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Notes7 SLR Spring 09

# Notes7 SLR Spring 09 - Notes 7.1 of 22 Notes7 Simple Linear...

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20 15 10 5 0 70 60 50 40 30 20 10 0 X Y Scatterplot of Y vs X 20 15 10 5 0 900 800 700 600 500 400 300 200 100 0 X Y_ Scatterplot of Y_ vs X Notes 7. 1 of 22 Notes7: Simple Linear Regression Text Sections 14.1 – 14.7 Regression analysis is a tool used to determine whether an independent variable Y is related to a dependent variable X . We assume that if there is a relation between these variables it takes the form of a linear relation. If there is a relation, we can use it for example to predict future values of Y if we know the value of X . Linear relation example: Y = 2 + 3 X Curvilinear relation example: Y = 2 - 20 X + 3 X 2

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20 15 10 5 0 70 60 50 40 30 20 10 0 X Y Scatterplot of Y vs X 20 15 10 5 0 60 50 40 30 20 10 0 X Y Notes 7. 2 of 22 The linear relation Y = 2 + 3 X is an example of an exact or deterministic relation. If Y and X are related in this way, observed data points would fall on a line: In the real world, data points are more likely to randomly scatter around a line. This is a probabilistic relation . A mathematical ‘model’ or statement of this relation can be expressed as: Y = 2 + 3 X + ε where ε (‘epsilon’) represents a random error component of the model.
Notes 7. 3 of 22 In general, the simple linear regression model is expressed as: y = β 0 + β 1 x + ε where: β 0 and β 1 are the y -intercept and slope, respectively (these are unknown parameters in the model) y = the dependent or response variable x = the independent or predictor variable ε = the random error component Example: Suppose that the Steamboat Springs Chamber of Commerce is interested in the relation, if any exists, between a visitor’s income ( x , measured in \$1k units) and the amount that he or she spends per day on a vacation at Steamboat Springs ( y , measured in \$). (Why is y the natural choice for the dependent variable?) Data is obtained for ten randomly selected visitors. Case # x y x 2 xy y 2 1 14 54 196 756 2916 2 27 104 729 2808 10816 3 38 168 1444 6384 28224 4 19 82 361 1558 6724 5 43 188 1849 8084 35344 6 26 101 676 2626 10201 7 59 207 3481 12213 42849 8 37 141 1369 5217 19881 9 29 106 841 3074 11236 10 45 172 2025 7740 29584 Sums 337 1323 12971 50460 197775

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60 50 40 30 20 10 225 200 175 150 125 100 75 50 x y Scatterplot of y vs x Notes 7.
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