Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Motivating ExampleRelationshipsCausal RelationshipAssociationQuantitative VariablesIndependent VariableDependent VariableExample 23Example 23Lurking VariablesLurking Variables - ExampleLurking VariablesDescribing RelationshipsScatterplotExample 24Example 24Example 24Example 24Example 24Example 24Example 24Description of RelationshipDirectionPositive AssociationDirectionNegative AssociationFormLinear AssociationFormQuadratic AssociationFormExponential GrowthStrengthWeak AssociationModerate AssociationStrong AssociationExample 25Example 25Correlation CoefficientCorrelation CoefficientCorrelation CoefficientPropertiesPropertiesPropertiesPropertiesr near 0r near 0Propertiesr = 0.10r = 0.50r = 0.87r = 0.97Propertiesr = +1.0r = +1.0r = -1.0r = -1.0PropertiesSlide 63Correlation CoefficientCorrelation CoefficientPropertiesExample 26Example 26Example 26Example 26Example 26Example 26Example 26TI-83/84 CalculatorSlide 75Motivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExampleSTAT 210Lecture 14Scatterplots and CorrelationSeptember 27, 2017Practice ProblemsPages 130 through 137Relevant problems: V.1, V.2 (a), (b), (c), V.3 (a), V.4 (a) and (b), V.5, V.6 (a), V.8 (a), V.9 (a), V.10 (a) and V.11Recommended problems: V.1, V.5, V.8 (a), V.9 (a), V.10 (a), and V.11Additional Reading and ExamplesRead pages 109 and 110Top HatMotivating ExampleWatching television also often means watching or dealing with commercials, and of interest is to describe the relationship between the number of hours of television watched per day and the number of commercials watched.RelationshipsSuppose we have two variables, and our goal is to describe the relationship between the two variables.Causal RelationshipThe question: does one variable cause changes or explain changes in the other variable? This would imply a causalrelationship.Example: in young children, as they get older they gain weight and grow taller. Hence changes in age cause (explain) changes in weight and height.AssociationOften two variables are associated, and yet one variable does not cause changes in the other variable.Example: students taking the same math and chemistry classes, those students who did well on the first chemistry test also tended to do well on the first math test.Quantitative VariablesSuppose we have two quantitative variables X and Y.We want to explain the causal relationship between X and Y by writing Y as a linear function of X.This linear function will then be used to predict values of Y for specified values of X.Independent VariableX is called the independent or explanatory variable, which is a measurement variable that has no restraints placed on it and attempts to explain the observed outcomes of Y.Example: X = age of childDependent VariableY is called the dependent or response variable, and is the measurement variable that measures an outcome of a process that is the effect or consequence of the independent variable.Example: Y = weight of child As a child ages they gain weight, hence the process is the aging processand growing older causes weight gain.Example 231. X = number of TV ads run Y = number of cars sold or2. X = number of cars sold Y = number of TV ads run Top HatExample 231. X = number of TV ads run Y = number of cars soldLurking VariablesA variable which has an important effect on the relationship between the independent (X) and dependent (Y) variables but which is not included in the list of variables being studied is called a lurking variable.When a lurking variable exists we see an association between the two variables, but we cannot say that one variable is causing changes in the other.Lurking Variables - ExampleX = score on first chemistry testY = score on first math testScoring well on the first chemistry test does not cause you to score well on the first math test, but it is often associated with a good score on the first math test.There are potential lurking variables, such as the overall ability of the student or the amount of time and effort spent studying, that would explain the relationship that we see.Lurking VariablesWhen a lurking variable exists, then we say that the effect that X is having on Y is confounded with the effect of the lurking variable.So it appears that X is causing changes in Y, but really the lurking variable is involved in the relationship and hence confounds the results.Describing RelationshipsWe now turn our attention to describing the relationship between the independent variable X and the dependent variable Y. A complete description of this relationship includes specifying the direction, form and strength of the relationship, and to accurately describe these three things we need both a graph and a numerical descriptor. In the two sections that follow we learn about the scatterplot (a graph) and the correlation coefficient (a numerical descriptor).ScatterplotGraphical procedure for displaying the relationship between two quantitative variables.Label X along the horizontal axis.Label Y along the vertical axis.Plot each (X, Y) observation on the plot.Example 24X = number of TV ads runY = number of cars soldExample 240 5 10 15 20 25 30 35 Number of ads runExample 240 5 10 15 20 25 30 35 Number of ads runNumber of cars sold 40 36 32 28 24 20 16 12 8 4Example 240 5 10 15 20 25 30 35 Number of ads runNumber of cars sold 40 36 32 28 24 20 16 12 8 4Example 240 5 10 15 20 25 30 35 Number of ads runNumber of cars sold 40 36 32 28 24 20 16 12 8 4Example 240 5 10 15 20 25 30 35 Number of ads runNumber of cars sold 40 36 32 28 24 20 16 12 8 4Example 240 5 10 15 20 25 30 35 Number of ads runNumber of cars sold 40
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