Where Are We?Chapter 14Thought Question 1Thought Question 2Thought Question 3Statistical versus Deterministic RelationshipsDistance versus SpeedIncome versus AssetsStrength and Statistical SignificanceWarnings about Statistical SignificanceLinear RelationshipExamples of RelationshipsMeasuring Strength & Direction of a Linear RelationshipCorrelation CoefficientExamples of CorrelationsNot all Relationships are Linear Miles per Gallon versus SpeedSlide 17Problems with CorrelationsOutliers and CorrelationPrice of Books versus SizeKey ConceptsCorrelation CalculationCase StudySlide 24Slide 25Slide 26Slide 27Chapter 6 1Where Are We?Population and SampleGetting Data: Experiments and Observational StudiesQuantitative versus Categorical VariablesRelated Variables (Causation, Non-Causal, Other)Ex1: Experiment to look at relationship between nicotine patches and smoking cessationEx2: Obs study to look at relationship between IQ and number of spankingsDescribing Data: graphs, summary statistics (mean&stdev, median&quartiles), distributions (stemplots, histograms, boxplots)Percent quitting Nicotine Placebo Smoker at home 31% 20% No smoker at home 58% 20%Chapter 14 2Chapter 14Describing Relationships: Scatterplots and CorrelationChapter 14 3Thought Question 1For all cars manufactured in the U.S., there is a positive correlation between the size of the engine and horsepower. There is a negative correlation between the size of the engine and gas mileage. What does it mean for two variables to have a positive correlation or a negative correlation?Chapter 14 4Thought Question 2What type of correlation would the following pairs of variables have – positive, negative, or none?1. Temperature during the summer and electricity bills2. Temperature during the winter and heating costs3. Number of years of education and height4. Frequency of brushing and number of cavities5. Number of churches and number of bars in cities in your state6. Height of husband and height of wifeChapter 14 5Thought Question 3Consider the two scatterplots below. How does the outlier impact the correlation for each plot?–does the outlier increase the correlation, decrease the correlation, or have no impact?Chapter 14 6Statistical versus Deterministic RelationshipsDistance versus Speed (when travel time is constant).Income (in millions of dollars) versus total assets of banks (in billions of dollars).Chapter 14 7Distance versus Speed Distance = Speed  TimeSuppose time = 1.5 hoursEach subject drives a fixed speed for the 1.5 hrs–speed chosen for each subject varies from 10 mph to 50 mphDistance does not vary for those who drive the same fixed speedDeterministic relationship010203040506070800 20 40 60speeddistanceChapter 14 8Income versus Assets0501001502002503000 20 40 60assets (billions)income (millions)Income =a + bAssetsAssets vary from 3.4 billion to 49 billionIncome varies from bank to bank, even among those with similar assetsStatistical relationshipChapter 14 9Strength and Statistical SignificanceA strong relationship seen in the sample may indicate a strong relationship in the population.The sample may exhibit a strong relationship simply by chance and the relationship in the population is not strong or is zero.The observed relationship is considered to be statistically significant if it is stronger than a large proportion of the relationships we could expect to see just by chance.Chapter 14 10Warnings aboutStatistical Significance“Statistical significance” does not imply the relationship is strong enough to be considered “practically important”.Even weak relationships may be labeled statistically significant if the sample size is very large.Even very strong relationships may not be labeled statistically significant if the sample size is very small.Chapter 14 11Linear RelationshipSome relationships are such that the points of a scatterplot tend to fall along a straight line — linear relationshipChapter 14 12Examples of Relationships0102030405060$0 $10 $20 $30 $40 $50 $60 $70IncomeHeath Status Measure0102030405060700 20 40 60 80 100AgeHeath Status Measure0246810121416180 20 40 60 80 100AgeEducation Level30354045505560650 20 40 60 80Physical Health ScoreMental Health ScoreChapter 14 13Measuring Strength & Directionof a Linear RelationshipHow closely does a non-horizontal straight line fit the points of a scatterplot?The correlation coefficient (often referred to as just correlation): r–measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r.–measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship.Click for ComputationChapter 14 14Correlation Coefficientspecial values for r:-a perfect positive linear relationship would have r = +1-a perfect negative linear relationship would have r = -1-if there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0 -Note: r must be between -1 and +1, inclusiver > 0: as one variable changes, the other variable tends to change in the same directionr < 0: as one variable changes, the other variable tends to change in the opposite directionPlotChapter 14 15Examples of CorrelationsHusband’s versus Wife’s agesr = .94Husband’s versus Wife’s heightsr = .36Professional Golfer’s Putting Success: Distance of putt in feet versus percent successr = -.94PlotClick for Graphical ExamplesChapter 14 16Not all Relationships are LinearMiles per Gallon versus SpeedLinear relationship?MPG = a + bSpeedSpeed chosen for each subject varies from 20 mph to 60 mphMPG varies from trial to trial, even at the same speedStatistical relationshipy = - 0.013x + 26.9r = - 0.06051015202530350 50 100speedmiles per gallonChapter 14 17Not all Relationships are LinearMiles per Gallon versus SpeedCurved relationship(r is misleading)Speed chosen for each subject varies from 20 mph to 60 mphMPG varies from trial to trial, even at the same speedStatistical relationship051015202530350 50 100speedmiles per gallonChapter 14 18Problems with CorrelationsOutliers can inflate or deflate correlationsGroups combined inappropriately may mask relationships (a third variable)–groups may have different relationships when separatedPlotChapter 14 19Outliers and