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S  0d  ^ `   >*  0f       * Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. BWDg C H  0޽h ? 3380___PPT10.p3q Default Design w0$(  $ $ 0 `0G  w3Chapter 12 Simple Linear Regression and Correlation(4(2 0)<H $ 0޽h ? 3380___PPT10.^ }@((  ( ( 6 8c"`  i'12.1 The Simple Linear Regression Model&((20#<H ( 0޽h ? 3380___PPT10. Pۘ qiP ,(  , , 0Ⱥ  GLinear Relationship 2(3 , 0<ͺ (  kThe simplest deterministic mathematical relationship between two variables x and y is a linear relationshipLl 2K$$$$$` , c $A  ?? Y    , 0xӺ 0   GThe set of pairs (x,y) for which determines a straight line. LH 2$$$$3$`  , c $A ??`   H , 0޽h ? 3380___PPT10.! {`0(  0 0 0ߺ P  ? Terminology 2 (4 0 0 P0 The variable whose value is fixed by the experimenter, denoted x, is the independent (predictor, explanatory) variable. For a fixed x, the second variable will be a random variable Y with observed value y, referred to as the dependent (response) variable. 2?$$ $-$$$/$$$$$$$H 0 0޽h ? 3380___PPT10.@,L p 4L(  4 4 00  Z V"The Simple Linear Regression Model# 2#( 4 0  KThere exists parameters 2$- 4 0 p  esuch that for any fixed value of x, the dependent variable is related to x through the model equationLf 2!$$'$$$` 4 c $A ?? "i  ` 4 c $A ??P @   4 0  0 n: is a random variable (called the random deviation) with; 2;$`  4 c $A ??0 4  `  4 c $A ??  H 4 0޽h ? 3380___PPT10.Cb     C (  ~  s *WQ  `}  Q  RB  s *DjJ RB  s *DjJ @ `2  s *"` P RB   s *D8c ^B  6DjJ   0Q  P7  B 2   6 Q jJ"` l  LTrue regression line 2 LB  c $D   Xr  0    0-Q ppM  (x1,y1)h 2         `  c $A ?? f  RB  s *D`    0&Q P p Vx18 2     `2  s *"` @p `2  s *"` p `2  s *"` P `2  s *"`P `2  s *"` 0 `2  s *"`@ `  0 TA ? ?    Q H  0޽h ? 33___PPT10i.LgU+D=' = @B +m  <m(  < < 0 @ DDistribution of  2(` < c $A !??@ Q !x < <A "??;  "XB < 0DV @  < 0  ` M  50 2 RB  < s *DV  RB  < s *DV 0 0  `  < c $A #??   #`  < c $A &??    &  < 0 P p X$Normal, mean = 0, standard deviation% 2% ` < c $A '?? 'H < 0޽h ? 3380___PPT10.    @(  @ @ 0( * +Distribution of Y for Different Values of xL, 2(((((x @ <A ,??  ,.  @  AP ` .  @  A + RB  @ s *DV p XB @ 0DV@@XB @ 0DV@@XB @ 0DjJ@XB @ 0DjJ0 0 @XB @ 0DjJP XB @@ 0DjJ  XB @@ 0DjJ ` XB @@ 0DjJ  @ 0* p}  x1 x2 x3 2             f @ s *A 0? ? D  0RB @ s *D@  f @ s *A 3? ? `0  3f @ s *A 6? ?pY8  6f @ s *A 9? ?``?s 9H @ 0޽h ? 3380___PPT10.a ~vD(  D D 6l9 8c"`  b 12.2 Estimating Model Parameters&!(20<H D 0޽h ? 3380___PPT10. Pۘ   =(  ~  s *!Q   Q    0Q `N JThe vertical deviation of the point (xi,yi) from the line y = b0 + b1x isK 2%$$$$$$$$$$$$$,%!   0<$Q pP  qThe sum of squared vertical deviations from the points to the line is:r 2r$`  c $A ??  h     B0 $ 8 yi  (b0 + b1xi)$$$$$$$$$$$$`   c $A [?? p [H  0޽h ? 33___PPT10i.MC+D=' = @B +=  TL H(  H~ H s *<    H0  `A ^? ?"`  p  ^  H0  `A `? ?"`  `  H 0D>0 <The least-squares (regression) line for the data is given by>= 2$$$ H 0   ;and 2`  H c $A \??0 9 \  H 04 9where 2$H H 0޽h ? 33___PPT10i.MC+D=' = @B +2 y$%(    0B ` 5Ex. Find the equation of least-squares for the data <6 2 1 `  c $A ??@@    0\H  0 0 M ,$  0 O= 2.5. 2    0A i?>?"` ` 8 i$D  0 0  `A e? ?"`P  + H e$D  0   0  `A g? ?"` PH g$D  0  v   # #""PPPP   <\T ? @  K9 @`   <8\ ? @  L21 @`   <c ?P@  K7 @`   <k ?@P  K3 @`   <t ? @ K4 @`  <o ?  @ K6 @`  <} ?P @ K3 @`  <P ?P@ K2 @`  < ? @ K1 @`  <ܛ ? @  K2 @`  <0 ?P@  K2 @`  < ?@P K1 @`  <x ? @ dx20  @`  < ?  @ hxy @`  < ?P @ My @`  <8 ?P@ Mx @``B  0o ?ZB  s *1 ?@@ZB  s *1 ?ZB  s *1 ?@@`B  0o ?  `B  0o ? ZB  s *1 ?PP ZB   s *1 ?  ZB ! s *1 ?  `B " 0o ?  # 0  ,$  0 d*Sum: 6 12 29 14+0 2+ $ B,  },$  0 D=  1 0 H  0޽h ? 33/'___PPT10.MPQ+D3' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(left)*<3<*#D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*++0+ ++0+# ++0+$ + { L(  L L 0  ` OFitted Values and Residuals 2( L 0P  P  pThe fitted (predicted) values are obtained by substituting into the equation of the estimated regression line: The residuals are the vertical deviations T 2$$$ $%$ L 0L  0 t  Lfrom the estimated line. 2$` L c $A j??p0? j` L c $A m?? V m`  L c $A v??n  v`  L c $A y?? P z  yH L 0޽h ? 3380___PPT10. # T(  T T 0 Pp  HError Sum of Squares 2( T 0  {)The error sum of squares, denoted SSE, is6* 2$$$` T c $A! ??=l   T 0    Pand the estimate of is 2$` T c $A" ?? N `  T c $A$ ??  p  H T 0޽h ? 3380___PPT10.+  |t\ (  \ \ 0D 0: IComputational Formula 2( \ 0$ Ppt ['A computational formula for the SSE, is( 2($` \ c $A# ??cv  H \ 0޽h ? 3380___PPT10.++ `+(  ` ` 0 - pj HTotal Sum of Squares 2( ` 01 Pp {)The total sum of squares, denoted SST, is6* 2$$$` ` c $A% ??p H  H ` 0޽h ? 3380___PPT10.+  X(  X X 0:  ` PCoefficient of Determination 2( X 0=  ~ <The coefficient of determination, denoted by r2, is given by^= 2$$ $$ $$` X c $A' ??  & X 0E P P0 xIt is interpreted as the proportion of observed y variation that can be explained by the simple linear regression model.2y 20$$H$H X 0޽h ? 3380___PPT10.MW+ t+(  t t 0P P0  MRegression Sum of Squares 2( t 0|T   T SSR = SST  SSE  2$ t 0W `  lRegression sum of squares is interpreted as the amount of variation that is explained by the model. We have2m 2I$$"$` t c $A+ ??p  f H t 0޽h ? 3380___PPT10. $^  d(  d d 68b 8c"`@  Q12.3 Inferences About the Slope Parameter &R(20M<` d c $A( ?? p{P  H d 0޽h ? 3380___PPT10. Pۘ 0 x(  x` x c $A- ??@  x x <A* ?? x   x 0l P` E1. The mean of  2$x x <A, ?? 0 V % x 0q PH@8___PPT9 u) The variance and standard deviation are* 2*$8X  x 0y  KH@8___PPT9 n" has a normal distribution. # 2#$8Xx  x <A* ?? P` `  x c $AD <?? 0  <H x 0޽h ? 3380___PPT10.0 @p(  p p 0l @j Z T Variable4 2( ((` p c $A) ??JX   p 0č P ZThe assumptions of the simple linear regression model imply that the standardized variable[ 2[$0 p 0\   Fhas a t distribution with n  2 df.L$ 2$$$$ $ H p 0޽h ? 3380___PPT10.%~i P|i(  | | 0H 0 GConfidence Interval 2(` | c $A/ ??p\7  | 0 d Rof the true regression line is 2$` | c $A0 ??`  H | 0޽h ? 3380___PPT10.XO ` (    0X @ JNull hypothesis:0 2$`  c $A: ??    0    OTest statistic value:0 2$`  c $A.  ??` S     6  0y ? "0 2,  BT  z `  F 0,   0l `z QHypothesis-Testing Procedures 2(H  0޽h ? ̙33 ;3p(  `  c $A1  ??`L    0Ŀ 0@ PAlternative Hypothesis0 2$  0$ 0 p f,Rejection Region for Approx. Level Test-0 2-$`  c $A5 ??0 `  c $A2  ??`# g  `   c $A; ??@ H    0D P   <or0 2   0 `z QHypothesis-Testing Procedures 2(`  c $A6 ??  `  c $A7 ??   `  c $A< ?? '  `  c $A3 ??P !X  p  0P   A P-value based on n  2 df can be calculated as in Chap 8 and 9. LD 2$$$$/$(H  0޽h ? ̙33\ \(    0P  @ FHypothesis-Testing 2(  0 pD w%The model utility test is the test of6& 2$$$`  c $A4 ??pc   0 Pp>  g3in which case the test statistic value is the ratio4 24$`  c $A8 !??  !H  0޽h ? 3380___PPT10.F @8$$(    0  : E ANOVA Table 0 2 (v pp  # #"" 0  < ? p  T  @`  <  ?t p  T  @`  < ? t p  MSST @`  < ?  p  ` n  1$ @`  < ? p  OTotal @`   < ?p  T  @`   <( ?t p  T  @`   <* ? pt  MSSE @`   <2 ?p  ` n  2$ @`   <4 ?p  OError @`  < I ?p T  @`  <Q ?t p MSSR @`  <Y ? t p MSSR @`  <a ? p X1 $ @`  <\ ?p T Regression   @`  <8l ?p Mf @`  <Pz ?t p U Mean Square   @`  <{ ? pt  XSum of squares @`  <h ?p  fdf @`  <ؒ ?p ]Source of Variation @``B  0o ?ppZB  s *1 ?ZB  s *1 ?ppZB  s *1 ?  `B  0o ?p p `B  0o ?pp ZB  s *1 ?pp ZB  s *1 ? p p ZB   s *1 ?t pt p ZB ! s *1 ?pp `B " 0o ?pp ` # c $A= "??   "` $ c $A9 %??Q  %H  0޽h ? ̙3380___PPT10.( G?h(  h7 h 6Z 8c"`@@w g12.4 Inferences Concerning and the Prediction of Future Y Values @h(20X<< <` h c $A& ??  H h 0޽h ? 3380___PPT10. Pۘ 4, (  `  c $AA 6?? `B 6  0\ ` L1. The mean of is  2$x  <A@ 3??0 3  0Ĭ 0 P H@8___PPT9 n" Variance and standard deviation:# 2#$8X`   c $A? /??  /  0ܵ   ;is some 2$  0и `  _fixed value of x.2 2$$$x   <A> 4?? )  4x  <A> =??P y @ =H  0޽h ? 3380___PPT10.0" "(  x  <A> >??   >  0  @4  w!3. has a normal distibution." 2"$ x  <A> ???p  ?`  c $AB @??pd @  0x `P`  C2. (continued) 2$H  0޽h ? 3380___PPT10.: :(    0 @j Z T Variable4 2( ((`  c $AC B??p  B0  0T  D  Fhas a t distribution with n  2 df.L$ 2$$$$ $   0 p  @ The variable 2 $H  0޽h ? 3380___PPT10.%~$ $(    0 0 GConfidence Interval 2(`  c $AF F??O7 F  0  #expected value of Y when x = x*, isf$ 2$$$$$$$`  c $AE H??0c  H`  c $AH ?? 4   H  0޽h ? 3380___PPT10.XC C(    0T `` GPrediction Interval 2(\  0 `0  A future value of Y is not a parameter but instead a random variable; its interval of plausible values is referred to as a prediction interval.P 2$$i$$$H  0޽h ? 3380___PPT10.@CJ: [S(    0\ `: GPrediction Interval 2(  0 p` to be made when x = x*, isL 2$$$$$`  c $AG ??PP  `  c $AJ ??@ p_    0p  P 4A PI for a future Y observation25 2'$$ $`  c $AI ??P  H  0޽h ? 3380___PPT10.X ogl(  l l 6\ 8c"`P  S12.5 Correlation &(20 <H l 0޽h ? 3380___PPT10. Pۘ< SK (    0$ @ YSample Correlation Coefficient  0 2 ,h  0+ p The sample correlation coefficient, denoted r, of n pairs (x1,y1),& ,(xn,yn) is S0 2$$ $$$$$$ $$$ $$$ $$$ $ $,E z  0AK ? ?"`F  H  0޽h ? ̙33___PPT10i.P֏5+D=' = @B + 0 k(      0= Ppp QEx. Find the correlation coefficient for the least-squares line from the points <R 2$M$ `   c $A  ?? p  z   0A  ? ?"`0`)   z   0A  ? ?"`P P     6XD 33>"`pP  B= 0.9449 0 2 $H  0޽h ? ̙33___PPT10i.*Z+D=' = @B + @(    0L   ]Properties of r2 2(((  0tR `p gImportant properties of r2 2$$$R  0W pP D<4___PPT9  The value of r does not depend on which of the two variables under study is labeled x and which is labeled y. The value of r is independent of the units in which x and y are measured.  2$$F$$$$$$&$$$$$8X`  c $A[ l?? P lH  0޽h ? 3380___PPT10.9M PM(    0g z ]Properties of r2 2(((P  0z 0 4. r = 1 iff all (xi, yi) pairs lie on straight line with positive slope, and r =  1 iff all (xi, yi) pairs lie on a straight line with negative slope. 5. The square of the sample correlation coefficient gives the value of the coefficient of determination that would result from fitting the simple linear regression model. F 2$$$$ $$ $6$$$$ $$ $$P  = 8XH  0޽h ? 3380___PPT10.PW> `((7(    0    iDifferent Values of r80 2(((RB  s *DRB  s *DPPPRB  s *D`P P PRB  s *D   RB  s *D  RB  s *D  RB  s *D` P P RB  s *D p R2  s *0`R2  s *`pR2  s *R2  s * @R2  s *P@R2  s *P R2  s * @R2  s *`@R2  s *R2  s *@PR2  s *0R2  s * 0R2  s *  R2  s * p R2  s *p ` R2  s *   R2  s * 0 R2  s * 0 R2  s * @ R2  s * @ R2  s * p R2  s *p p R2 ! s * 00 R2 " s *0 ` R2 # s *0 P R2 $ s *0 p  % 0T   ^r near 1: 0 2    & 0 P  _ r near -1: 0 2    ' 0 P p  or near 0, no relationship:0 2    ( 0  `  v r near 0, nonlinear relationship:!0 2   H  0޽h ? ̙334 pt(    0   `&The Population Correlation Coefficient'0 2'(`  c $AL ??@    0p @  ?where0 2$`  c $AM ??@} `  c $AN ??     0` Lp0 D0 2$   0   5depending on whether (X,Y) is discrete or continuous.860 2$$$H  0޽h ? ̙33 C(    0$ p C Estimator 0 2 (`  c $AO ??z  H  0޽h ? ̙33  4(    0P 0z D Assumption 0 2 (  0  =The joint probability distribution of (X,Y ) is specified by8>0 2($$$`  c $AP ??0 E  x  <AQ ??  `  c $AR ??    `  c $AS ?? @ 87    0D  P0 E is called the bivariate normal probability distribution.<F0 2$)$$ "x   <AT ?? - H  0޽h ? ̙33 :2 (    0 P0 Z&Testing for the Absence of Correlation' 2'(`  c $A_ }??  }`  c $A` ~??P  ~  0 p h4When is true, the test statistic:5 25$0  0@ 0   FHas a t distribution with n  2 df.L$ 2$$$$ $ H  0޽h ? 3380___PPT10.Rw :2(  `  c $AZ v??`L v  0 0@ PAlternative Hypothesis0 2$  0$ 0 p f,Rejection Region for Approx. Level Test-0 2-$`  c $A5 n??0 n`  c $A2 o??`# g o`  c $A; p??@ H  p  0 P   <or0 2   0T @z FHypothesis-Testing 2(`   c $A< s?? '  s`   c $A3 t??P !X  tz  0    A P-value based on n  2 df can be calculated as described previously. LI 2$$$$4$-`  c $A\ z??@,  z`  c $A] |??P <  |H  0޽h ? ̙333  3(    0   OOther Inferences Concerning 2(`  c $A^ ??B 2  0| P~ When (X1, Y1),& ,(Xn, Yn) is a sample from a bivariate normal distribution, the rvR 2$$ $$$ $$$ $$$ $;$P `  c $Ab ??    0&  P@~  r>has approximately a normal distribution with mean and variance? 2?$`   c $Aa ??  H  0޽h ? 3380___PPT10.9N ~(    0/ P RThe test statistic for testing 2$`  c $Ac ??    0D4 P@>  PAlternative Hypothesis0 2$`  c $Ad ??P  <  `   c $Ae ??0  `   c $Af ??    0p8   @  ^$Rejection Region for Level Test%0 2%$`   c $A5 ??   `   c $Ag ??P Z;  `  c $Ai ??   `  c $Aj ??  `  c $Ah ?? h   0=  PD 6or 2$`  c $Ak ??  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