Slide 1Test 3Practice ProblemsAdditional Reading and ExamplesSlide 5Motivating ExampleDescribing RelationshipsDescribing RelationshipsC. Regression LineEquation of a LineExample 27Example 27Motivating ExampleMotivating ExampleMotivating ExamplePredictionExtrapolationExtrapolationExample 28Example 28Example 28Example 28Example 28Example 28Example 28ResidualResidualExampleResidual PlotResidual PlotResidual PlotIdeal Residual PlotPatterned Residual PlotExample 29Example 29Example 29Example 29Example 29Example 29Example 29OutlierOutlierOutlierInfluential ObservationInfluential ObservationSlide 46Coefficient of DeterminationCoefficient of DeterminationCoefficient of DeterminationExample 30Example 30Example 30Example 30Example 30Example 30TI-83/84 CalculatorMotivating ExampleMotivating ExampleMotivating ExampleSTAT 210Lecture 16Regression LineOctober 2, 2017Test 3Friday, October 6Covers chapter 5, pages 99 – 138Combination of multiple choice questions and short answer questions and problems.Formulas provided, please bring calculator and writing instrument.Practice ProblemsPages 128 through 134Relevant problems: V.2 (d), V.3 (b) and (d), V.4 (c) and (d), V.6 (b), (c) and (d), V.7, V.8 (b) and (c), V.9 and V.10.Recommended problems: V.7 and V.9Additional Reading and ExamplesRead pages 120 and 121Top HatMotivating ExampleStudents like to make good grades, and making good grades is usually associated with studying and learning. We will conduct a study to analyze the relationship between time spent studying and grade on a test.Describing RelationshipsWe want to use the independent or explanatory variable X to predict the dependent or response variable Y.Top HatDescribing RelationshipsTo describe the relationship between two variables we must describe the direction, form, and strengthof the relationship.A scatterplot and the correlation coefficient are twostatistical tools that can be used to help describe therelationship.C. Regression LineNow our goal is to determine the equation of the line that best models (explains) the relationship between X and Y. This is referred to as the regression line.Equation of a LineY = intercept + slope(X)The intercept is the predicted value of Y when x = 0. If x = 0, the predicted y is the intercept value.The slope is the amount that Y changes (increases or decreases) when X is increased by one unit. If x increases by 1 unit, the predicted y increases or decreases by slope units.Example 27slope = Sxy / Sxx = 690 / 682.5 = 1.011This implies that as the number of ads run increases by one ad, the predicted number of cars sold increases by 1.011 cars.Increases since positive.Example 27intercept = y - slope x = 21.2 - 1.011 (14.5)= 21.2 - 14.66= 6.54This implies that if 0 ads are run (X = 0), then the dealer is predicted to sell 6.54 cars.Motivating ExampleA study is created to evaluate the effect that time spent studying has on the grade earned on a test.In this scenario, what are the independent (X) and dependent (Y) variables?X = ???Y = ???Motivating ExampleA study is created to evaluate the effect that time spent studying has on the graded earned on a test.In this scenario, what are the independent (X) and dependent (Y) variables?X = time spent studyingY = grade on testMotivating ExampleX = time spent studyingY = grade on a testRegression line: intercept slopePredicted grade on test = 36 + 10 (time spent studying)Top Hat 2PredictionWe can predict the value of Y for any value of X simply by substituting the value of X into the regression equation.Example: weight = 6 + 10 * ageAt age = 4, we predict weight = 6 + 10 (4) = 6 +40 = 46 poundsExtrapolationWhen predicting, it is important that the value of X at which we want to predict falls within the range of the original X data. The regression line describes the linear relationship between X and Y only for the range of data that we have.Predicting outside the range of the original X data is called extrapolation and should be avoided.Example: If the data used to determine the regression equation weight = 6 + 10 * age is only for kids between the ages of 2 and 10 (X between 2 and 10), then predicting the weight of a 45 year old is extrapolation: weight = 6 + 10(45) = 456 pounds.ExtrapolationWhen predicting, it is important that the value of X at which we want to predict falls within the range of the original X data. The regression line describes the linear relationship between X and Y only for the range of data that we have.Predicting outside the range of the original X data is called extrapolation and should be avoided.Example: If the data used to determine the regression equation weight = 6 + 10 * age is only for kids between the ages of 2 and 10 (X between 2 and 10), then predicting the weight of a 90 year old is extrapolation: weight = 6 + 10(90) = 906 pounds.Example 28Y = 6.54 + 1.011 Xx = 10: y = 6.54 + 1.011 (10) = 6.54 + 10.11 = 16.65Example 28Y = 6.54 + 1.011 Xx = 10: y = 6.54 + 1.011 (10) = 6.54 + 10.11 = 16.65 If the dealer runs 10 ads, he can expect to sell between 16 and 17 cars.Example 28Y = 6.54 + 1.011 Xx = 20: y = 6.54 + 1.011 (20) = 6.54 + 20.22 = 26.76Example 28Y = 6.54 + 1.011 XX = 20: Y = 6.54 + 1.011 (20) = 6.54 + 20.22 = 26.76If the dealer runs 20 ads, he can expect to sell between 26 and 27 cars.Example 28Y = 6.54 + 1.011 Xx = 200: y = 6.54 + 1.011 (200) = 6.54 + 202.2 = 208.74Example 28Y = 6.54 + 1.011 Xx = 200: y = 6.54 + 1.011 (200) = 6.54 + 202.2 = 208.74Run 200 ads, expect to sell 208 or 209 cars. This is very unrealistic and is an example of extrapolation. They only have 125 cars on the lot.Example 280 5 10 15 20 25 30 35 Number of ads runNumber of cars sold 40 36 32 28 24 20 16 12 8 4Intercept= 6.54***ResidualThe difference between an observed dependent variable (Y) value and a predicted dependent variable value. residual = y - yThis is the vertical deviation of a data point from the regression line.ResidualXYExampleIn example 26 (page 103), when x = 20 ads were run, y = 31 cars were sold.In example 28 (page 112), the regression line predicts that y = 26.760 cars will be sold.Hence the residual is y - y = 31 - 26.760 = 4.240.Residual PlotThe residuals can be used to analyze the
View Full Document
Unlocking...