [P( T 4 > 3.885)] ad P-value < (.5). Degrees of freedom of the resdual error 5 4. Sum of squares regresso Sum of square Total Sum of square resdual error SSRegresso 45. MS Regresso 45. MSR 45. F MS 7.3 E P-value s ear zero b) Because the P-value of the F-test s less tha α.5, we reject the ull hypothess that at the.5 level of sgfcace. Ths s the same result obtaed from the T test. If the assumptos are vald, a useful lear relatoshp exsts. c) ˆ σ MS E 7.3-6

17 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -3 a) ) The parameter of terest s the regressor varable coeffcet, ) H : 3) H : 4) α. 5) The test statstc s f MSR SSR / MS SS /( ) E E 6) Reject H f f > f α,, where f.,, ) Usg results from Exercse - SS ˆ.3987( ) R Sxy SSE S yy SSR f / 8) Sce > 9.33 reject H ad coclude that compressve stregth s sgfcat predctg trsc permeablty of cocrete at α.. We ca therefore coclude that the model specfes a useful lear relatoshp betwee these two varables. P-value. SSE.3 b) ˆ σ MSE.8436 ad ˆ.8436 ˆ σ se( ).696 S xx c) ˆ x 3.74 se( ˆ ) σ Sxx a) ) The parameter of terest s the regressor varable coeffcet,. ) H : 3) H : 4) α. 5) The test statstc s MSR SSR / f MS SS /( ) E E 6) Reject H f f > f α,,8 where f.,, ) Usg the results from Exercse - -7

18 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, SS ˆ R Sxy (.46)(4.445).5886 SSE S yy SSR.75 (8.86 ) f /8 8) Sce > 8.9, reject H ad coclude the model specfes a useful relatoshp at α.. P-value. b) ˆ.796 ˆ σ se( ) S xx 4 ˆ 73.9 ˆ σ x Sxx se( ) a) Regresso Aalyss: Ratg Pts versus Yds per Att The regresso equato s Ratg Pts Yds per Att Predctor Coef SE Coef T P Costat Yds per Att S R-Sq 67.% R-Sq(adj) 66.% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total Refer to the ANOVA H : H : α. Because the P-value. < α., reject H. If the assumptos are vald, we coclude that there s a useful lear relatoshp betwee these two varables. b) ˆ σ 7. -8

19 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, ˆ 7. ˆ σ se( ).87 S 6.4 xx ˆ 7 ˆ σ x Sxx se( ) c) ) The parameter of terest s the regressor varable coeffcet. ) H : 3) H : 4) α. 5) The test statstc s t ˆ se( ˆ ), 6) Reject H f t < t α/,- where t.5,3.75 or t > t.5,3.75 7) Usg the results from Exercse -6.9 t ) Because.75 <.75, fal to reject H. There s ot eough evdece to coclude that the slope dffers from at α.. -6 Refer to ANOVA for Exercse -4 a) ) The parameter of terest s the regressor varable coeffcet,. ) H : 3) H : 4) α.5, usg t-test ˆ 5) The test statstc s t se( ˆ ) 6) Reject H f t < t α/,- where t.5,.74 or t > t.5,.74 7) Usg the results from Exercse t ) Sce 8.58 >.74 reject H ad coclude the model s useful α.5. b) ) The parameter of terest s the slope, ) H : 3) H : 4) α.5 5) The test statstc s f MS SS R/ MS SS /( ) R E E -9

20 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, 6) Reject H f f > f α,, where f.,, ) Usg the results from Exercse / f / 8) Sce > 4.33, reject H ad coclude the model s useful at a sgfcace α.5. The F-statstc s the square of the t-statstc. The F-test s a restrcted to a two-sded test, whereas the t-test could be used for oe-sded alteratve hypotheses. c) ˆ ˆ σ se( ).397 S xx ˆ ˆ σ x Sxx se( ) d) ) The parameter of terest s the tercept,. ) H : 3) H : 4) α.5, usg t-test ˆ 5) The test statstc s t se( ˆ ) 6) Reject H f t < t α/,- where t.5,.74 or t > t.5,.74 7) Usg the results from Exercse t ) Sce 5.79 >.74 reject H ad coclude the tercept s ot zero at α Refer to the ANOVA for Exercse -65 a) ) The parameter of terest s the regressor varable coeffcet,. ) H : 3) H : 4) α.5 MSR SSR / 5) The test statstc s f MS SS /( ) E E 6) Reject H f f > f α,, where f.5,,.4 7) Usg the results from Exercse / f / 8) Sce >.4, reject H ad coclude the model s useful α.5. P-value <. b) se( ˆ ) , se( ˆ )

21 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, c) ) The parameter of terest s the regressor varable coeffcet,. ) H : 3) H : 4) α.5, α/.5 5) The test statstc s t ˆ se( ˆ ), 6) Reject H f t < t α/,- where t.5,.8 or t > t.5,.8 7) Usg the results from Exercse t ) Sce 3.37 <.8 reject H ad coclude the slope s ot at α.5. P-value. d) H : H : t P-value <.5. Reject H ad coclude that the tercept should be cluded the model. -8 Refer to the ANOVA for Exercse -6 (a) H : ; H : f F MSR MS 4..,,9 E 8.8 Reject the ull hypothess ad coclude that the slope s ot zero. The exact P-value s P.463 (b) From the computer output from Exercse -6: se( ˆ ).6, se( ˆ ).767 (c) H :.5; H : <.5 t ˆ ˆ,.46 (.5) se( ˆ ) t.539, sce t s ot less tha t.539, do ot reject H.,9.,9 P. (d) H : ; H : t ˆ ˆ, se( ˆ ).6 t.86, sce t > t reject H.5,9.5,9 P 4.95E 4 -

22 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -9 Refer to the ANOVA for Exercse -7 a) H : H: α.5 f f f ,, > f α,, Therefore, reject H. P-value.4. b) se( ˆ ).454 se( ˆ ) c) H : H : α.5 t.6778 t.5,. t < / t α /, Therefore, do ot reject H. P-value Refer to the ANOVA for Exercse -8 a) H : H : α.5 f 53.5 f.5,, f > f α,,8 Therefore, reject H. P-value.9. b) se( ˆ ).5663 se( ˆ ).356 c) H : H : α.5 -

23 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, t t ,8 t > t α /,8 Therefore, reject H. P-value Refer to ANOVA for Exercse - a) H : H : α. f f f ,,8 > f α,,8 Therefore, reject H. P-value <.. b) se( ˆ ) 3.8 se( ˆ ).34 c) H : 6.84 H : 6.84 α ( 6.84) t t.5,8.878 t > / t α /,8 Therefore, do ot reject H. P-value.53().36. d) H : H : α. t 58. t.878 t.5,8 t α /,8 >, therefore, reject H. P-value <.. e) H : H : > α. -3

24 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, t t t ,8 >, therefore reject H. P-value.64. t α,8-3 Refer to ANOVA for Exercse - a) H : H : α.5 f 9.4 f.5,, f > f α,,6 Therefore, reject H. b) P-value <. c) se( ˆ ).469 se( ˆ ).9359 d) H : H : α.5 t.43 t.5,6. t >/ t α /,6 Therefore, do ot reject H. There s ot suffcet evdece to coclude that the tercept dffers from zero. Based o ths test result, the tercept could be removed from the model. -33 a) Refer to ANOVA for Exercse -3. H : H : α. Because the P-value. < α., reject H. There s evdece of a lear relatoshp betwee these two varables. b) ˆ σ.83 The stadard errors for the parameters ca be obtaed from the computer output or calculated as follows. -4

25 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, ˆ ˆ σ.83 se( ).4 S 4.9 xx ˆ.9 ˆ σ x Sxx 4.9 se( ) c) ) The parameter of terest s the tercept. ) H : 3) H : 4) α. 5) The test statstc s t se( ) 6) Reject H f t < t α /, where t.5,9 3.5 or t > t α /, where t.5, ) Usg the results from Exercse t ) Because t 3.97 > 3.5 reject H ad coclude the tercept s ot zero at α a) Refer to ANOVA for Exercse -4 H : H : α. Because the P-value. < α., reject H. There s evdece of a lear relatoshp betwee these two varables. b) Yes c) ˆ σ.8 ˆ ˆ σ.8 se( ).436 S 588 xx ˆ ˆ σ x Sxx se( ) a) H : H : α. Because the P-value.3 > α., fal to reject H. There s ot suffcet evdece of a lear relatoshp betwee these two varables. The regresso equato s -5

26 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, BMI Age Predctor Coef SE Coef T P Costat Age S R-Sq 4.6% R-Sq(adj).4% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total b) ˆ σ 3.69, se( ˆ ).34, se( ˆ ) 9.4 from the computer output c) ˆ ˆ σ x Sxx se( ) ˆ -36 t After the trasformato ˆ b ˆ, ˆ σ / S a xx S ˆ ˆ, ad xx a Sxx, x ax, b ˆ σ b ˆ σ. Therefore, b ˆ / a t t. ( b ˆ σ ) / a Sxx -37 (.5) d /6.4 Assume α.5, from Chart VII (e) ad terpolatg betwee the curves for 3 ad 4, a) ˆ ˆ σ x has a t dstrbuto wth degree of freedom. b) From Exercse -7, ˆ.346, ˆ σ , ad x The t-statstc part (a) s.334 ad H : s rejected at usual α values. -6

27 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Sectos -5 ad t α/,- t.5,.79 a) 95% cofdece terval o. ˆ ± t ˆ α /, se( ).398 ± t.5, (.696).398 ±.79(.696) b) 95% cofdece terval o. ˆ ± t ˆ.5,se( ) 48.3 ±.79(.5959) c) 95% cofdece terval o μ whe x 5.. ˆ μ (.5) Yx ( x x) Yx ± t.5, σ + Sxx ˆ μ ˆ ( ) Yx ( ) ± (.79).844( + ) ±.79(.3943) ˆ μ d) 99% o predcto terval whe x 5...5, ( x x) Sxx yˆ ± t ˆ σ ( + + ) ( ) ± ( + + ) ± 3.55(.456) y It s wder because t depeds o both the errors assocated wth the ftted model ad the future observato. -4 t α/,- t.5,8.878 a) ˆ ± ( ) t se( ˆ ).5,8.46 ± (.878)(.484) b) ˆ ± ( ).5,8 t se( ˆ ) ± (.878)(.495) c) 99% cofdece terval o μ whe x 85 F. -7

28 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, ˆ μ Yx ( x x) Yx ± t.5,8 σ + Sxx ˆ μ ˆ ( ) Yx ( ) ± (.878).796( + ) ± ˆ μ d) 99% predcto terval whe x 9 F. yˆ ,8 ( x x) Sxx yˆ ± t ˆ σ ( + + ) (9 73.9) ± ( + + ) ± y tα /, t.5,3.4 a) 99% cofdece terval o ˆ ± t ˆ α /, se( ).9 ± t.5,3 (.87).9 ±.75(.87) b) 99% cofdece terval o ˆ ± t ˆ α /, se( ) 4.95 ± t.5,3(9.56) 4.95 ±.75(9.56).79 ˆ c) 99% cofdece terval for the mea ratg whe the average yards per attempt s 8. ˆ μ (8.) ( x x) ˆ μ± t.5,3 ˆ σ + Sxx ( 8 7) 94.93± μ d) 99% predcto terval o x 8. -8

29 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, yˆ± t.5,3 ˆ σ + + ( x x) ( 8 7) 94.93± μ 9.99 S xx -4 Regresso Aalyss: Prce versus Taxes The regresso equato s Prce Taxes Predctor Coef SE Coef T P Costat Taxes S.964 R-Sq 76.7% R-Sq(adj) 75.7% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total a) (.393) (.393) 4.34 b) (.393) (.398) ( ) c) ± (.74) ( + ) ± ˆ μ Yx ( ) d) ± (.74) ( + + ) ± y Regresso Aalyss: Usage versus Temperature The regresso equato s -9

30 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Usage Temperature Predctor Coef SE Coef T P Costat Temperature S.8343 R-Sq 99.9% R-Sq(adj) 99.9% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error 34 3 Total a) (.5798) (.5798) 7.7 b) (.77) (.77) (3 8.8).97 c) 8.67 ± (.8) 3( + ) 8.67 ± μˆ Y x (3 8.8).97 d) 8.67 ± (.8) 3( + + ) 8.67 ± y It s wder because the predcto terval cludes errors for both the ftted model ad for a future observato. -44 Refer to the ANOVA for Exercse -6. (a) t.5, ; (b) Descrptve Statstcs: x dsplacemet Sum of Varable Mea Sum Squares x

31 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, yˆ 33.5 whe x 5 (5 38.9) 33.5 ± , 436, ±.856 μ Yx (c) yˆ 33.5 whe x 5 (5 38.9) 33.5 ± , 436, ± Y a) b) c).64 (3.6 ) ( (9 939 ) ± + ) ± ) μ y x (9 939) d) ± (3.6) ( + + ) 46.64± y a) b) (85 8.3) 3. c) ± (.).983( + ) 4.763± μ y x (85 8.3) 3. d) ± (.).983( + + ) 4.763± y a) b) (3 4.5) c) 8.84 ± (.365) ( + ) -3

32 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, 8.84 ± μ yx -48 a) b) c).38 (.9) 3.89 ( (.86 ) ± + ) ± μ yx d).38 (.9) 3.89( (.86 ) ± + + ) ± y a) b) ( ) c) 999. ± ( + ) 999. ± μ yx ( ) d) 999. ± ( + + ) 999. ± y tα /, t.5,5 4.3 a) 99% cofdece terval o ˆ ˆ ± t ˆ α /, se( ).34 ± t.,5(.6).34 ± 4.3(.6).388 ˆ.78 b) 99% cofdece terval o -3

33 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, ˆ ± t ˆ α /, se( ) ± t.5,5(3.) ± 4.3(3.) ˆ 85. c) 99% cofdece terval for the mea legth whe x 5: ˆ μ (5) 4.63 ( x x) ˆ μ± t.5,5 ˆ σ + Sxx 4.63 ± ± 4.3(7.396) 5.9 μ ( ) d) 99% predcto terval whe x 5 ( x x).5,5 σ Sxx yˆ± t ˆ + + ( ) 4.63 ± ± 4.3(.49) 4.7 y 5.96 It s wder because t depeds o both the error assocated wth the ftted model as well as that of the future observato. -5 Refer to the Mtab output Exercse -3 tα /, t.5,9 3.5 a) 99% cofdece terval for ˆ ˆ ± t ˆ α /, se( ).78 ± t.5,9 (.4).78 ± 3.5(.4).35 ˆ.35 b) 99% cofdece terval o ˆ ± t ˆ α /, se( ).6578 ± t.5,9 (.657).6578 ± 3.5(.657).9 ˆ.96 c) 95% cofdece terval o μ whe x 8-33

34 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, ˆ μ (8).8 ˆ μ yx ± t ˆ σ + yx.5,9 yx ( x x) ( 8.9).8 ± μ.9 S xx -34

35 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Secto -7 ˆ S S 59.7 XX -5 R ( ) YY The model accouts for 86.7% of the varablty the data. -53 Refer to the Mtab output Exercse -3. a) R.67 The model accouts for 67.% of the varablty the data. b) There s o major departure from the ormalty assumpto the followg graph. c) The assumpto of costat varace appears reasoable. -35

36 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -54 Use the results from Exercse -5 to aswer the followg questos. a) SalePrce Taxes Predcted Resduals b) Assumpto of ormalty does ot seem to be volated sce the data appear to fall alog a straght le. Normal Probablty Plot cumulatve percet Resduals -36

37 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, c) There are o serous departures from the assumpto of costat varace. Ths s evdet by the radom patter of the resduals. Plot of Resduals versus Predcted Plot of Resduals versus Taxes Resduals 4 Resduals Predcted Values Taxes d) R 76.73% ; -55 Use the results of Exercse -5 to aswer the followg questos a) R %; The proporto of varablty explaed by the model. b) Yes, ormalty seems to be satsfed because the data appear to fall alog the straght le. Normal Probablty Plot cumulatve percet Resduals c) There mght be lower varace at the mddle settgs of x. However, ths data does ot dcate a serous departure from the assumptos. Plot of Resduals versus Predcted Plot of Resduals versus Temperature Resduals.4 Resduals Predcted Values Temperature -37

38 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -56 a) R.% b) These plots mght dcate the presece of outlers, but o real problem wth assumptos. c) The ormalty assumpto appears margal. Normal Probablty Plot of the Resduals (respose s y) Resdual Normal Score -57 a) R b) No departures from costat varace are oted. Resduals Versus x (respose s y) Resduals Versus the Ftted Values (respose s y) 3 3 Resdual Resdual x Ftted Value c) Normalty assumpto appears reasoable. Normal Probablty Plot of the Resduals (respose s y) 3 Resdual Normal Score -38

39 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -58 a) R 7.7% b) No major departure from ormalty assumptos. Normal Probablty Plot of the Resduals (respose s y) 3 Resdual Normal Score c) Assumpto of costat varace appears reasoable. Resduals Versus x (respose s y) Resduals Versus the Ftted Values (respose s y) 3 3 Resdual Resdual x Ftted Value -59 a) R 85. % b) Assumptos appear reasoable, but there s a suggesto that varablty creases slghtly wth ŷ. Resduals Versus x (respose s y) Resduals Versus the Ftted Values (respose s y) 5 5 Resdual Resdual x Ftted Value c) Normalty assumpto may be questoable. There s some bedg away from a le the tals of the ormal probablty plot. -39

40 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Normal Probablty Plot of the Resduals (respose s y) 5 Resdual Normal Score -6 a) The regresso equato s Compressve Stregth Desty Predctor Coef SE Coef T P Costat Desty S R-Sq 86.% R-Sq(adj) 85.6% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total b) Because the P-value. < α., the model s sgfcat. c) ˆ σ 569 SSR SSE d) R % SS SS T T The model accouts for 85.97% of the varablty the data. e) No major departure from the ormalty assumpto. f) -4

41 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Assumpto of costat varace appears reasoable. -6 a) R % of the varablty s explaed by the model. b) Yes, the two pots wth resduals much larger magtude tha the others seem uusual. c) R ew model.8799 Smaller, because the older model s better able to accout for the varablty the data wth these two outlyg data pots removed. old model d) ˆ σ 4699 ew model ˆ σ Yes, reduced more tha 5%, because the two removed pots accouted for a large amout of the error. -6 a) -4

42 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, yˆ x b) H: H: α.5 c) f 7.4 f 4.75 f.5,, > f α,, Reject H. ˆ σ 6.97 org d) ˆ σ 7.5 The ew estmate s larger because the ew pot added addtoal varace that was ot accouted for by the model. e) y ˆ (86) e y yˆ Yes, e 4 s especally large compared to the other resduals. f) The oe added pot s a outler ad the ormalty assumpto s ot as vald wth the pot cluded. -4

43 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, g) Costat varace assumpto appears vald except for the added pot. x -43

44 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -63 Yes, whe the resduals are stadardzed the uusual resduals are easer to detfy For two radom varables X ad X, V( X + X ) V( X ) + V( X ) + Cov( X, X ) The, VY ( ˆ ˆ ˆ Y) VY ( ) + VY ( ) CovYY (, ) ˆ ˆ ( x x) σ + V( + x ) σ + S xx ( x x) ( x x) σ + σ + σ + Sxx Sxx ( x x) σ ( + ) S xx a) Because e s dvded by a estmate of ts stadard error (whe σ s estmated by ˆ σ ), r has approxmately ut varace. b) No, the term brackets the deomator s ecessary. c) If x s ear x ad s reasoably large, r s approxmately equal to the stadardzed resdual. d) If x s far from x, the stadard error of e s small. Cosequetly, extreme pots are better ft by least squares regresso tha pots ear the mddle rage of x. Because the studetzed resdual at ay pot has varace of approxmately oe, the studetzed resduals ca be used to compare the ft of pots to the regresso le over the rage of x. -65 Usg ( )( ) S SS S SS SSE SS S E yy yy E yy E R, F S yy SSE SSE ˆ σ Syy Also, S yy SS E SS E ( y ( y ( y ( y ˆ Therefore, F ˆ ˆ x ) y ˆ ( x x)) y) + ˆ ( x x) y) ˆ ˆ ˆ σ / Sxx t ( x x) ˆ ( x x) ( y y)( x x) Because the square of a t radom varable wth degrees of freedom s a F radom varable wth ad degrees of freedom, the usual t-test that compares t to t α /, s equvalet to comparg f t to fα,,. /, t α -44

45 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3,.9(3) a) f 7.9. Reject H :. b) Because f.,,3 7.88, H s rejected f That s, H s rejected f 3R > 7.88( R ) 7.8R > 7.88 R >.89 3R R >

46 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Secto a) H : ρ H : ρ α.5.8 t t.5,8. t > t.5,8 Reject H. P-value (<.5)() <. b) H : ρ.5 H : ρ.5 α.5 z z z.5 (arctah (.8) arctah (.5))(7).96 > z α / Reject H. P-value (.)().4. /.65 z.5 z.5 c) ρ tah(arctah.8 ) tah(arctah.8 ) where z ρ Because ρ ad ρ.5 are ot the terval, so reject H. -67 a) H : ρ H : ρ > α.5 t t t.5, > t.5,8 Reject H. P-value <.5 b) H : ρ.5 H : ρ >.5 α.5 z z z.5 (arctah (.75) arctah (.5))(7).65 > z α Reject H. P-value.4 z.5 c) 7 / ρ tah(arctah.75 ) where z.5.64 ρ Because ρ ad ρ.5 are ot the terval, reject the ul hypotheses from parts (a) ad (b) r.83-46

47 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, a) H : ρ H : ρ α.5 t r.83 8 r (.83) t t.48.5,8 > t α /,8 Reject H. P-value z.5 z.5 b) ρ tah(arctah.83 ) tah(arctah.83 ) where z ρ.7. a) H : ρ.8 H : ρ.8 α.5 z z.96 z / (arctah.83 arctah.8)(7) >/ z α / Do ot reject H. P-value (.3)() r.6 a) H : ρ H : ρ α. t r.6 48 r (.6) t.5,48.68 t > t.5,48 Reject H. P-value z.5 z.5 b) ρ tah(arctah.6 ) tah(arctah.6+ ) where z ρ.87. c) Yes..5-7 a) r.9333 b) H : ρ H : ρ α.5 t.6 r r (.879) t t.3.5,5 > t α /,5 Reject H. c) yˆ x -47

48 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, H : H : α.5 f f f ,,5 >> f α,,5 Reject H. Coclude that the model s sgfcat at α.5. Ths test ad the oe part b) are detcal. d) No problems wth model assumptos are oted. Resduals Versus x (respose s y) Resduals Versus the Ftted Values (respose s y) 3 3 Resdual Resdual x Ftted Value Normal Probablty Plot of the Resduals (respose s y) 3 Resdual Normal Score -7 a) yˆ x b) H : H : α.5 f f 4.4.5,,8 f >> f α,,8-48

49 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Reject H. c) r d) H : ρ H : ρ α.5 R t R.86 t. t.5,8 > t α /,8 Reject H. e) H : ρ.5 H : ρ.5 α.5 z z.96 z.5 > z α / Reject H. tah(arctah.9334 ) tah(arctah ) where z z.5 z.5 f) ρ ρ a) yˆ x b) H : H : α.5 f f f ,,4 > f α,,4 Reject H. c) r d) H : ρ H : ρ α.5 t t t ,4 > t α /,4 Reject H e) H : ρ.6 H : ρ.6 α.5-49

50 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, z z.96 z / (arctah arctah.6)(3).65.5 >/ z α / Do ot reject H. z.5 z.5 f) ρ 3 3 tah(arctah ) tah(arctah ) where z ρ a) The regresso equato s Curret WthoutElect(mA) Supply Voltage Predctor Coef SE Coef T P Costat Supply Voltage S R-Sq 89.9% R-Sq(adj) 88.6% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total yˆ x Yes, because the P-value, the regresso model s sgfcat at α.5. b) r c) H : ρ -5

51 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, H : ρ r.948 t 8.45 r.948 t.36.5,8 t 8.45 > t.36.5,8 Reject H. d) zα / zα / tah arcta h r ρ tah arcta h r tah arcta h.948 ρ tah arcta h ρ a) The regresso equato s Y Y Predctor Coef SE Coef T P Costat Y S.37 R-Sq 99.5% R-Sq(adj) 99.4% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error.. Total 3.39 yˆ.4 +.x Yes, because the P-value, the regresso model s sgfcat at α.5. -5

52 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, b) r c) H : ρ.9 H : ρ.9 z (arcta h R arcta h ρ ) 3.5 ( ) / ( ) z (arcta h.9975 arcta h.9) 3 z zα / z.5.96 z > z Reject H. P-value ( )().. / d) zα / zα / tah arcta h r ρ tah arcta h r tah arcta h.9975 ρ tah arcta h ρ Refer to the Mtab output Exercse -3. a) r.67.8 b) H : ρ H : ρ r.8 3 t r.8 t.5,3.4 t > t.5,3 Reject H. The P-value < c) tah arcta h (.8) ρ tah arcta h (.8) ρ.99 d) H : ρ.7 H : ρ.7 z (arcta h R arcta h ρ ) 3 / ( ) ( ) z (arcta h.8 arcta h.7) 3 3 / z.56 z z.96 α /.5 z < z.5 Fal to Reject H. The P-value (.594).88-5

53 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -76 Here r. The correlato coeffcet does ot detect the relatoshp betwee x ad y because the relatoshp s ot lear. See the graph above. -53

54 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Secto a) Yes, l y l + l x+ l ε b) No c) Yes, l y l + xl + l ε d) Yes, + + ε y x -78 a) There s curvature the data. b) y x c) Source DF SS MS F P Regresso Resdual Error Total 6589 d) There s a curve the resduals. -54

55 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, e) The data are lear after the trasformato to y* l y ad x* /x. l y.6 585(/x) Aalyss of Varace Source DF SS MS F P Regresso Resdual Error 9.. Total There s stll curvature the data, but ow the plot s covex stead of cocave. -79 a) -55

56 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, 5 y x b) yˆ x c) H : H : α.5 f.3 f > f.5,,48 Reject H. Coclude that regresso model s sgfcat at α.5 a) No, t seems the varace s ot costat, there s a fuel shape. Resduals Versus the Ftted Values (respose s y) 3 Resdual Ftted Value e) yˆ x. Yes, the trasformato stablzes the varace. -56

57 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Secto - -8 a) The ftted logstc regresso model s yˆ + exp[ ( x)] The Mtab result s show below Bary Logstc Regresso: Home Owershp Status versus Icome Lk Fucto: Logt Respose Iformato Varable Value Cout Home Owershp Status (Evet) 9 Total Logstc Regresso Table Odds 95% CI Predctor Coef SE Coef Z P Rato Lower Upper Costat Icome Log-Lkelhood -.63 Test that all slopes are zero: G 5., DF, P-Value.3 b) The P-value for the test of the coeffcet of come s.44 < α.5. Therefore, come has a sgfcat effect o home owershp status. c) The odds rato s chaged by the factor exp( ) exp(.7). for every ut crease come. More realstcally, f come chages by $, the odds rato s chaged by the factor exp( ) exp(.7) a) The ftted logstc regresso model s yˆ + exp[ ( x)] The Mtab result s show below Bary Logstc Regresso: Number Falg, Sample Sze, versus Load (kn/m ) Lk Fucto: Logt Respose Iformato Varable Value Cout Number Falg Falure 337 Success 353 Sample Sze Total 69 Logstc Regresso Table Odds 95% CI -57

58 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Predctor Coef SE Coef Z P Rato Lower Upper Costat load (kn/m ) Log-Lkelhood Test that all slopes are zero: G.459, DF, P-Value. b) The P-value for the test of the coeffcet of load s ear zero. Therefore, load has a sgfcat effect o falg performace. -8 a) The ftted logstc regresso model s yˆ + exp[ ( x)] The Mtab results are show below Bary Logstc Regresso: Number Redee, Sample sze, versus Dscout, x Lk Fucto: Logt Respose Iformato Varable Value Cout Number Redeemed Success 693 Falure 397 Sample Sze Total 66 Logstc Regresso Table Odds 95% CI Predctor Coef SE Coef Z P Rato Lower Upper Costat Dscout, x Log-Lkelhood Test that all slopes are zero: G 74.36, DF, P-Value. b) The P-value for the test of the coeffcet of dscout s ear zero. Therefore, dscout has a sgfcat effect o redempto. -58

59 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, c) d) The P-value of the quadratc term s.95 >.5, so we fal to reject the ull hypothess of the quadratc coeffcet at the.5 level of sgfcace. There s o evdece that the quadratc term s requred the model. The Mtab results are show below Bary Logstc Regresso: Number Redee, Sample sze, versus Dscout, x Lk Fucto: Logt Respose Iformato Varable Value Cout Number Redeemed Evet 693 No-evet 397 Sample Sze Total 66 Logstc Regresso Table 95% Odds CI Predctor Coef SE Coef Z P Rato Lower Costat Dscout, x Dscout, x* Dscout, x Predctor Upper Costat Dscout, x. Dscout, x*dscout, x. Log-Lkelhood Test that all slopes are zero: G , DF, P-Value. e) The expaded model does ot vsually provde a better ft to the data tha the orgal model. Comparso of Data ad two Logstc Regresso -59

60 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -83 a) The Mtab results are show below Bary Logstc Regresso: y versus Icome x, Age x Lk Fucto: Logt Respose Iformato Varable Value Cout y (Evet) Total Logstc Regresso Table Odds 95% CI Predctor Coef SE Coef Z P Rato Lower Upper Costat Icome x Age x Log-Lkelhood -.43 Test that all slopes are zero: G 6.88, DF, P-Value.3 b) Because the P-value.3 < α.5 we ca coclude that at least oe of the coeffcets (of come ad age) s ot equal to zero at the.5 level of sgfcace. The dvdual z-tests do ot geerate P-values less tha.5, but ths mght be due to correlato betwee the depedet varables. The z-test for a coeffcet assumes t s the last varable to eter the model. A model mght use ether come or age, but after oe varable s the model, the coeffcet z-test for the other varable may ot be sgfcat because of ther correlato. c) The odds rato s chaged by the factor exp( ) exp(.833).8 for every ut crease come wth age held costat. Smlarly, odds rato s chaged by the factor exp( ) -6

61 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, exp(.663).894 for every ut crease age wth come held costat. More realstcally, f come chages by $, the odds rato s chaged by the factor exp( ) exp(.833).87 wth age held costat. d) At x 45 ad x 5 from part (a) yˆ + exp[ ( x x )].78 e) The Mtab results are show below Bary Logstc Regresso: y versus Icome x, Age x Lk Fucto: Logt Respose Iformato Varable Value Cout y (Evet) Total Logstc Regresso Table Odds 95% CI Predctor Coef SE Coef Z P Rato Lower Upper Costat Icome x Age x Icome x*age x Log-Lkelhood -8. Test that all slopes are zero: G.53, DF 3, P-Value.9 Because the P-value.4 there s o evdece that a teracto term s requred the model. -6

62 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, -6 Supplemetal Exercses -84 a) ˆ ˆ ( ) y y y y ad ˆ ˆ y x + from the ormal equatos The, ˆ ˆ ˆ ( ) ˆ ˆ ˆ ˆ ( ) ˆ ˆ ˆ ˆ x y x x x x b) x y x y x y y ˆ ) ˆ ( ad + x x x y ˆ ˆ from the ormal equatos. The, ˆ ˆ ˆ ˆ ) ˆ ˆ ( ˆ ˆ x x x x x x x x c) y y ˆ ) ˆ ˆ ( ˆ x y + y x x y x x y x x y x x y ˆ ˆ ) ˆ ˆ ( ) ˆ ) ˆ ( ( ) ˆ ˆ ( ) ˆ ˆ ( ˆ -85 a)

63 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Plot of y vs x..9.6 y x Yes, a lear relatoshp seems plausble. b) Model fttg results for: y Idepedet varable coeffcet std. error t-value sg.level CONSTANT x R-SQ. (ADJ.). SE.79 MAE.63 DurbWat.843 Prevously:.... observatos ftted, forecast(s) computed for mssg val. of dep. var. yˆ x c) Aalyss of Varace for the Full Regresso Source Sum of Squares DF Mea Square F-Rato P-value Model Error Total (Corr.) R-squared Std. error of est..7976e-3 R-squared (Adj. for d.f.) Durb-Watso statstc.8439 ) H : 3) H : 4) α.5 SSR / k 5) The test statstc s f SS /( p) 6) Reject H f f > f α,,8 where f.,,8.6 7) Usg the results from the ANOVA table E -63

64 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3,.9663/ f / 8 8) Because 564 >.6 reject H ad coclude that the regresso model s sgfcat at α.5. P-value d) 99 percet cofdece tervals for coeffcet estmates Estmate Stadard error Lower Lmt Upper Lmt CONSTANT x e) ) H : 3) H : 4) α. 5) The test statstc s t ˆ se( ˆ ) 6) Reject H f t < t α/,- where t.5, or t > t.5, ) Usg the results from the table above.9668 t ) Sce < reject H ad coclude the tercept s sgfcat at α a) yˆ x b) H : H : α.5 f.87 f.5,,4 4.6 f > f.5,,4 Reject H. Coclude that at α.5. c) ( ) d) ( ) e) y ˆ (.5)

65 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, ± ± ˆ μ Yx.5 (.5.35) yˆ x where y / y. No, the model does ot seem reasoable. The resdual plots dcate a possble outler. -88 yˆ x, r.99, R 98.43% The model appears to be a excellet ft. The R s large ad both regresso coeffcets are sgfcat. No, the exstece of a strog correlato does ot mply a cause ad effect relatoshp. -89 yˆ.796x Eve though y should be zero whe x s zero, because the regressor varable does ot usually assume values ear zero, a model wth a tercept fts ths data better. Wthout a tercept, the resduals plots are ot satsfactory. -9 a) 9 8 days dex 8 b) The regresso equato s yˆ x Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total

66 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Do ot reject H. We do ot have evdece of a relatoshp. Therefore, there s ot suffcet evdece to coclude that the seasoal meteorologcal dex (x) s a relable predctor of the umber of days that the ozoe level exceeds. ppm (y). c) 95% CI o ˆ ± t α /,.5, se( ˆ ) 5.96 ± t (9.4) 5.96 ±.45(9.4) d) The ormalty plot of the resduals s satsfactory. However, the plot of resduals versus ru order exhbts a strog dowward tred. Ths could dcate that there s aother varable should be cluded the model ad t s oe that chages wth tme. Resdual Observato Order 4 6 Normal Score Resdual a) y x.8.9. b) yˆ x -66

67 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, c) Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total R.47% Because the P-value >.5, reject the ull hypothess ad coclude that the model s sgfcat. d) There appears to be curvature the data. There s a dp the mddle of the ormal probablty plot ad the plot of the resduals versus the ftted values shows curvature..5.. Normal Score Resdual Resdual Ftted Value -9 a) 94 y x b) yˆ x c) Predctor Coef SE Coef T P Costat Therm S R-Sq 7.% R-Sq(adj) 67.6% Aalyss of Varace Source DF SS MS F P Regresso

68 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Resdual Error Total Reject the ull hypothess ad coclude that the model s sgfcat. Here 77.3% of the varablty s explaed by the model. d) H : H : α.5 ˆ.999 t.3354 se( ˆ ).9 t a/, t.5,8.36 Because t > t a/,, we caot reject H ad we coclude that there s ot eough evdece to reject the clam that the devces produce dfferet temperature measuremets. Therefore, we assume the devces produce equvalet measuremets. e) The resdual plots to ot reveal ay major problems. Normal Probablty Plot of the Resduals (respose s IR) Normal Score Resdual -68

69 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Resduals Versus the Ftted Values (respose s IR) 5 Resdual Ftted Value -93 a) b) yˆ. +.7x c) Source DF SS MS F P Regresso Resdual Error Total Reject the ull hypothess ad coclude that the model s sgfcat. d) x 4.5 ˆ μ yx ( ) ± ±.36(.356) μ yx

70 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, e) The ormal probablty plot of the resduals appears lear, but there are some large resduals the lower ftted values. There may be some problems wth the model. -94 a) The regresso equato s No. Of Atoms (x E9) power(mw) Predctor Coef SE Coef T P Costat power(mw) S R-Sq 98.9% R-Sq(adj) 98.8% Aalyss of Varace -7

71 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Source DF SS MS F P Regresso Resdual Error Total 4.38 b) Yes, there s a sgfcat regresso at α.5 because p-value. < α. c) r d) H : ρ H : ρ r t r.994 t.5,3.6 t > t.6..5,3 Reject H. P-value e) 95% cofdece terval for ˆ ˆ ± t ˆ α /, se( ). ± t.5,3(.66). ±.6(.66).6 ˆ a) -7

72 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, The relatoshp betwee carat ad prce s ot lear. Yes, there s oe outler, observato umber 33. b) The perso obtaed a very good prce hgh carat damod at low prce. c) All the data The regresso equato s prce carat Predctor Coef SE Coef T P Costat carat S 33.9 R-Sq 78.5% R-Sq(adj) 77.9% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total t α/,- t.5, % cofdece terval o. ˆ ± t ˆ α /, se( ) 9349 ± t.5,38(794.) 9349 ±.73(794.) Wth uusual data omtted The regresso equato s prce_ carat_ Predctor Coef SE Coef T P Costat

73 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, carat_ S 96.8 R-Sq 83.3% R-Sq(adj) 8.8% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total t α/,- t.5, % cofdece terval o. ˆ ± t ˆ α /, se( ) 989 ± t.5,37 (7.5) 989 ±.78(7.5) The wdth for the outler removed s arrower tha for the frst case. -96 The regresso equato s Populato Cout Predctor Coef SE Coef T P Costat Cout S 838 R-Sq 33.9% R-Sq(adj) 8.4% Aalyss of Varace Source DF SS MS F P Regresso.7763E+.7763E Resdual Error E Total E+ yˆ x Yes, the regresso s sgfcat at α.. Care eeds to be take makg cause ad effect statemets based o a regresso aalyss. I ths case, t s surely ot the case that a crease the stork cout s causg the populato to crease, fact, the opposte s most lkely the case. However, uless a desged expermet s performed, cause ad effect statemets should ot be made o regresso aalyss aloe. The exstece of a strog correlato does ot mply a cause ad effect relatoshp. -73

74 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Md-Expadg Exercses -97 The correlato coeffcet for the pars of data (x, z ) wll ot be ear uty. It wll be ear zero. The data for the pars (x, z ) where z y wll ot fall alog the le y x whch has a slope ear uty ad gves a correlato coeffcet ear uty. These data wll fall o a le y x that has a slope ear zero ad gves a much smaller correlato coeffcet. -98 a) ˆ SxY, ˆ Y ˆ x S xx Cov( ˆ, ˆ ) Cov( Y, ˆ ) xcov( ˆ, ˆ ) ˆ Cov( Y, S ) Cov(, ( )) ( ) xy Y Y x x x x σ. Therefore, Sxx Sxx Sxx Cov( Y, ) ˆ ˆ ˆ σ Cov(, ) V( ) S xx ˆ ˆ xσ Cov(, ) S xx b) The requested result s show part a). -99 a) MS E ( Y ˆ ˆ x) e Ee ( ) EY ( ) E( ˆ ) E( ˆ ) x ( x x) σ Sxx V( e ) [ ( + )] Therefore, E( MS E ) E( e ) V ( e ) ( x x) S xx σ [ ( + σ [ ] σ )] b) Usg the fact that SS R MS R, we obta { } EMS ( ) E( ˆ S ) S V( ˆ ) + [ E( ˆ )] R xx xx σ Sxx + σ + S Sxx xx -74

75 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, S - ˆ S x Y xx E Y ( x x ) ( + x + x )( x x ) ˆ E( ) S S xx xx S + x ( x x ) S + xx xx Sxx S xx No, ˆ s o loger ubased. - ˆ σ V ( ). To mmze V( ˆ ), Sxx should be maxmzed. Because S xx xx ( ), xx S x x S s maxmzed by choosg approxmately half of the observatos at each ed of the rage of x. From a practcal perspectve, ths allocato assumes the lear model betwee Y ad x holds throughout the rage of x ad observg Y at oly two x values prohbts verfyg the learty assumpto. It s ofte preferable to obta some observatos at termedate values of x. - Oe mght mmze a weghted some of squares w( y x) whch a Y wth small varace (w large) receves greater weght the sum of squares. w y x w y x ( ) ( ) w( y x) w( y x) x Settg these dervatves to zero yelds ˆ w + ˆ wx w y ˆ ˆ wx + wx wxy ad these equatos are solved as follows ˆ ( wxy )( w) wy ( w)( wx ) ( wx ) wy wx w w ˆ ˆ. sy -3 yˆ y + r ( x x) s x -75

76 Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, Sxy y y x x y + S S ( x x) Sxy y + ( x x) S xx ( ) ( ) xx yy y + ˆ x ˆ x ˆ + ˆ x -4 a) ( y x) ( y x) x Upo settg the dervatve to zero, we obta Therefore, x + x xy xy x x( y ) ˆ x x ( ) b) ( ˆ x Y x σ σ V ) V x x x c) ˆ ± t ˆ σ α /, x Ths cofdece terval s shorter because x ( x x). Also, the t value based o - degrees of freedom s slghtly smaller tha the correspodg t value based o - degrees of freedom. -76