• Ch 5 Relationships Between Quantitative Variables (pg 150)

– Will use 3 tools to describe, picture, and quantify

• 1) scatterplot 2) correlation 3) regression equation

– Sec 5.1 Looking for Patterns with Scatterplots (pg 152)

• Scatterpot- two-dimensional graph of the two variables’ measurements

• Questions to ask:

– Average pattern – straight line? curved line?

– Direction of pattern

• Ex 5.1 Height and Handspan

– Data on 167 students -- see slide2

– Fig 5.1 (slide 3) looks like a “linear relationship” and “positive association”

• Def’n (pg 153)– Positive association --- values of 1 variable increases as the values of

the other variable increases

– Negative association – values of 1 increases while the other decreases

– Linear relationship – the pattern seems to approximate a straight line

• Ex 5.2 Driver Age and Maximum Legibility Distance of Highway Signs

– See table 5.2 (pg 154) -- see slide 4

– Figure 5.2 ( slide 5) -- appears linear with negative association

• Curvilinear Patterns (pg 154)– Ex 5.3 Development of Musical Preference

» Study of 108 people (ages 16 to 86) who scored 28 songs

» -- scored from 1 (disliked a lot) to 10 (liked a lot)

» -- adjusted so each persons mean score set = to 0

» Fig 5.3 (slide 6) shows results are a curvilinear relationship

• Indicating Groups Within Data on Scatterplots pg 155

– Fig 5.4 ( slide 7) shows subgroups (male and female) for Fig 5.1(slide 3)

• Look for Outliers (pg 156)

– Ex 5.4 Heights and Foot Lengths – see slide 8

» Potential “outliers” were actually 3 errors in computer entries

– Sec 5.2 Describing Patterns with Regression Line (pg 157)

• Regression Analysis – relationship between a quantitative response variable and 1 or more explanatory variables

– use an equation to predict

– simplest is a “straight line”

• Def’n Regression Line (pg 158)– Describes how values of a quantitative response variable (y) are

related “ on average” to values of quantitative explanatory variable (x)

• Ex 5.5 Height and Handspan Regression Line (pg 158) --slide 9

– For a specific x-value, can estimate a corresponding y –value

– Using regression equation( discussed on pg 159)

» Handspan= -3+ 0.35( Height) or ŷ = -3 + 0.35x

» So, when x=60 y= -3 + .35(60) = 18 cm

» when x=70 y= -3 + .35 (70) = 21.5 cm

– Recall equation of a straight line

0 1y b b x

• Statistical Relationship versus Deterministic Relationship (pg160)– Deterministic --- no variation– Statistical -- there is variation from an “average pattern”

• The Equation of a Regression line (pg 160)– – Ex 5.6 Regression for Driver age and maximum Legibility Distance

» Fig 5.7(slide 10) shows regression line for data in Fig 5.2 (pg 154)» Chart on pg 161(slide 11) shows Average Maximum Legibility

Distance for 3 selected ages» SPSS tip: Analyze>Regression>Linear

• Prediction Errors and Residuals (pg 163)– Residual = – Ex 5.7 Prediction Errors for Highway Sign data

» Chart on page 163 (slide 12) shows “residuals” or “prediction error”» Fig 5.8 (slide 13) shows graphically the residual for x = 27

• Least Squares Criterion (pg 163)– Find the regression line with the smallest “sum of squared errors”– See formulas on pg 164 : – where

0 1y b b x

ˆ( )y y

1 2

( )( )

( )i i

i

x x y yb

x x

0 1b y b x 0 1y b b x