Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear: To calculate the coeffcet of correlato The meag of the regresso coeffcets b ad db. How to use regresso aalyss to predct the value of a depedet varable based o a depedet varable. Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-2 Correlato vs. Regresso A scatter plot ca be used to show the relatoshp betwee two varables Correlato aalyss s used to measure the stregth of the assocato (lear relatoshp) betwee two varables Correlato s oly cocered wth stregth of the relatoshp No causal effect s mpled wth correlato Scatter plots were frst preseted Ch. 2 Correlato was frst preseted Ch. 3 Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-3
Chapter 3 3-2 Coeffcet of Correlato Measures the relatve stregth of the lear relatoshp betwee two umercal varables Sample coeffcet of correlato: r x x y y S S Where x ad y are the meas of x ad y-values S x ad S y are the stadard devatos of x ad y- values x y Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-4 3-4 Shortcut Formula r x y xy 2 2 2 x x y 2 y Where x ad y are the meas of x ad y-values Questo : Wll outlers effect the correlato? ES Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-5 3-5 Features of the Coeffcet of Correlato The populato coeffcet of correlato s referred as ρ. The sample coeffcet of correlato s referred to as r. Ether ρ or r have the followg features: Ut free Rages betwee ad The closer to, the stroger the egatve lear relatoshp The closer to, the stroger the postve lear relatoshp The closer to, the weaker the lear relatoshp Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-6 3-6
Chapter 3 3-3 Scatter Plots of Sample Data wth Varous Coeffcets of Correlato r = - r = -.6 r = + r = +.3 Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-7 3-7 r = Correlato Coeffcet Example: Real estate aget A real estate aget wshes to exame the relatoshp betwee the sellg prce of a home ad ts sze (measured square feet) A radom sample of houses s selected Depedet varable () = house prce $s Idepedet varable () = square feet Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-8 Correlato Coeffcet Example: Data House Prce $s () Square Feet () 245 4 32 6 279 7 38 875 99 29 55 45 235 324 245 39 425 255 7 Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-9
Chapter 3 3-4 Smple Lear Regresso Example: Scatter Plot House prce model: Scatter Plot s) House Prce ($ 45 4 35 3 25 2 5 5 5 5 2 25 3 Square Feet Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2- Calculatos x y x y xy 2 2 r 75, 2865, 398375, 853423, 585975 585975 75286.5.762 2 2 398375 75 853423 286.5 There s postve relatoshp betwee the sellg prce of a home ad ts sze. Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2- Itroducto to Regresso Aalyss Regresso aalyss s used to: Predct the value of a depedet varable based o the value of at least oe depedet varable Expla the mpact of chages a depedet varable o the depedet varable Depedet varable: the varable we wsh to predct or expla Idepedet varable: the varable used to predct or expla the depedet varable Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-2
Chapter 3 3-5 Smple Lear Regresso Model Oly oe depedet varable, Relatoshp betwee ad s descrbed by a lear fucto Chages are assumed to be related to chages Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-3 Types of Relatoshps Lear relatoshps Curvlear relatoshps Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-4 Types of Relatoshps (cotued) Strog relatoshps Weak relatoshps Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-5
Chapter 3 3-6 Types of Relatoshps (cotued) No relatoshp Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-6 Smple Lear Regresso Model Depedet Varable Populato tercept β Populato Slope Coeffcet β Idepedet Varable ε Radom Error term Lear compoet Radom Error compoet Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-7 Observed Value of for Smple Lear Regresso Model β β ε (cotued) Predcted Value of for ε Radom Error for ths value Slope = β Itercept = β Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-8
Chapter 3 3-7 Smple Lear Regresso Equato (Predcto Le) The smple lear regresso equato provdes a estmate of the populato regresso le Estmated (or predcted) value for observato Estmate of the regresso tercept Estmate of the regresso slope Ŷ b b Value of for observato Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-9 Fdg the Regresso Equato The coeffcets b ad b are gve by: The Slope b : The Itercept b : where S b S r S y b y b x x 2 2 x x y y, S Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-2 Iterpretato of the Slope ad the Itercept b s the estmated mea value of whe the value of s zero b s the estmated chage the mea value of as a result of a oe-ut chage Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-2
Chapter 3 3-8 Smple Lear Regresso Example: Recall the real estate example - Scatter Plot House prce model: Scatter Plot s) House Prce ($ 45 4 35 3 25 2 5 5 5 5 2 25 3 Square Feet Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-22 Calculatos S y S x 6. 854 b r b 762..977 47. 8649 b yb x b 286. 5.97775=98.248 Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-23 Smple Lear Regresso Example: Graphcal Represetato House prce model: Scatter Plot ad Predcto Le Itercept = 98.248 s) House Prce ($ 45 4 35 3 25 2 5 5 5 5 2 25 3 Square Feet Slope =.977 house prce 98.24833.977 (square feet) Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-24
Chapter 3 3-9 Smple Lear Regresso Example: Iterpretato of b o house prce 98.24833.977 (square feet) b s the estmated mea value of whe the value of s zero (f = s the rage of observed values) Because a house caot have a square footage of, b has o practcal applcato Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-25 Smple Lear Regresso Example: Iterpretg b house prce 98.24833.977 (square feet) b estmates the chage the mea value of as a result of a oe-ut crease Here, b =.977 tells us that the mea value of a house creases by.977($) = $9.77, o average, for each addtoal oe square foot of sze Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-26 Smple Lear Regresso Example: Makg Predctos Predct the prce for a house wth 2 square feet: house prce 98.25.98 (sq.ft.) 98.25.98(2) 37.85 The predcted prce for a house wth 2 square feet s 37.85($,s) = $37,85 Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-27
Chapter 3 3- Coeffcet of Determato, r 2 The coeffcet of determato s the porto of the total varato the depedet varable that s explaed by varato the depedet varable The coeffcet of determato s also called r-squared ad s deoted as r 2 2 r Correlato betwee ad 2 ote: r 2 Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-28 Examples of r 2 Values r 2 = r 2 = Perfect lear relatoshp betwee ad : % of the varato s explaed by varato r 2 = Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-29 Examples of r 2 Values < r 2 < Weaker lear relatoshps betwee ad : Some but ot all of the varato s explaed by varato Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-3
Chapter 3 3- Examples of r 2 Values r 2 = No lear relatoshp betwee ad : r 2 = The value of does ot deped o. (Noe of the varato s explaed by varato ) Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-3 Smple Lear Regresso Example: Coeffcet of Determato, r 2 2 2 r.762.588 58.8% of the varato house prces s explaed by varato square feet Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-32 Chapter Summary Itroduced Correlato coeffcet. Itroduced types of regresso models Dscussed determg the smple lear regresso equato Descrbed measures of varato Dscussed resdual aalyss Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic.. Chap 2-33