Slide 1Learning ObjectivesSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Positive TrendNegative TrendSlide 16No TrendStrength of AssociationStrength of AssociationLinear TrendsSlide 21Other ShapesSlide 23Summary of Analysis of the ScatterplotWriting Clear Descriptions Based on AssociationThe Correlation Coefficient rPositive CorrelationNegative CorrelationWeak or No CorrelationSlide 30Slide 31Slide 32Slide 33Interpreting CorrelationSlide 35Switching x and yCorrelation, Arithmetic, and UnitsCorrelation and Linearity and OutliersLeast Squares Regression LineSlide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Using the Regression LineInterpreting the SlopeInterpreting the y-interceptInterpreting the SlopeCorrelation is Not CausationSlope and CausationWhy Not to Use the y-interceptSlide 59Nonlinear DataSlide 61Slide 62Beware of OutliersExample of an Influential PointSlide 65Regression of Aggregate DataAggregate DataDon’t ExtrapolateSlide 69Slide 70Slide 71Slide 72Coefficient of Determination ExampleSlide 74Slide 75Slide 76Slide 77Are Brinks Employees Stealing from Parking Meters?Predicted vs. Actual Brinks CollectionComparing Brinks vs. Honest EmployeesSlide 81Does the Cost of a Flight Depend on the Distance?Create a ScatterplotThe Regression LineAnswer the QuestionSlide 86Test Scores: SlopeTest Scores: y-interceptTest Scores: Regression Line1 - 1Chapter 4Regression Analysis:Exploring Associationsbetween Variables1 - 2Bivariate DataqData in which two variables are measured on each individual.qOften the purpose of studying them is:I. Find association between two categorical variables.II. Analyze correlation between two numerical variables, least-squares regression problem, making predictions.1 - 3Two Categorical Variables (Example I)qBehavioral Risk Factor Surveillance System (BRFSS) survey in 2000, sample size = 20,000qExample: gender and smoking habitsq 0 1 m 4547 5022 f 6012 44191 - 4Compare: Percentages of Smokers (Similar Example in Lab 1)q 0 1 total m 4547 5022 9569 f 6012 4419 10431total 10559 9441 20000qOut of male, 5022/9569 = 52.5% are smokers.qOut of female, 4419/10431 = 42.4% are smokers.qConclusion: Male is more likely to smoke than a female in this data set.1 - 5Clicker Question: Which of the following segmented bar chart better illustrates the previous comparison?AB1 - 6Clicker Question: Which of the following segmented bar chart better illustrates the previous comparison? (Answer)BPlot (B) shows that male has a higher tendency to be a smoker in this data set because its proportion of gray area out of the column total is larger than that of female.1 - 7Titanic Survivors (Example II)qFirst Second Third Crew Total Count 203 118 178 212 711Row% 28.6% 16.6% 25.0% 29.8% 100%Col % 62.5% 41.4% 25.2% 24.0% 32.3%Cell% 9.2% 5.4% 8.1% 9.6% 32.3%Count 122 167 528 673 1490Row% 8.2% 11.2% 35.4% 45.2% 100%Col% 37.5% 58.6% 74.8% 76.0% 67.7%Cell% 5.6% 7.6% 24.0% 30.6% 67.7%Count 325 285 706 885 2201Row% 14.8% 12.9% 32.1% 40.2% 100%Col% 100% 100% 100% 100% 100%Cell% 14.8% 12.9% 32.1% 40.2% 100%AliveDeadTotal1 - 8What Kind of Association Do You Observe? What Should You Ask?1 - 9Clicker Question: Which Class Had the 2nd Highest Survival Rate?A BC D1 - 10Clicker Question: Which Class Had the 2nd Highest Survival Rate? (Answer)A BC D62.5% 41.4%25.2% 24.0%1 - 11Two Numerical VariablesqY: response variable (dependent variable)qX: explanatory/predictor variable (independent variable)qGraphical tool: Scatter plot – each individual in the data set is represented by a point in the scatter plot. (We do not connect the dots.)1 - 12Example: Is Hand Size a Good Predictor for Height?qY: HeightqX: Hand length1 - 13Example: Scatter Plot●I. Is the relation positive?●II. How strong is the relationship?1 - 14Positive TrendqOlder cars tend to have more miles than newer cars.qNewer cars tend to have fewer miles than older cars.qThere is a positive association between car age and miles the car has been driven.1 - 15Negative TrendqCountries with higher literacy rates tend to have fewer births per woman.qCountries with lower literacy rates tend to have more births per woman.qThere is a negative association between literacy rate and births per woman.1 - 16Positive Vs. Negative Trendas x increases, y increasesNE-SW directionas x increases, y decreasesNW-SE direction1 - 17No TrendqThere is no trend between the speed and age of a marathon runner.qKnowing the age of a marathon runner does not help predict the runner’s speed.qThere is no association between a marathon runner’s age and speed.1 - 18Strength of AssociationqIf for each value of x, there is a small spread of y values, then there is a strong association between x and y.qIf for each value of x, there is a large spread of y values, then there is a weak or no association between x and y.qIf there is a strong (weak) association between x and y, then x is a good (bad) predictor of y.1 - 19Strength of AssociationWeak Association Strong Association1 - 20Linear TrendsqA trend is linear if there is a line such that the points in general do not stray far from the line.qLinear trends are the easiest to work with.qThere is a positive linear association between number of searches for “Vampire” and number for “Zombie”.1 - 21Positive and Negative Linear Trend1 - 22Other ShapesqNonlinear association can also occur, but this is covered in a more advanced statistics course.qIn terms of the strength of a linear relationship, which one is stronger? Why?AB1 - 23Nonlinear Trend1 - 24Summary of Analysis of the Scatter PlotqLook to see if there is a trend or association.qDetermine the strength of trend. Is the association strong or weak?qLook at the shape of the trend. Is it linear? Is it nonlinear?1 - 25Writing Clear Descriptions Based on Association (NOT Causation)qGood:qPeople who have higher salaries tend to travel farther on vacation.qA person who has a high salary is predicted to travel far on vacation.qBad:qBecause they have higher salaries, they travel farther.qA person with a high salary will travel farther on vacation.1 - 26Quantifying the Strength of Linear Relationship: Correlation Coefficient rqThe Pearson correlation coefficient is a number, r, that