1 STAT 110 – Introduction to Descriptive Statistics Chapter 15 Describing Relationships: Regression, Prediction, and Causation2 STAT 110 – Introduction to Descriptive Statistics Regression Line regression 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 other3 STAT 110 – Introduction to Descriptive Statistics Supposed Fertility Enhancer The 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 People 140 55.5 148 55.5 175 64.9 195 67.5 245 69.0 250 72.0 250 75.54 STAT 110 – Introduction to Descriptive Statistics Example (cont’d) r = 0.9415 STAT 110 – Introduction to Descriptive Statistics Equation 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=06 STAT 110 – Introduction to Descriptive Statistics Least Squares Regression Line least-squares regression line – the line that makes the sum of the squared vertical distances to the line as small as possible We’ll go to an applet to see what this means… http://www.nctm.org/standards/content.aspx?id=267877 STAT 110 – Introduction to Descriptive Statistics Least Squares Regression Line8 STAT 110 – Introduction to Descriptive Statistics Example – Fitting the Least Squares Line 150 200 250 X 60 65 70 75 P E O P L E Interpretation of the Slope: People = 35.49 + 0.1507 x9 STAT 110 – Introduction to Descriptive Statistics Prediction 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.10 STAT 110 – Introduction to Descriptive Statistics Prediction Example People = 35.49 + 0.1507 x Using this equation to estimate the mean population of Oldenburg Germany for an X level of 200, we have 35.49 + 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 200.11 STAT 110 – Introduction to Descriptive Statistics Correlation and Regression r2 - the fraction of the variation in the values of y that is explained by the least-squares regression of y on x In 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 E12 STAT 110 – Introduction to Descriptive Statistics What 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?13 STAT 110 – Introduction to Descriptive Statistics Causation What are the criteria for giving evidence about causation when we can’t do an experiment? 1. Strong association 2. Consistent association 3. Higher doses associated with stronger responses. 4. Alleged cause precedes the effect in time. 5. Alleged cause is
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