Basic Business Statistics, 10/e

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1 Chapter 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 Prentce Hall, Inc.

2 Chapter 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 Prentce Hall, Inc.

3 Chapter 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 Prentce Hall, Inc.

4 Chapter 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 Prentce Hall, Inc.

5 Chapter 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 Prentce Hall, Inc.

6 Chapter 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 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.

7 Chapter Calculaton Example Tree Heght Trunk Dameter y x xy y x =31 =73 =314 =14111 =713 Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap Tree Heg ht, y 70 Calculaton Example r (contnued) n xy x y [n( x ) ( x) ][n( y ) ( y) ] (314) (73)(31) [8(713) (73) ][8(14111) (31) ] Trunk Dameter, x r = relatvely strong postve lnear assocaton between x and y Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap Prentce Hall, Inc.

8 Chapter Excel Output Excel Correlaton Output Tools / data analyss / correlaton Tree Heght Trunk Dameter Tree Heght 1 Trunk Dameter Correlaton between Tree Heght and Trunk Dameter Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap 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 Prentce Hall, Inc.

9 Chapter 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 Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap Example: Test Soluton r t 1 r n d.f. = 8- = 6 /= /=.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/ Busness Statstcs: A Decson-Makng Approach, 6e 005 Prentce-Hall, Inc. Chap Prentce Hall, Inc.

10 Chapter 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 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 Prentce Hall, Inc.

11 Chapter 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 Prentce Hall, Inc.

12 Chapter 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 Prentce Hall, Inc.

13 Chapter 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 Prentce Hall, Inc.

14 Chapter 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 Prentce Hall, Inc.

15 Chapter 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 Prentce Hall, Inc.

16 House Prce ($1000s) Chapter Smple Lnear Regresson Example: Data House Prce n $1000s () Square Feet (X) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Smple Lnear Regresson Example: Scatter Plot House prce model: Scatter Plot Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

17 Chapter Smple Lnear Regresson Example: Usng Excel Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Smple Lnear Regresson Example: Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 The regresson equaton s: houseprce (squarefeet) ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

18 House Prce ($1000s) Chapter Smple Lnear Regresson Example: Mntab Output The regresson equaton s Prce = Square Feet Predctor Coef SE Coef T P Constant Square Feet The regresson equaton s: house prce = (square feet) S = R-Sq = 58.1% R-Sq(adj) = 5.8% Analyss of Varance Source DF SS MS F P Regresson Resdual Error Total Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Smple Lnear Regresson Example: Graphcal Representaton House prce model: Scatter Plot and Predcton Lne Intercept = Square Feet Slope = houseprce (squarefeet) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

19 Chapter Smple Lnear Regresson Example: Interpretaton of b o houseprce (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 Smple Lnear Regresson Example: Interpretng b 1 houseprce (squarefeet) b 1 estmates the change n the average value of as a result of a one-unt ncrease n X Here, b 1 = 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 Prentce Hall, Inc.

20 House Prce ($1000s) Chapter Smple Lnear Regresson Example: Makng Predctons Predct the prce for a house wth 000 square feet: houseprce (sq.ft.) (000) The predcted prce for a house wth 000 square feet s ($1,000s) = $317,850 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Do not try to extrapolate beyond the range of observed X s Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

21 Chapter 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 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 Prentce Hall, Inc.

22 Chapter SST = ( - ) Measures of Varaton SSE = ( - ) _ SSR = ( - ) (contnued) _ X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Prentce Hall, Inc.

23 Chapter 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 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 Prentce Hall, Inc.

24 Chapter 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 Smple Lnear Regresson Example: Coeffcent of Determnaton, r n Excel Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 r SSR SST % of the varaton n house prces s explaned by varaton n square feet ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

25 Chapter Smple Lnear Regresson Example: Coeffcent of Determnaton, r n Mntab The regresson equaton s Prce = Square Feet Predctor Coef SE Coef T P Constant Square Feet S = R-Sq = 58.1% R-Sq(adj) = 5.8% Analyss of Varance Source DF SS MS F P Regresson Resdual Error Total r SSR SST % of the varaton n house prces s explaned by varaton n square feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Prentce Hall, Inc.

26 Chapter Smple Lnear Regresson Example: Standard Error of Estmate n Excel Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 S X ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Smple Lnear Regresson Example: Standard Error of Estmate n Mntab The regresson equaton s Prce = Square Feet Predctor Coef SE Coef T P Constant Square Feet S = R-Sq = 58.1% R-Sq(adj) = 5.8% S X Analyss of Varance Source DF SS MS F P Regresson Resdual Error Total Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

27 Chapter 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 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 Prentce Hall, Inc.

28 resduals resduals Chapter 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 Resdual Analyss for Lnearty x x x x Not Lnear Lnear Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

29 resduals resduals resduals Chapter Resdual Analyss for Independence Not Independent Independent X X X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Prentce Hall, Inc.

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

31 Resduals Chapter Smple Lnear Regresson Example: Excel Resdual Output RESIDUAL OUTPUT Predcted House Prce Resduals House Prce Model Resdual Plot Square Feet Does not appear to volate any regresson assumptons Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Prentce Hall, Inc.

32 Resduals Chapter 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 Tme (t) Volates the regresson assumpton that resduals are random and ndependent Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Prentce Hall, Inc.

33 Sales Chapter 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 Testng for Postve Autocorrelaton (contnued) Suppose we have the followng tme seres data: y = x R = Tme Is there autocorrelaton? Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

34 Sales Chapter Example wth n = 5: Excel/PHStat output: Durbn-Watson Calculatons Sum of Squared Dfference of Resduals Sum of Squared Resduals Durbn-Watson Statstc Testng for Postve Autocorrelaton y = x R = Tme (contnued) D n (e e n 1 e ) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 = < d L = 1.9, so reject H 0 and conclude that sgnfcant postve autocorrelaton exsts Decson: reject H 0 snce D = < 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 Prentce Hall, Inc.

35 Chapter 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 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 Prentce Hall, Inc.

36 Chapter Inferences About the Slope: t Test Example House Prce n $1000s (y) Square Feet (x) Estmated Regresson Equaton: houseprce (sq.ft.) The slope of ths model s Is there a relatonshp between the square footage of the house and ts sales prce? Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Square Feet From Mntab output: b 1 S b1 Predctor Coef SE Coef T P Constant Square Feet b 1 S b 1 t STAT b β 1 1 S b Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

37 Chapter 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/ Decson: Reject H 0 There s suffcent evdence that square footage affects house prce Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Square Feet From Mntab output: Predctor Coef SE Coef T P Constant Square Feet 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 Prentce Hall, Inc.

38 Chapter 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 F-Test for Sgnfcance Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 ANOVA F STAT df SS MS F Sgnfcance F Regresson Resdual Total MSR MSE Wth 1 and 8 degrees of freedom p-value for the F-Test Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

39 Chapter F-Test for Sgnfcance Mntab Output Analyss of Varance Source DF SS MS F P Regresson Resdual Error Total p-value for the F-Test Wth 1 and 8 degrees of freedom F STAT MSR MSE Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 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 Prentce Hall, Inc.

40 Chapter 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 Square Feet At 95% level of confdence, the confdence nterval for the slope s (0.0337, ) Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Confdence Interval Estmate for the Slope (contnued) Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet 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 $ 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 Prentce Hall, Inc.

41 Chapter 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 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 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

42 Chapter t-test For A Correlaton Coeffcent (contnued) t STAT d.f. = 10- = 8 /=.05 r ρ 1 r n /=.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/ Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 X X 006 Prentce Hall, Inc.

43 Chapter 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 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 Prentce Hall, Inc.

44 Chapter 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 = ($1,000s) 1 (X X) Ŷ t 0.05S X n (X X) The confdence nterval endponts are and , or from $80,660 to $354,900 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 = ($1,000s) 1 (X X) Ŷ t 0.05S X n (X X) The predcton nterval endponts are and 40.07, or from $15,500 to $40,070 Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap Prentce Hall, Inc.

45 Chapter 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 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 Prentce Hall, Inc.

46 Chapter 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 (80.7, 354.9) (15.5, 40.1) Values of Predctors for New Observatons New Square Obs Feet Input values Predcton Interval Estmate for X=X Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc.. Chap 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 Prentce Hall, Inc.

47 Chapter 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 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 Prentce Hall, Inc.

48 Chapter 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 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 Prentce Hall, Inc.