Basic Business Statistics, 10/e

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Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable The meanng of the regresson coeffcents b 0 and b 1 How to evaluate the assumptons of regresson analyss and know what to do f the assumptons are volated To make nferences about the slope and correlaton coeffcent To estmate mean values and predct ndvdual values Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-006 Prentce Hall, Inc.

Chapter 13 13- Correlaton vs. Regresson A scatter plot can be used to show the relatonshp between two varables Correlaton analyss s used to measure the strength of the assocaton (lnear relatonshp) between two varables Correlaton s only concerned wth strength of the relatonshp No causal effect s mpled wth correlaton Scatter plots were frst presented n Ch. Correlaton was frst presented n Ch. 3 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-3 Introducton to Regresson Analyss Regresson analyss s used to: Predct the value of a dependent varable based on the value of at least one ndependent varable Explan the mpact of changes n an ndependent varable on the dependent varable Dependent varable: the varable we wsh to predct or explan Independent varable: the varable used to predct or explan the dependent varable Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-4 006 Prentce Hall, Inc.

Chapter 13 13-3 Smple Lnear Regresson Model Only one ndependent varable, X Relatonshp between X and s descrbed by a lnear functon Changes n are assumed to be related to changes n X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-5 Types of Relatonshps Lnear relatonshps Curvlnear relatonshps X X X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-6 006 Prentce Hall, Inc.

Chapter 13 13-4 Types of Relatonshps (contnued) Strong relatonshps Weak relatonshps X X X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-7 Types of Relatonshps (contnued) No relatonshp X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-8 006 Prentce Hall, Inc.

Chapter 13 13-5 Correlaton Coeffcent The populaton correlaton coeffcent ρ (rho) measures the strength of the assocaton between the varables The sample correlaton coeffcent r s an estmate of ρ and s used to measure the strength of the lnear relatonshp n the sample observatons (contnued) Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-9 Features of ρ and r Unt free Range between -1 and 1 The closer to -1, the stronger the negatve lnear relatonshp The closer to 1, the stronger the postve lnear relatonshp The closer to 0, the weaker the lnear relatonshp Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-10 006 Prentce Hall, Inc.

Chapter 13 13-6 Examples of Approxmate r Values y y y x r = -1 r = -.6 r = 0 y y x x x r = +.3 r = +1 Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-11 x Calculatng the Correlaton Coeffcent Sample correlaton coeffcent: r r or the algebrac equvalent: [n( [ (x (x x)(y y) x) ][ where: r = Sample correlaton coeffcent n = Sample sze x = Value of the ndependent Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-1 (y y) ] n xy xy x ) ( x) ][n( y ) ( y) ] 006 Prentce Hall, Inc.

Chapter 13 13-7 Calculaton Example Tree Heght Trunk Dameter y x xy y x 35 8 80 15 64 49 9 441 401 81 7 7 189 79 49 33 6 198 1089 36 60 13 780 3600 169 1 7 147 441 49 45 11 495 05 11 51 1 61 601 144 =31 =73 =314 =14111 =713 Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-13 Tree Heg ht, y 70 Calculaton Example r (contnued) n xy x y [n( x ) ( x) ][n( y ) ( y) ] 60 50 40 8(314) (73)(31) [8(713) (73) ][8(14111) (31) ] 30 0 0.886 10 0 0 4 6 8 10 1 14 Trunk Dameter, x r = 0.886 relatvely strong postve lnear assocaton between x and y Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-14 006 Prentce Hall, Inc.

Chapter 13 13-8 Excel Output Excel Correlaton Output Tools / data analyss / correlaton Tree Heght Trunk Dameter Tree Heght 1 Trunk Dameter 0.88631 1 Correlaton between Tree Heght and Trunk Dameter Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-15 Sgnfcance Test for Correlaton Hypotheses H 0 : ρ = 0 H A : ρ 0 (no correlaton) (correlaton exsts) Test statstc t r (wth n degrees of freedom) 1 r n Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-16 006 Prentce Hall, Inc.

Chapter 13 13-9 Example: Produce Stores Is there evdence of a lnear relatonshp between tree heght and trunk dameter at the.05 level of sgnfcance? H 0 : ρ = 0 H 1 : ρ 0 (No correlaton) (correlaton exsts) =.05, df = 8 - = 6 t r 1 r n.886 1.886 8 4.68 Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-17 Example: Test Soluton r t 1 r n d.f. = 8- = 6 /=.05.886 1.886 8 4.68 /=.05 Decson: Reject H 0 Concluson: There s evdence of a lnear relatonshp at the 5% level of sgnfcance Reject H 0 -t α/ Do not reject H 0 0 Reject H t 0 α/ -.4469.4469 4.68 Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-18 006 Prentce Hall, Inc.

Chapter 13 13-10 Introducton to Regresson Analyss Regresson analyss s used to: Predct the value of a dependent varable based on the value of at least one ndependent varable Explan the mpact of changes n an ndependent varable on the dependent varable Dependent varable: the varable we wsh to explan Independent varable: the varable used to explan the dependent varable Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-19 Smple Lnear Regresson Model Only one ndependent varable, x Relatonshp between x and y s descrbed by a lnear functon Changes n y are assumed to be caused by changes n x Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-0 006 Prentce Hall, Inc.

Chapter 13 13-11 Types of Regresson Models Postve Lnear Relatonshp Relatonshp NOT Lnear Negatve Lnear Relatonshp No Relatonshp Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-1 Smple Lnear Regresson Model Dependent Varable Populaton ntercept β 0 Populaton Slope Coeffcent β X 1 Independent Varable ε Random Error term Lnear component Random Error component Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-006 Prentce Hall, Inc.

Chapter 13 13-1 Observed Value of for X Smple Lnear Regresson Model β 0 β X 1 ε (contnued) Predcted Value of for X ε Random Error for ths X value Slope = β 1 Intercept = β 0 X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-3 Smple Lnear Regresson Equaton (Predcton Lne) The smple lnear regresson equaton provdes an estmate of the populaton regresson lne Estmated (or predcted) value for observaton Estmate of the regresson ntercept Estmate of the regresson slope Ŷ b 0 b 1 X Value of X for observaton Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-4 006 Prentce Hall, Inc.

Chapter 13 13-13 Least Squares Crteron b 0 and b 1 are obtaned by fndng the values of b 0 and b 1 that mnmze the sum of the squared resduals e (y ŷ) (y (b 0 b 1 x)) Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-5 The Least Squares Equaton The formulas for b 1 and b 0 are: b 1 ( x x)( y y) ( x x) algebrac equvalent: x y xy b n 1 ( x) x n and b0 y b1 x Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 13-6 006 Prentce Hall, Inc.

Chapter 13 13-14 The Least Squares Method mn b 0 and b 1 are obtaned by fndng the values of that mnmze the sum of the squared dfferences between and Ŷ : ( Ŷ ) mn ( (b0 b1x )) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-7 Fndng the Least Squares Equaton The coeffcents b 0 and b 1, and other regresson results n ths chapter, wll be found usng Excel or Mntab Formulas are shown n the text for those who are nterested Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-8 006 Prentce Hall, Inc.

Chapter 13 13-15 Interpretaton of the Slope and the Intercept b 0 s the estmated average value of when the value of X s zero b 1 s the estmated change n the average value of as a result of a one-unt change n X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-9 Smple Lnear Regresson Example A real estate agent wshes to examne the relatonshp between the sellng prce of a home and ts sze (measured n square feet) A random sample of 10 houses s selected Dependent varable () = house prce n $1000s Independent varable (X) = square feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-30 006 Prentce Hall, Inc.

House Prce ($1000s) Chapter 13 13-16 Smple Lnear Regresson Example: Data House Prce n $1000s () Square Feet (X) 45 1400 31 1600 79 1700 308 1875 199 1100 19 1550 405 350 34 450 319 145 55 1700 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-31 Smple Lnear Regresson Example: Scatter Plot House prce model: Scatter Plot 450 400 350 300 50 00 150 100 50 0 0 500 1000 1500 000 500 3000 Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-3 006 Prentce Hall, Inc.

Chapter 13 13-17 Smple Lnear Regresson Example: Usng Excel Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-33 Smple Lnear Regresson Example: Excel Output Regresson Statstcs Multple R 0.7611 R Square 0.5808 Adjusted R Square 0.584 Standard Error 41.3303 Observatons 10 The regresson equaton s: houseprce 98.4833 0.10977(squarefeet) ANOVA df SS MS F Sgnfcance F Regresson 1 18934.9348 18934.9348 11.0848 0.01039 Resdual 8 13665.565 1708.1957 Total 9 3600.5000 Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept 98.4833 58.03348 1.6996 0.189-35.5770 3.07386 Square Feet 0.10977 0.0397 3.3938 0.01039 0.03374 0.18580 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-34 006 Prentce Hall, Inc.

House Prce ($1000s) Chapter 13 13-18 Smple Lnear Regresson Example: Mntab Output The regresson equaton s Prce = 98. + 0.110 Square Feet Predctor Coef SE Coef T P Constant 98.5 58.03 1.69 0.19 Square Feet 0.10977 0.0397 3.33 0.010 The regresson equaton s: house prce = 98.4833 + 0.10977 (square feet) S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 5.8% Analyss of Varance Source DF SS MS F P Regresson 1 18935 18935 11.08 0.010 Resdual Error 8 13666 1708 Total 9 3600 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-35 Smple Lnear Regresson Example: Graphcal Representaton House prce model: Scatter Plot and Predcton Lne Intercept = 98.48 450 400 350 300 50 00 150 100 50 0 0 500 1000 1500 000 500 3000 Square Feet Slope = 0.10977 houseprce 98.4833 0.10977(squarefeet) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-36 006 Prentce Hall, Inc.

Chapter 13 13-19 Smple Lnear Regresson Example: Interpretaton of b o houseprce 98.4833 0.10977(squarefeet) b 0 s the estmated average value of when the value of X s zero (f X = 0 s n the range of observed X values) Because a house cannot have a square footage of 0, b 0 has no practcal applcaton Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-37 Smple Lnear Regresson Example: Interpretng b 1 houseprce 98.4833 0.10977(squarefeet) b 1 estmates the change n the average value of as a result of a one-unt ncrease n X Here, b 1 = 0.10977 tells us that the mean value of a house ncreases by.10977($1000) = $109.77, on average, for each addtonal one square foot of sze Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-38 006 Prentce Hall, Inc.

House Prce ($1000s) Chapter 13 13-0 Smple Lnear Regresson Example: Makng Predctons Predct the prce for a house wth 000 square feet: houseprce 98.5 0.1098(sq.ft.) 98.5 0.1098(000) 317.85 The predcted prce for a house wth 000 square feet s 317.85($1,000s) = $317,850 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-39 Smple Lnear Regresson Example: Makng Predctons When usng a regresson model for predcton, only predct wthn the relevant range of data Relevant range for nterpolaton 450 400 350 300 50 00 150 100 50 0 0 500 1000 1500 000 500 3000 Do not try to extrapolate beyond the range of observed X s Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-40 006 Prentce Hall, Inc.

Chapter 13 13-1 Measures of Varaton Total varaton s made up of two parts: SST SSR SSE Total Sum of Squares Regresson Sum of Squares Error Sum of Squares SST ( SSR (Ŷ SSE ( Ŷ ) ) ) where: = Mean value of the dependent varable = Observed value of the dependent varable ˆ = Predcted value of for the gven X value Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-41 Measures of Varaton (contnued) SST = total sum of squares (Total Varaton) Measures the varaton of the values around ther mean SSR = regresson sum of squares (Explaned Varaton) Varaton attrbutable to the relatonshp between X and SSE = error sum of squares (Unexplaned Varaton) Varaton n attrbutable to factors other than X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-4 006 Prentce Hall, Inc.

Chapter 13 13- SST = ( - ) Measures of Varaton SSE = ( - ) _ SSR = ( - ) (contnued) _ X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-43 Coeffcent of Determnaton, r The coeffcent of determnaton s the porton of the total varaton n the dependent varable that s explaned by varaton n the ndependent varable The coeffcent of determnaton s also called r-squared and s denoted as r r SSR regresson sum of squares SST total sum of squares note: 0 r 1 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-44 006 Prentce Hall, Inc.

Chapter 13 13-3 Examples of Approxmate r Values r = 1 r = 1 X Perfect lnear relatonshp between X and : 100% of the varaton n s explaned by varaton n X r = 1 X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-45 Examples of Approxmate r Values X 0 < r < 1 Weaker lnear relatonshps between X and : Some but not all of the varaton n s explaned by varaton n X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-46 006 Prentce Hall, Inc.

Chapter 13 13-4 Examples of Approxmate r Values r = 0 No lnear relatonshp between X and : r = 0 X The value of does not depend on X. (None of the varaton n s explaned by varaton n X) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-47 Smple Lnear Regresson Example: Coeffcent of Determnaton, r n Excel Regresson Statstcs Multple R 0.7611 R Square 0.5808 Adjusted R Square 0.584 Standard Error 41.3303 Observatons 10 r SSR SST 18934.9348 0.5808 3600.5000 58.08% of the varaton n house prces s explaned by varaton n square feet ANOVA df SS MS F Sgnfcance F Regresson 1 18934.9348 18934.9348 11.0848 0.01039 Resdual 8 13665.565 1708.1957 Total 9 3600.5000 Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept 98.4833 58.03348 1.6996 0.189-35.5770 3.07386 Square Feet 0.10977 0.0397 3.3938 0.01039 0.03374 0.18580 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-48 006 Prentce Hall, Inc.

Chapter 13 13-5 Smple Lnear Regresson Example: Coeffcent of Determnaton, r n Mntab The regresson equaton s Prce = 98. + 0.110 Square Feet Predctor Coef SE Coef T P Constant 98.5 58.03 1.69 0.19 Square Feet 0.10977 0.0397 3.33 0.010 S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 5.8% Analyss of Varance Source DF SS MS F P Regresson 1 18935 18935 11.08 0.010 Resdual Error 8 13666 1708 Total 9 3600 r SSR 18934.9348 0.5808 SST 3600.5000 58.08% of the varaton n house prces s explaned by varaton n square feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-49 Standard Error of Estmate The standard devaton of the varaton of observatons around the regresson lne s estmated by S Where X SSE n 1 SSE = error sum of squares n = sample sze n ( ˆ n ) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-50 006 Prentce Hall, Inc.

Chapter 13 13-6 Smple Lnear Regresson Example: Standard Error of Estmate n Excel Regresson Statstcs Multple R 0.7611 R Square 0.5808 Adjusted R Square 0.584 Standard Error 41.3303 Observatons 10 S X 41.3303 ANOVA df SS MS F Sgnfcance F Regresson 1 18934.9348 18934.9348 11.0848 0.01039 Resdual 8 13665.565 1708.1957 Total 9 3600.5000 Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept 98.4833 58.03348 1.6996 0.189-35.5770 3.07386 Square Feet 0.10977 0.0397 3.3938 0.01039 0.03374 0.18580 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-51 Smple Lnear Regresson Example: Standard Error of Estmate n Mntab The regresson equaton s Prce = 98. + 0.110 Square Feet Predctor Coef SE Coef T P Constant 98.5 58.03 1.69 0.19 Square Feet 0.10977 0.0397 3.33 0.010 S = 41.3303 R-Sq = 58.1% R-Sq(adj) = 5.8% S X 41.3303 Analyss of Varance Source DF SS MS F P Regresson 1 18935 18935 11.08 0.010 Resdual Error 8 13666 1708 Total 9 3600 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-5 006 Prentce Hall, Inc.

Chapter 13 13-7 Comparng Standard Errors S X s a measure of the varaton of observed values from the regresson lne smalls X X large S X X The magntude of S X should always be judged relatve to the sze of the values n the sample data.e., S X = $41.33K s moderately small relatve to house prces n the $00K - $400K range Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-53 Assumptons of Regresson L.I.N.E Lnearty The relatonshp between X and s lnear Independence of Errors Error values are statstcally ndependent Normalty of Error Error values are normally dstrbuted for any gven value of X Equal Varance (also called homoscedastcty) The probablty dstrbuton of the errors has constant varance Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-54 006 Prentce Hall, Inc.

resduals resduals Chapter 13 13-8 Resdual Analyss e The resdual for observaton, e, s the dfference between ts observed and predcted value Check the assumptons of regresson by examnng the resduals Examne for lnearty assumpton Evaluate ndependence assumpton Evaluate normal dstrbuton assumpton Examne for constant varance for all levels of X (homoscedastcty) Graphcal Analyss of Resduals Can plot resduals vs. X Ŷ Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-55 Resdual Analyss for Lnearty x x x x Not Lnear Lnear Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-56 006 Prentce Hall, Inc.

resduals resduals resduals Chapter 13 13-9 Resdual Analyss for Independence Not Independent Independent X X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-57 Checkng for Normalty Examne the Stem-and-Leaf Dsplay of the Resduals Examne the Boxplot of the Resduals Examne the Hstogram of the Resduals Construct a Normal Probablty Plot of the Resduals Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-58 006 Prentce Hall, Inc.

resduals resduals Chapter 13 13-30 Resdual Analyss for Normalty When usng a normal probablty plot, normal errors wll approxmately dsplay n a straght lne Percent 100 0-3 - -1 0 1 3 Resdual Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-59 Resdual Analyss for Equal Varance x x x x Non-constant varance Constant varance Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-60 006 Prentce Hall, Inc.

Resduals Chapter 13 13-31 Smple Lnear Regresson Example: Excel Resdual Output RESIDUAL OUTPUT Predcted House Prce Resduals 1 51.9316-6.9316 73.87671 38.139 3 84.85348-5.853484 4 304.0684 3.93716 5 18.9984-19.9984 6 68.3883-49.3883 7 356.051 48.79749 8 367.1799-43.1799 9 54.6674 64.3364 10 84.85348-9.85348 80 60 40 0-40 -60 House Prce Model Resdual Plot 0-0 0 1000 000 3000 Square Feet Does not appear to volate any regresson assumptons Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-61 Measurng Autocorrelaton: The Durbn-Watson Statstc Used when data are collected over tme to detect f autocorrelaton s present Autocorrelaton exsts f resduals n one tme perod are related to resduals n another perod Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-6 006 Prentce Hall, Inc.

Resduals Chapter 13 13-3 Autocorrelaton Autocorrelaton s correlaton of the errors (resduals) over tme Tme (t) Resdual Plot Here, resduals show a cyclc pattern, not random. Cyclcal patterns are a sgn of postve autocorrelaton 15 10 5 0-5 -10-15 0 4 6 8 Tme (t) Volates the regresson assumpton that resduals are random and ndependent Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-63 The Durbn-Watson Statstc The Durbn-Watson statstc s used to test for autocorrelaton H 0 : resduals are not correlated H 1 : postve autocorrelaton s present D n (e e n 1 e ) 1 The possble range s 0 D 4 D should be close to f H 0 s true D less than may sgnal postve autocorrelaton, D greater than may sgnal negatve autocorrelaton Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-64 006 Prentce Hall, Inc.

Sales Chapter 13 13-33 Testng for Postve Autocorrelaton H 0 : postve autocorrelaton does not exst H 1 : postve autocorrelaton s present Calculate the Durbn-Watson test statstc = D (The Durbn-Watson Statstc can be found usng Excel or Mntab) Fnd the values d L and d U from the Durbn-Watson table (for sample sze n and number of ndependent varables k) Decson rule: reject H 0 f D < d L Reject H 0 Inconclusve Do not reject H 0 0 d L d U Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-65 Testng for Postve Autocorrelaton (contnued) Suppose we have the followng tme seres data: 160 140 10 100 80 60 y = 30.65 + 4.7038x R = 0.8976 40 0 0 0 5 10 15 0 5 30 Tme Is there autocorrelaton? Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-66 006 Prentce Hall, Inc.

Sales Chapter 13 13-34 Example wth n = 5: Excel/PHStat output: Durbn-Watson Calculatons Sum of Squared Dfference of Resduals 396.18 Sum of Squared Resduals 379.98 Durbn-Watson Statstc 1.00494 Testng for Postve Autocorrelaton 160 140 10 100 80 60 40 0 y = 30.65 + 4.7038x R = 0.8976 0 0 5 10 15 0 5 30 Tme (contnued) D n (e e n 1 e ) 1 396.18 1.00494 379.98 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-67 Testng for Postve Autocorrelaton (contnued) Here, n = 5 and there s k = 1 one ndependent varable Usng the Durbn-Watson table, d L = 1.9 and d U = 1.45 D = 1.00494 < d L = 1.9, so reject H 0 and conclude that sgnfcant postve autocorrelaton exsts Decson: reject H 0 snce D = 1.00494 < d L Reject H 0 Inconclusve Do not reject H 0 d L =1.9 0 d U =1.45 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-68 006 Prentce Hall, Inc.

Chapter 13 13-35 Inferences About the Slope The standard error of the regresson slope coeffcent (b 1 ) s estmated by Sb 1 S X SSX S X (X X) where: S b1 = Estmate of the standard error of the slope S X SSE n = Standard error of the estmate Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-69 Inferences About the Slope: t Test t test for a populaton slope Is there a lnear relatonshp between X and? Null and alternatve hypotheses H 0 : β 1 = 0 (no lnear relatonshp) H 1 : β 1 0 (lnear relatonshp does exst) Test statstc t STAT b 1 S β b 1 d.f. n 1 where: b 1 = regresson slope coeffcent β 1 = hypotheszed slope S b1 = standard error of the slope Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-70 006 Prentce Hall, Inc.

Chapter 13 13-36 Inferences About the Slope: t Test Example House Prce n $1000s (y) Square Feet (x) 45 1400 31 1600 79 1700 308 1875 199 1100 19 1550 405 350 34 450 319 145 55 1700 Estmated Regresson Equaton: houseprce 98.5 0.1098(sq.ft.) The slope of ths model s 0.1098 Is there a relatonshp between the square footage of the house and ts sales prce? Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-71 Inferences About the Slope: t Test Example From Excel output: H 0 : β 1 = 0 H 1 : β 1 0 Coeffcents Standard Error t Stat P-value Intercept 98.4833 58.03348 1.6996 0.189 Square Feet 0.10977 0.0397 3.3938 0.01039 From Mntab output: b 1 S b1 Predctor Coef SE Coef T P Constant 98.5 58.03 1.69 0.19 Square Feet 0.10977 0.0397 3.33 0.010 b 1 S b 1 t STAT b β 1 1 S b 1 0. 10977 0 3. 3938 0. 0397 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-7 006 Prentce Hall, Inc.

Chapter 13 13-37 Inferences About the Slope: t Test Example Test Statstc: t STAT = 3.39 H 0 : β 1 = 0 H 1 : β 1 0 d.f. = 10- = 8 /=.05 /=.05 Reject H 0 Reject H 0 Do not reject H -t 0 α/ t 0 α/ -.3060.3060 3.39 Decson: Reject H 0 There s suffcent evdence that square footage affects house prce Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-73 Inferences About the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1 0 From Excel output: Coeffcents Standard Error t Stat P-value Intercept 98.4833 58.03348 1.6996 0.189 Square Feet 0.10977 0.0397 3.3938 0.01039 From Mntab output: Predctor Coef SE Coef T P Constant 98.5 58.03 1.69 0.19 Square Feet 0.10977 0.0397 3.33 0.010 Decson: Reject H 0, snce p-value < α There s suffcent evdence that square footage affects house prce. p-value Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-74 006 Prentce Hall, Inc.

Chapter 13 13-38 F Test for Sgnfcance F Test statstc: F STAT MSR MSE where MSR SSR k SSE MSE n k 1 where F STAT follows an F dstrbuton wth k numerator and (n k - 1) denomnator degrees of freedom (k = the number of ndependent varables n the regresson model) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-75 F-Test for Sgnfcance Excel Output Regresson Statstcs Multple R 0.7611 R Square 0.5808 Adjusted R Square 0.584 Standard Error 41.3303 Observatons 10 ANOVA F STAT df SS MS F Sgnfcance F Regresson 1 18934.9348 18934.9348 11.0848 0.01039 Resdual 8 13665.565 1708.1957 Total 9 3600.5000 MSR MSE Wth 1 and 8 degrees of freedom 18934.9348 11.0848 1708.1957 p-value for the F-Test Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-76 006 Prentce Hall, Inc.

Chapter 13 13-39 F-Test for Sgnfcance Mntab Output Analyss of Varance Source DF SS MS F P Regresson 1 18935 18935 11.08 0.010 Resdual Error 8 13666 1708 Total 9 3600 p-value for the F-Test Wth 1 and 8 degrees of freedom F STAT MSR MSE 18934.9348 11.0848 1708.1957 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-77 0 H 0 : β 1 = 0 H 1 : β 1 0 =.05 df 1 = 1 df = 8 Do not reject H 0 Crtcal Value: F = 5.3 F Test for Sgnfcance =.05 Reject H 0 F.05 = 5.3 F Test Statstc: MSR F STAT 11.08 MSE Decson: Reject H 0 at = 0.05 Concluson: (contnued) There s suffcent evdence that house sze affects sellng prce Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-78 006 Prentce Hall, Inc.

Chapter 13 13-40 Confdence Interval Estmate for the Slope Confdence Interval Estmate of the Slope: b 1 t α / S b 1 d.f. = n - Excel Prntout for House Prces: Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept 98.4833 58.03348 1.6996 0.189-35.5770 3.07386 Square Feet 0.10977 0.0397 3.3938 0.01039 0.03374 0.18580 At 95% level of confdence, the confdence nterval for the slope s (0.0337, 0.1858) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-79 Confdence Interval Estmate for the Slope (contnued) Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept 98.4833 58.03348 1.6996 0.189-35.5770 3.07386 Square Feet 0.10977 0.0397 3.3938 0.01039 0.03374 0.18580 Snce the unts of the house prce varable s $1000s, we are 95% confdent that the average mpact on sales prce s between $33.74 and $185.80 per square foot of house sze Ths 95% confdence nterval does not nclude 0. Concluson: There s a sgnfcant relatonshp between house prce and square feet at the.05 level of sgnfcance Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-80 006 Prentce Hall, Inc.

Chapter 13 13-41 t Test for a Correlaton Coeffcent Hypotheses H 0 : ρ = 0 (no correlaton between X and ) H 1 : ρ 0 (correlaton exsts) Test statstc r - ρ tstat 1 r n (wth n degrees of freedom) where r r f b 0 1 r r f b 0 1 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-81 t-test For A Correlaton Coeffcent Is there evdence of a lnear relatonshp between square feet and house prce at the.05 level of sgnfcance? H 0 : ρ = 0 H 1 : ρ 0 (No correlaton) (correlaton exsts) =.05, df = 10 - = 8 (contnued) t STAT r ρ 1 r n.76 0 1.76 10 3.39 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-8 006 Prentce Hall, Inc.

Chapter 13 13-4 t-test For A Correlaton Coeffcent (contnued) t STAT d.f. = 10- = 8 /=.05 r ρ 1 r n.76 0 1.76 10 3.39 /=.05 Decson: Reject H 0 Concluson: There s evdence of a lnear assocaton at the 5% level of sgnfcance Reject H 0 Reject H t 0 α/ Do not reject H -t 0 α/ 0 -.3060.3060 3.39 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-83 Estmatng Mean Values and Predctng Indvdual Values Goal: Form ntervals around to express uncertanty about the value of for a gven X Confdence Interval for the mean of, gven X = b 0 +b 1 X Predcton Interval for an ndvdual, gven X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-84 X X 006 Prentce Hall, Inc.

Chapter 13 13-43 Confdence Interval for the Average, Gven X Confdence nterval estmate for the mean value of gven a partcular X Confdencentervalfor μ ˆ t α / S X h XX : Sze of nterval vares accordng to dstance away from mean, X 1 (X X) 1 (X X) h n SSX n (X X) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-85 Predcton Interval for an Indvdual, Gven X Confdence nterval estmate for an Indvdual value of gven a partcular X Confdencentervalfor ˆ t α / S X 1 XX h : Ths extra term adds to the nterval wdth to reflect the added uncertanty for an ndvdual case Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-86 006 Prentce Hall, Inc.

Chapter 13 13-44 Estmaton of Mean Values: Example Confdence Interval Estmate for μ X=X Fnd the 95% confdence nterval for the mean prce of,000 square-foot houses Predcted Prce = 317.85 ($1,000s) 1 (X X) Ŷ t 0.05S X 317.85 37.1 n (X X) The confdence nterval endponts are 80.66 and 354.90, or from $80,660 to $354,900 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-87 Estmaton of Indvdual Values: Example Predcton Interval Estmate for X=X Fnd the 95% predcton nterval for an ndvdual house wth,000 square feet Predcted Prce = 317.85 ($1,000s) 1 (X X) Ŷ t 0.05S X 1 317.85 10.8 n (X X) The predcton nterval endponts are 15.50 and 40.07, or from $15,500 to $40,070 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-88 006 Prentce Hall, Inc.

Chapter 13 13-45 Fndng Confdence and Predcton Intervals n Excel From Excel, use PHStat regresson smple lnear regresson Check the confdence and predcton nterval for X= box and enter the X-value and confdence level desred Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-89 Fndng Confdence and Predcton Intervals n Excel (contnued) Input values Confdence Interval Estmate for μ X=X Predcton Interval Estmate for X=X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-90 006 Prentce Hall, Inc.

Chapter 13 13-46 Fndng Confdence and Predcton Intervals n Mntab Predcted Values for New Observatons Confdence Interval Estmate for μ X=X New Obs Ft SE Ft 95% CI 95% PI 1 317.8 16.1 (80.7, 354.9) (15.5, 40.1) Values of Predctors for New Observatons New Square Obs Feet 1 000 Input values Predcton Interval Estmate for X=X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-91 Ptfalls of Regresson Analyss Lackng an awareness of the assumptons underlyng least-squares regresson Not knowng how to evaluate the assumptons Not knowng the alternatves to least-squares regresson f a partcular assumpton s volated Usng a regresson model wthout knowledge of the subject matter Extrapolatng outsde the relevant range Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-9 006 Prentce Hall, Inc.

Chapter 13 13-47 Strateges for Avodng the Ptfalls of Regresson Start wth a scatter plot of X vs. to observe possble relatonshp Perform resdual analyss to check the assumptons Plot the resduals vs. X to check for volatons of assumptons such as homoscedastcty Use a hstogram, stem-and-leaf dsplay, boxplot, or normal probablty plot of the resduals to uncover possble non-normalty Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-93 Strateges for Avodng the Ptfalls of Regresson If there s volaton of any assumpton, use alternatve methods or models (contnued) If there s no evdence of assumpton volaton, then test for the sgnfcance of the regresson coeffcents and construct confdence ntervals and predcton ntervals Avod makng predctons or forecasts outsde the relevant range Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-94 006 Prentce Hall, Inc.

Chapter 13 13-48 Chapter Summary Introduced types of regresson models Revewed assumptons of regresson and correlaton Dscussed determnng the smple lnear regresson equaton Descrbed measures of varaton Dscussed resdual analyss Addressed measurng autocorrelaton Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-95 Chapter Summary (contnued) Descrbed nference about the slope Dscussed correlaton -- measurng the strength of the assocaton Addressed estmaton of mean values and predcton of ndvdual values Dscussed possble ptfalls n regresson and recommended strateges to avod them Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 13-96 006 Prentce Hall, Inc.

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