Simple Linear Regression. Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapter Chapter. β0 β1. β β = 1. a. b.

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1 Chapter 3 3- Simple Liear Regressio Chapter 3 3. a. b. c. d. 3.3 a. Usig the techique explaied i Exercise 3.: b. () = β + β β β 6 = + () 4 () = β + β β β 6 = + () β = β y= + x = β = 4 y= 4+ x β = c. = β + β() β = 6 = β + β( ) β = 4 y= + 4x d. 4 = β + β() 7 β β(3) = + β = 4 y= 4 x β = 3.5 Slope (β ) y-itercept (β ) a. 3 b. c. 3 d. 5 e. 4 Copyright Pearso Educatio, Ic. Publishig as Pretice all.

2 3- Simple Liear Regressio 3.7 a. To compute ˆ β ad ˆ β, we first costruct the followig table: x y xy x y x = y = xy = x = y = 36 The, ( x) () SSxx = x = = 5 ( x)( y) () SSxy = xy = = 5 ( y) ( ) SS yy = y = 36 = 6 5 y x y = = = x = = = 5 5 Thus, the least squares estimates of β ad β are: ˆ SSxy = = =. SSxx ˆ = y ˆ βx = (.)() = β β ad the equatio of the least squares predictio lie is yˆ =. x. b. 3.9 a. Yes, a positive wide scattered bad sice the year 4. b. Positive slope which implies that the older composers had a tedecy to produce a certai pitch more times. c. The lie is based o the sample data. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

3 Chapter a. No, there does ot appear to ay tred for cooperatio use versus the average payoff. b. No, there does ot appear to ay tred for defective use versus the average payoff. c. Yes, there appears to be somewhat of a liear relatioship for average payoff ad puishmet use. d. Negative relatioship; the more puishmet use, the average payoff decreases. e. Yes, wiers ted to puish less tha o-wiers. 3.3 a. Some prelimiary calculatios are: xi i= xi i= = 667 = 645 yi i= xy i i i= = 35.8 = 4 = x i y i i= i= ( 667)( 35.8) = = = SSxy xi yi i= SSxx b xi i= x i i = ( 667) = = 645 = SS ˆ xy = = = SS xx ˆ ˆ (.3769 b = y b x = ) = The least squares lie is yˆ = x. b. b ˆ = 6.5 Sice x = is ot i the observed rage, b has o iterpretatio other tha beig the y-itercept. b ˆ =.3. For each additioal icrease of part per millio of pecti, the mea sweetess idex is estimated to decrease by.3. c. y ˆ = 6.5.3( 3) = Copyright Pearso Educatio, Ic. Publishig as Pretice all.

4 3-4 Simple Liear Regressio 3.5 Some prelimiary calculatios are: x 3.7 y = = = y 79 x = = = x y 79(3.7) SSxy = xy = = ( x) 79 SSxx = x = 5, = ˆ SSxy = = = SSxx 756 β ˆ ˆ o = y βx = (.64957) = β The estimated regressio lie is yˆ = x; sice x = is osesical, o practical iterpretatio of ˆ β =.574; for each oe-positio icrease i order, estimated recall proportio icreases by ˆ β = a. SSE.9 s =.33 = 9 = b. s =.33 =.769 = =. =. of their 3.9 We expect that 95% of the actual etropy values (y) will fall withi d s ( ) predicted values; i.e. y ±.. 3. a. y= β+ βx+ ε b. yˆ = x c. Assumptio : The mea of the probability distributio of ε is. Assumptio : The variace of the probability distributio of ε is costat for all settigs of the idepedet variable x. Assumptio 3: The probability distributio of ε if ormal. Assumptio 4: The errors associated with ay two differet observatios are idepedet. d. s = 635. yˆ± s yˆ± 635. yˆ± 7.4. e. ( ) Copyright Pearso Educatio, Ic. Publishig as Pretice all.

5 Chapter a. : β = a : β b Test statistic: t = ˆ =.857 = 6.7 s SS xx Rejectio regio: α =.5, df = = 4, t.5 =.776 Reject if t <.776 or t >.776 Coclusio: Reject. There is sufficiet evidece (at α =.5 ) to idicate that x cotributes iformatio for the predictio of y usig a liear model. b. : β= a : β b Test statistic: ˆ t = =. = 5. s SSxx.733 Rejectio regio: α =.5, df = = 3, t.5 = 3.8 Reject if t < 3.8 or t > 3.8 Coclusio: Reject. There is sufficiet evidece (at α =.5 ) to idicate that x provides iformatio for predictig y usig a liear model. 3.5 Some prelimiary calculatios are: y i i = ( 35.8) SS yy = yi = = i= 4 SSE = SS b ˆ SS = = yy xy ( )( ) SSE s = = =.4639 s bˆ s.4639 = = =.96 SS xx For cofidece level.9, α =. ad α =. =.5. From Table, Appedix D with df = = 4 =, t =.77. The cofidece iterval is:.5 bˆ.5 b ˆ ± t s.3 ± ,.8 ( ) ( ) Copyright Pearso Educatio, Ic. Publishig as Pretice all.

6 3-6 Simple Liear Regressio We are 9% cofidet that the chage i the mea sweetess idex for each oe uit chage i the pecti is betwee.39 ad a. To determie if body plus head rotatio ad active head movemet are positively liearly related, we test: β : = a : β > The test statistic is bˆ.88 t = = = 6.9 s ˆ.4 b The rejectio regio requires α =.5 i each tail of the t distributio with df = = 39 = 37. From Table, Appedix D, t The rejectio regio is t >.687. Sice the observed value of the test statistic falls i the rejectio regio ( t = 6.86 >.687, is rejected. There is sufficiet evidece to idicate that the two variables are positively liearly related at α =.5. b. For cofidece level.9, α =. ad α =. =.5. From Table, Appedix D, with df = = 39 = 37, t The cofidece iterval is: b ˆ ± t.5sˆ.88 ±.687 (.4 ).88 ±.4 (.64,. ) b We are 9% cofidet that the true value of β is betwee.64 ad.. c. Because the iterval i part b cotais the value, there is o evidece that the true slope of the lie differs from. 3.9 Usig the calculatios from Exercise 3.5 ad these calculatios: ( ) y 3.7 SSyy = y = = ( xy ) ( )( ) SSE = SS ˆ yy β SS = = SSE s = = = s= s = = To determie if there is a liear tred betwee the proportio of ames recalled ad positio, we test: : β = a : β The test statistic is bˆ bˆ.64 t = = = =.86 s bˆ s SSxx Copyright Pearso Educatio, Ic. Publishig as Pretice all.

7 Chapter The rejectio regio requires α =. =.5 i each tail of the t distributio. From Table, Appedix D, with df = = 44 = 4, t The rejectio regio is t <.576 or t >.576. Sice the observed test statistic falls i the rejectio regio ( t =.86 >.576 ), is rejected. There is sufficiet evidece to idicate the proportio of ames recalled is liearly related to positio at α = The MINITAB pritout below shows that there is a sigificat liear relatioship betwee empathic cocer ad brai activity sice the p-value for empathy of.9 / =.45 is less tha a sigificace level of.5. Thus, the researchers were correct i their theory that β >. The regressio equatio is ACTIVITY = EMPATY Predictor Coef SE Coef T P Costat EMPATY S =.684 R-Sq = 39.4% R-Sq(adj) = 35.% 3.33 a. b ˆ =.55 b ˆ =. b. Yes, there is sufficiet evidece to show that there is a positive liear relatioship betwee elevatio ad sluggig percetage sice the p-value for the elevatio is.8 / =.4 which is smaller tha the sigificace level of.. c..65 Scatterplot of SLUGPCT vs ELEVATION SLUGPCT ELEVATION Dever s elevatio is very high. d. b ˆ =.554 b ˆ =. There is sufficiet evidece to show that there is ot a positive liear relatioship betwee elevatio ad sluggig percetage sice the p-value for the elevatio is.33 / =.66 which is larger tha the sigificace level of.. The thi air theory is false. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

8 3-8 Simple Liear Regressio 3.35 a. From Exercise 3.6, SS xx = 7.5, SS yy = 4 ad SS xy = 5 SSxy 5 r = = =.9583 SS SS xx yy ( )( ) From Exercise 3.8, SSE =.435 SSyy SSE r = = =.983. SS 4 yy There is a strog positive correlatio betwee x ad y. We ca explai 9.83% of the variatio i the sample y s usig the liear model with x. b. I Exercise 3.7, SS xx =, SS yy = 6 ad SS xy = SSxy r = = = SSxxSS yy ( 6) = ˆ = 6. =.6. I Exercise 3.7, SSE SS β SS ( )( ) SSyy SSE 6.6 r = = =.9. SS 6 yy yy xy There is a strog positive liear correlatio betwee x ad y. We ca explai 9% of the variatio i the sample y s usig the liear model with x We would expect the GPA of a college studet to be correlated to his/her I.Q. As the I.Q. score icreases, we would expect the GPA to icrease. Thus, the correlatio would be positive a. y= β+ βx+ ε b. Moderate positive liear relatioship betwee RMP ad SET ratigs. c. Positive slope sice the correlatio is positive. r d. Sice r =.68, = 46 the test statistic t value is t = = = 9.97 r.68 So, the correspodig P-value is <. <.5 ad has to be rejected. e. ( ) r =.68 = % of the sum of squares of deviatio of the sample SET ratigs about their mea ca be explaied by the RMP ratigs as a liear predictor. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

9 Chapter a. Piao: r =.447 Because this value if ear.5, there is s slight positive liear relatioship betwee recogitio exposure time ad goodess of view for piao. Bech: r =.57 Because this value is extremely close to, there is a extremely weak egative liear relatioship betwee recogitio exposure time ad goodess of view for bech. Motorbike: r =.69 Because this value is ear.5, there is a moderate positive liear relatioship betwee recogitio exposure time ad goodess of view for motorbike. Armchair: r =.94 Because this value is fairly close to, there is a weak positive liear relatioship betwee recogitio exposure time ad goodess of view for armchair. Teapot: r =.949 Because this value is very close to, there is a strog positive liear relatioship betwee recogitio exposure time ad goodess of view for teapot. b. Piao: ( ) r =.447 = % of the total sample variability aroud the sample mea recogitio exposure time is explaied by the liear relatioship betwee the recogitio exposure time ad the goodess of view for piao. Bech: ( ) r =.57 =.3.3% of the total sample variability aroud the sample mea recogitio exposure time is explaied by the liear relatioship betwee the recogitio exposure time ad the goodess of view for bech. Motorbike: ( ) r =.69 = % of the total sample variability aroud the sample mea recogitio exposure time is explaied by the liear relatioship betwee the recogitio exposure time ad the goodess of view for motorbike. Armchair: ( ) r =.94 = % of the total sample variability aroud the sample mea recogitio exposure time is explaied by the liear relatioship betwee the recogitio exposure time ad the goodess of view for armchair. Teapot: ( ) r =.949 = % of the total sample variability aroud the sample mea recogitio exposure time is explaied by the liear relatioship betwee the recogitio exposure time ad the goodess of view for teapot. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

10 3- Simple Liear Regressio c. The test is: : β= a : β Followig are the values of α ad t α that correspod to df = = 5 = 3. α t α/ Piao: t =.4.69 <.4 <.5, p.5 For levels of sigificace greater tha α =.5, ca be rejected. There is sufficiet evidece to idicate that there is a liear relatioship betwee goodess of view ad recogitio exposure time for piao for α >.5. Bech: t =.7.7 <.39, p >. is ot rejected. There is isufficiet evidece to idicate that there is a liear relatioship betwee goodess of view ad recogitio exposure time for bech for α >.. Motorbike: t = > 3.767, p <. ca be rejected for α.. There is sufficiet evidece to idicate that there is a liear relatioship betwee goodess of view ad recogitio exposure time for motorbike for α.. Armchair: t = <.47 <.77, p.5 caot be rejected for levels of sigificace α <.5. There is isufficiet evidece to idicate that there is a liear relatioship betwee goodess of view ad recogitio exposure time for armchair for α <.5. Teapot: t = > 3.767, p <. ca be rejected for α.. There is sufficiet evidece to idicate that there is a liear relatioship betwee goodess of view ad recogitio exposure time for teapot for α a. Sice the p-value of.7 is smaller tha a sigificace level of.5, we ca coclude that there is o sigificat relatioship betwee baselie ad follow-up physical activity for obese youg adults; fail to reject : ρ = at.5. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

11 Chapter 3 3- b Scatterplot of Baselie vs Follow-up 65. Baselie Follow-up 6 7 c. ( ) r =.5 =.5, thus 5% of the variability aroud the sample mea for the total of follow-up umber of movemets is explaied by the baselie total umber of movemets for the obese adults. d. Sice the correlatio value itself is close to zero ad the p-value of.66 is smaller tha.95 correspodig to a sigificace level of.5 we ca coclude that there is o sigificat relatioship betwee baselie ad follow-up physical activity for ormal weight youg adults; fail to reject : ρ = at α =.5. e. Scatterplot of baselie vs follow-up 5 4 baselie follow-up 5 6 f. ( ) r =. =.44, thus.44% of the variability aroud the sample mea for the total of follow-up umber of movemets is explaied by the baselie total umber of movemets for the ormal weight youg adults i a simple liear regressio model SSxy 48.5 r = = =.57 SS SS (8.75) xx yy Because r is moderately small, there is a rather weak positive liear relatioship betwee blood lactate cocetratio ad perceived recovery. ( ) r =.57 = % of the sample variace of blood lactate cocetratio aroud the sample mea is explaied by the liear relatioship betwee blood lactate cocetratio ad perceived recovery. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

12 3- Simple Liear Regressio 3.47 a. This relatioship will have a egative correlatio sice the researchers claim a iverse relatioship. r b. Solvig t = r for r usig the smallest value of t that leads to a statistically sigificat t result gives: r =. So if t =.645 leads to a rejectio of t + : r =, the (.645) r = =.8 ad thus r =.736 =.895 sice r is egative. (.645) SS a. ˆ xy b = = = SSxx 4.77 SSE = SS b ˆ SS = = 4.6 yy xy SSE 4.55 s =.3. = = ( ) b. For x yˆ ( ) =.5, = =.6 The form of the 95% cofidece iterval is yˆ ± t s + α ( x x) SS xx For cofidece coefficiet.95, α =.5 ad α =.5 =.5. From Table, Appedix D, with df = = = 8, t.5 =.. The 95% cofidece iterval is: (.5.5).6 ± ± , ( ) We are 95% cofidet the mea value of y whe x =.5 is betwee.377 ad.83. c. For x yˆ ( ) =., = = 8.9. The 95% cofidece iterval is: (.5.5) 8.9 ± ± , ( ) We are 95% cofidet the mea value of y whe x =.5 is betwee 8.58 ad 9.9. d. For x yˆ ( ) = 3., = =.3. The 95% cofidece iterval is: ( 3..5).3 ± ±.39.98, ( ) Copyright Pearso Educatio, Ic. Publishig as Pretice all.

13 Chapter We are 95% cofidet the mea value of y whe x =.5 is betwee.98 ad.69. e. The width of the iterval i (b) is =.446 The width of the iterval i (c) is =.638 The width of the iterval i (d) is =.638 As the value of x moves away from x =.5, the cofidece iterval gets wider. f. The 95% predictio iterval is yˆ tα /s ± + + ( x x) SS xx ( 3..5).3± ±.46 (.54,3.346 ) a. Predictio iterval for y with x = b. If x x, p ( xp x) yˆ ± tα /s + + SS = = the the formula i part a reduces to y ( t ) iterval for E ( y ) with x = For x =.95% the cofidece iterval for E( y ) is ( 5.65,5.84 ). xx s ˆ ± α /, which is the cofidece 3.55 a. x =.87, SS xx = s =.8573, ad yˆ = 5..4 x. For x= 5, yˆ = 5..4( 5) = 3.5. For cofidece coefficiet.9, α =. ad α =. =.5. From Table, Appedix D, with df = = 3 =, t.5 =.7. The 9% cofidece iterval is: ( xp x) ( ) ( 5.87) yˆ ± tα /s + 3.5±.7 (.8573) + SSxx ± ,3.85. We are 9% cofidet that the mea mass of all spills with a elapsed time of 5 miutes is betwee 3.7 ad b. For cofidece coefficiet.9, α =. ad α =. =.5. From Table, Appedix D, with df = = 3 =, t.5 =.7. The 9% cofidece iterval is: ( xp x) ( 5.87) yˆ tα /s (.8573) SSxx ± + + ± + + ( ) 3.5±.54., 5.. We are 9% cofidet that the mass of a sigle spill with a elapsed time of 5 miutes is betwee. ad 5.. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

14 3-4 Simple Liear Regressio 3.57 The MINTAB results are preseted below: The regressio equatio is SDArea = NumberSD Predictor Coef SE Coef T P Costat NumberSD S = R-Sq = 38.7% R-Sq(adj) = 37.4% Aalysis of Variace Source DF SS MS F P Regressio Residual Error Total Uusual Observatios Obs NumberSD SDArea Fit SE Fit Residual St Resid R X X X R deotes a observatio with a large stadardized residual. X deotes a observatio whose X value gives it large leverage. Predicted Values for New Observatios New Obs Fit SE Fit 95% CI 95% PI (.7, 459.9) (-53.7, 535.3) Values of Predictors for New Observatios New Obs NumberSD 35 Based o the pritout, we are 95% cofidet that the total area of structurally deficiet bridges i a state is betwee (egative umber of bridges does ot make sese) to bridges; (,535.3 ) A scatterplot is give below to check visually for a liear relatioship: 7 Scatterplot of EATRATE vs RPM EATRA TE RPM The MINITAB pritout idicates that the measure of RPM is a sigificat liear predictor of the heat rate ad ca reject : β = ; t=.69 sice the p-value =. is less tha ay sigificace level. Thus the liear equatio is yˆ = x where 7.% of the sample variace i the heat rate is explaied by the RPM. s = 734; a 95% PI for y whe..33,3.58 x = is ( ) Copyright Pearso Educatio, Ic. Publishig as Pretice all.

15 Chapter The regressio equatio is EATRATE = RPM Predictor Coef SE Coef T P Costat RPM S = 86.7 R-Sq = 7.% R-Sq(adj) = 7.8% 3.6 A scatterplot of the umber of hours worked per week ad work/life balace scale score for MBA alumi is preseted below: 8 Scatterplot of WLB-SCORE vs OURS 7 6 WLB-SCORE OURS 8 As you ca see, the data is wildly spread aroud the straight-lie. A MINITAB aalysis was coducted ad the results are as follows: The regressio equatio is WLB-SCORE = OURS Predictor Coef SE Coef T P Costat OURS S =.845 R-Sq = 7.% R-Sq(adj) = 7.% Notice that whe testig : β =, p value=, thus we reject the ull hypothesis ad coclude that the umber of hours worked is sigificat as a useful predictor of the work/life balace score ad results i a regressio equatio of y= x. Also otice that oly 7%, R =.7 of the variability aroud the sample mea for work/life score. There could be other factors that could explai the variability or more data could be take. s = 4.6; ad a 95% PI for y whe x = 5 is (.,69.3 ). Thus, eve though the regressio lie is sigificat it should be used cautiously due to the variability of the data a. The results of the prelimiary calculatios are provided below: = 5, x = 3, xy = 78, y = 589 Substitutig ito the formula for ˆ β, we have ˆ xy 78 β = = = x 3 squares lie is yˆ = x. ad the least Copyright Pearso Educatio, Ic. Publishig as Pretice all.

16 3-6 Simple Liear Regressio = ˆ = =.8667 b. SSE y β xy ( )( ) SSE.8667 s = 3.67 = 5 = s= s = 3.67 =.7935 c. : β= a : β < ˆ β Test statistic: t = = = 8.3 s/ x.7935 / 3 Rejectio regio: α =.5, df = = 4, t.5 =.3 Reject if t <.3 Coclusio: Reject. Yes, there is sufficiet evidece at α =.5 to idicate that x ad y are egatively liearly related. d. A 95% cofidece iterval for β is ˆ β ± t s/ x.5 where t.5 =.776 is based o = 4 df. Substitutig, we have: ± ±.99 (.76, ). 3 =, = ˆ = 9.67 = e. Whe x yˆ β x ( )( ) p x p A 95% cofidece iterval for E( y ) is yˆ ± t.5s x 9.67 ±.776(.7935) 9.67 ±.99 (.76, ). 3 f. From part (e), we predict y ˆ = 9.67 whe x p =. A 95% predictio iterval for y is x p yˆ ± t.5s + x ( ) ( ) 9.67 ± ± , Copyright Pearso Educatio, Ic. Publishig as Pretice all.

17 Chapter a. Prelimiary calculatios yield: = 8, x = 59.75, xy = 3.5, y =738 The, ˆ xy 3.5 β = = = 5.364, ad the least squares lie is yˆ = x. x b. : β = a : β ˆ β Test statistic: t = s/ x where SSE y ˆ βxy 738 (5.364)(3.5) s= s = = = =.64 7 Substitutig, we have t = = / Rejectio regio: α =., df = = 7, tα / = t.5 =.895 Reject if t >.895 or t <.895 Coclusio: Reject at α =.. Yes, there is sufficiet evidece to idicate that the straight-lie model through the origi is useful for predictig decrease i pulse rate y. c. We wat to predict the decrease i pulse rate y correspodig to a drug dosage of x p = 3.5 cubic cetimeters. First, we obtai the poit estimate: yˆ = ˆ β x= = 8.77 ( ) A 99% predictio iterval for y is = 7 df. xp.5 yˆ ± t s + x where t.5 = is based o Substitutio yields: (3.5) ( ) ( ) 8.77 ± ± , Therefore, we predict the decrease i pulse rate correspodig to a dosage of 3.5 c.c. to fall betwee.47 ad 5.7 beats/miute with 99% cofidece. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

18 3-8 Simple Liear Regressio 3.67 a. Prelimiary calculatios yield: =, x =93354, xy = , y = The, ˆ xy β = = = 5.8, x y ˆ = 5.8x. ad the least squares predictio equatio is b. : β = a : β ˆ β Test statistic: t = s/ x where ( y ˆ βxy) SSE s= s = = ( ) = = Substitutig, we have 5.8 t = = / Rejectio regio: α =., df =, 9, t.5 = 3.5 Reject if t < 3.5 or t > 3.5 Coclusio: Reject. Yes, there is sufficiet evidece at α =. to idicate x, populatio i service area, cotributes iformatio for the predictio of y, residetial electric customers i service area. c. We eed the followig additioal iformatio: x = 486, y=97, SS = , SS = , SS 3898 xx xy yy = ˆ β = 47.7, ˆ β = , SSE = s = , s = The least squares predictio equatio is yˆ = x. : β = a : β ˆ β Test statistic: 47.7 t = = = s / SS / xx Copyright Pearso Educatio, Ic. Publishig as Pretice all.

19 Chapter Rejectio regio: α =., df = = 8, t.5 = Reject if t < or t > Coclusio: Reject. Yes, there is sufficiet evidece at α =. to idicate that x cotributes iformatio for the predictio of y. d. Without ruig a formal test, we ca compare the two models. The value of s for the model y= βx+ ε is while the value of s for the model y= β+ βx+ ε is Sice the value of s is much smaller for the secod model, it appears that the secod model should be used. For a formal test, refer to part (d) of Exercise 3.6. : β = : β a Test statistic: ˆ β t = = = x 48.6 s SSxx Rejectio regio: α =., df = = 8, t.5 = Reject if t < or t > Coclusio: Reject. There is sufficiet evidece at α =. to idicate that β should be icluded i the model a. The straight-lie model is y= β+ β x+ ε. Based o the theory, we would expect the metal level to decrease as the distace icreases. Thus, the slope should be egative. b. No. As the distace from the plat icreases, the arseic cocetratio of does ot seem to be clearly supported by the give scatterplot. 3.7 Usig MINITAB, the scattergram of the data is:.5 Scatterplot of Time vs Vehicles.4.3 Time Vehicles 4 6 Copyright Pearso Educatio, Ic. Publishig as Pretice all.

20 3- Simple Liear Regressio Some prelimiary calculatios: x = y =.37 xy = 3.86 x =4 y =.7 x y (.37) SSxy = xy = 3.86 =.9 5 ( x) SSxx = x = 4 = 8 5 x x = = = 8 5 y.37 y = = = ˆ SS xy.9 β = = = SSxx 8 ˆ = y ˆ βx = () 8 = β The least squares lie is yˆ = x. I calculatig we fid that t = 7.43, so we ca reject : β =. r.8; s.4 x = 8.85, ( ) = = 95% PI for y whe ( ) r % of sample variatio i ESLR score ca be explaied by x (SG, SR, or ER score) i liear model as give below: a..% of the liear relatioship betwee the ESLR scores were explaied by the Spaish grammar scores. b. 9.9% of the liear relatioship betwee the ESLR scores were explaied by the Spaish readig scores. c. 7.8% of the liear relatioship betwee the ESLR scores were explaied by the Eglish grammar scores. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

21 Chapter a. A scattergram of the data is: From the graph, it appears that as est box tit occupacy icreases the umber of flycatchers killed also icreases. b. ˆ β = 3.5. Sice x = is ot i the observed rage, ˆ β has o iterpretatio other tha the y-itercept. ˆ β =.8. For each additioal est box tit occupacy, the mea umber of flycatchers killed is estimated to icrease by.8. yˆ = x; t= 4., Reject r s : β = ; =.57; = a. b. Some prelimiary calculatios are: x = 53 x = y = 59. y = 57. xy =736.7 ( x) ( 53) SSxx = x = = x y (53)(59.) SSxy = xy = = Copyright Pearso Educatio, Ic. Publishig as Pretice all.

22 3- Simple Liear Regressio ( y) 59. SSyy = y = 57. = x 53 x = = = y 59. y = = = ˆ SSxy SSxx β = = = ˆ β ˆ = y βx = 78.5 The least squares lie is yˆ = x. c. The least squares lie is plotted o the graph above. d. SSE = SS ˆ yy β SS xy = ( )( ) = SSE s =.6396 = 7 = s =.6396 =.65 To determie if temperature is a useful liear predictor of chage i free eergy, we test: : β= a : β ˆ β Test statistic:.39 t = = =.3 s / SSxx.65 / Rejectio regio: α =., = 7 = 5, t.5 = 4.3 Reject if t < 4.3 or t > 4.3 Coclusio: Do ot reject. There is isufficiet evidece at α =. to idicate that temperature is a useful liear predictor of chage i free eergy. e. It looks like observatio #5 is a outlier. f. x = 75 x = xy = y = 5.7 y = ( x)( y) (75)(5.7) SSxy = xy = = ( x) (75) SSxx = x = = Copyright Pearso Educatio, Ic. Publishig as Pretice all.

23 Chapter x 75 x = = = 9 6 y 5.7 y = = = ˆ SSxy 36. β = = = SS 8.5 xx ˆ β = y ˆ β x = 8.45 ( ) 9 = ( ) The ew least squares lie is yˆ = x. ( y) (5.7) SSyy = y = = SSE=SS - ˆ β SS = ( )( 36.) = yy xy SSE s =.696 = 6 = s =.696 =.68 To determie if temperature is a useful predictor of chage i free eergy, we test: : β = : β a ˆ β Test statistic:.4497 t = = = 5.35 s / SS.68 / 8.5 xx Rejectio regio: α =., df = = 4, t.5 = 4.64 Reject if t < 4.64 or t > 4.64 Coclusio: Reject. Yes, there is sufficiet evidece to idicate temperature is a useful predictor of chage i free eergy at α = a. A straight lie model is y= β+ β x+ ε. b. The researcher hypothesized that therapists with more years of formal dace traiig will report a higher perceived success rate i cotherapy relatioships. This idicates that β >. c..6. r = Because this value is fairly close to, there is a weak egative liear relatioship betwee years of formal traiig ad reported success rate. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

24 3-4 Simple Liear Regressio d. To determie if there is a positive liear relatioship betwee years of formal traiig ad reported success rate, we test: : β = : > a β The test statistic is r.6 t = = = 3. ( r ) / ( ) (.6 ) / (36 ) ( ) The rejectio regio requires α =.5 i the upper tail of the t distributio with df = =36 = 34. From Table, Appedix D, t The rejectio regio is t>.645. Sice the observed value of the test statistic does ot fall i the rejectio regio ( t = 3. <.645 ), so, is ot rejected. There is isufficiet evidece to idicate that there is a positive liear relatioship betwee years of formal traiig ad perceived success rates at α = a. : ρ = : ρ > a The test statistic is r.6 3 t = = = 3.5 r.6 The rejectio regio requires α =. i the upper tail of the t distributio with df = = 3 = 8. From Table i Appedix D, t The rejectio regio is t >.358. Sice the observed value of the test statistic falls i the rejectio regio ( t = 3.5 >.358 ), is rejected. We agree that mother ad daughter loeliess scores were positively correlated at α =.. b. Usig the rejectio regio from (a), the rejectio regio is t >.358 whe testig for positive correlatio ad t <.358 whe testig for egative correlatio. We will begi by testig the ext strogest correlatio, r =.. The test statistic is: r. 3 t = = =.43 r (.) Sice -.43 <.358, there is sufficiet evidece to idicate mother umber of frieds ad daughter loeliess scores are egatively correlated at α =.. We ext test r =.9. The test statistic is: r.9 3 t = = =.9 r.9 Sice.9 <.358, there is isufficiet evidece to reject :. ρ = All other correlatio values will lead to the same coclusio at.. α = Copyright Pearso Educatio, Ic. Publishig as Pretice all.

25 Chapter c. Both a mother's loeliess ad a daughter's loeliess may be caused by other factors such as a abset husbad/father, livig locatios, etc. d. I part (b), we were testig oly for liear correlatios. It may be that some of the variables are correlated, but ot liearly correlated a. = 5 SSxx = ˆ β = x = SSxy = ˆ β = 4.67 y =694 SSyy = SSE = x =53.43 x = s = y = y = s = xy = The least squares lie is yˆ = x. Sice we wat to determie if there is a iverse liear relatioship betwee raise ad ratig, we test: β : = a : β> ˆ β Test statistic: 783 t = = =.98 s / SSxx / Rejectio regio: α =.5, df = = 3, t.5 =.77 Reject if t <.77. Coclusio: No, do ot reject at α =.5. There is isufficiet evidece of a iverse liear relatioship betwee raise ad ratig. b. = 4 SSxx = ˆ β = x = 4.9 SSxy = ˆ β = y =55 SSyy = SSE = x =34.53 x = 3.74 s = y = y = s = xy = The least squares lie is yˆ = x. Copyright Pearso Educatio, Ic. Publishig as Pretice all.

26 3-6 Simple Liear Regressio Sice we wat to determie if there is a iverse liear relatioship betwee raise ad ratig, we test: β : = a : β< ˆ β Test statistic: 3887 t = = =.9 s / SSxx / Rejectio regio: α =.5, df = =, t.5 =.78 Reject if t <.78. Coclusio: Reject at α =.5. Yes, there is sufficiet evidece of a iverse liear relatioship betwee raise ad ratig. c. Sice we oly kow the relatioship betwee raise, y, ad ratig, x, withi the observed rage of values of ratig, it would be very risky to use this least squares lie for the data i the table. The observed rage of values for ratig, x, is.5 to 4.. We should ot predict outside the observed rage of (.5 4.). d. As stated i the problem, typically oly those with axes to grid respod to surveys. Thus, the reported average ratigs may be quite differet tha the average ratigs of the faculty as a whole. With such a low respose rate, the chace that the sample is represetative of the etire populatio is probably quite small. The results should be viewed with cautio. e. Because of the small respose rate ad the probable biased resposes, the claim of the UFF should be viewed with skepticism. Copyright Pearso Educatio, Ic. Publishig as Pretice all.