Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 131STAT 110 – Introduction to Descriptive StatisticsChapter 15Describing Relationships:Regression, Prediction, and Causation2STAT 110 – Introduction to Descriptive StatisticsRegression Lineregression line – a straight line that describes how a response variable y changes as an explanatory variable x changes• regression line summarizes a linear relationship between two variables• one variable helps explain or predict the other3STAT 110 – Introduction to Descriptive StatisticsSupposed Fertility EnhancerThe data to the right concerns the relationship between the prevalence of a supposed fertility enhancer and the population of Oldenburg Germany in thousands of people between 1930 and 1936. The original data can be found in: Ornithologische Monatsberichte, 44, No.2, Jahrgang, 1936, Berlin, and 48, No.1, Jahrgang, 1940, Berlin, and Statistiches Jahrbuch Deutscher Gemeinden, 27-33, Jahrgang, 1932-1938, Gustav Fischer, Jena. X People140 55.5148 55.5175 64.9195 67.5245 69.0250 72.0250 75.54STAT 110 – Introduction to Descriptive StatisticsExample (cont’d)r = 0.9415STAT 110 – Introduction to Descriptive StatisticsEquation of a Line• The equation of a line is y = mx + b • m is the slope of the line• slope = the amount by which y changes when x increases one unit - a slope of zero means that there is no linear relationship between x and y• b is the intercept of the line• intercept = the value of y when x=06STAT 110 – Introduction to Descriptive StatisticsLeast Squares Regression Lineleast-squares regression line – the line that makes the sum of the squared vertical distances to the line as small as possibleWe’ll go to an applet to see what this means…http://www.nctm.org/standards/content.aspx?id=267877STAT 110 – Introduction to Descriptive StatisticsLeast Squares Regression Line8STAT 110 – Introduction to Descriptive StatisticsExample – Fitting the Least Squares LineInterpretation of the Slope:People = 35.49 + 0.1507 x 150 200 250 X 60 65 70 75 P E O P L E9STAT 110 – Introduction to Descriptive StatisticsPrediction Three Things to Understand about Prediction:• Prediction is based on fitting some “model” to a set of data.• Prediction works best when the model fits the data closely.• Prediction outside the range of the available data is risky. This is called an extrapolation.10STAT 110 – Introduction to Descriptive StatisticsPrediction ExamplePeople = 35.49 + 0.1507 xUsing this equation to estimate the mean population of Oldenburg Germany for an X level of 200, we have35.49 + 0.1507(200) = 65.63 So, we estimate the mean population of Oldenburg Germany (1930-1936) to be 65.53 thousand people for an X level of 200.11STAT 110 – Introduction to Descriptive StatisticsCorrelation and Regressionr2 - the fraction of the variation in the values of y that is explained by the least-squares regression of y on xIn the example, (0.941)2=0.8857 of the variation in the population is explained by the regression line using X. 150 200 250 X 60 65 70 75 P E O P L E12STAT 110 – Introduction to Descriptive StatisticsWhat is “X”?•The moral of the story is:Only experimentation can show causation!• When dealing with regression and/or correlation, NEVER say that one variable causes another.• Snake bites and ice cream sales are highly correlated. Does that mean that one causes the other?13STAT 110 – Introduction to Descriptive StatisticsCausationWhat are the criteria for giving evidence about causation when we can’t do an experiment?1. Strong association2. Consistent association3. Higher doses associated with stronger responses.4. Alleged cause precedes the effect in time.5. Alleged cause is
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