Slide 1Test 2 ResultsPractice ProblemsAdditional Reading and ExamplesTest 3Slide 6Describing RelationshipsDescribing RelationshipsC. Regression LineEquation of a LineLeast Squares EstimatesTI-83/84 CalculatorMotivating ExampleMotivating ExampleMotivating ExampleMotivating ExamplePredictionExtrapolationExtrapolationExample 28Example 28Example 28Example 28Example 28Example 28Example 28ResidualResidualExampleResidual PlotIdeal Residual PlotPatterned Residual PlotUnequal VariabilityExample 29Slide 35Example 29OutlierOutlierOutlierInfluential ObservationInfluential ObservationInfluential ObservationInfluential ObservationInfluential ObservationInfluential ObservationInfluential ObservationCoefficient of DeterminationCoefficient of DeterminationCoefficient of DeterminationExample 30Example 30Example 30Slide 53STAT 210Lecture 16Regression LineOctober 3, 2016Test 2 ResultsMean: 82.0 Median: 86Max: 100 Min: 37 n: 142Score Frequency Relative Frequency90’s 53 37.32%80’s 38 26.76%70’s 27 19.01%60’s 13 9.15% <60 11 7.75%Practice ProblemsPages 130 through 137Relevant 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 through 121Test 3Thursday, October 6Questions for the first 10 minutes, then test – papers due promptly at the end of class!Covers Chapter 5 (pages 99 – 129)Combination of multiple choice questions and written/short answer questions and problems.Formulas provided; Bring a calculator!Practice Tests and Formula Sheet on Blackboard.ClickerDescribing RelationshipsWe want to use the independent or explanatory variable X to predict the dependent or response variable Y.ClickerDescribing 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 = 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.Least Squares Estimatesslope = Sxy = r sy Sxx sxintercept = y - slope x where Sxx and Sxy are defined on pages 106 - 107 and r is the correlation between X and Y. Again x and y are the means of the X and Y data, respectively, and sx and sy are the standard deviations of the X and Y data, respectively.TI-83/84 CalculatorSee pages 129 and 130 for instructions on using a calculator to compute the regression line (intercept and slope).Follow the same steps as for finding r. On the output screen, a = intercept and b = slope.Motivating ExampleWatching television also often means watching or dealing with commercials, and of interest is to describe the relationship between the number of hours of television watched per day and the number of commercials watched.The goal is to use X = number of hours of television watched to predict Y = the number of commercials watched.Motivating ExampleThe scatterplot on the next slide depicts the relationship for a random sample of 23 people. The regression line isY = -0.8 + 3.75*XInterpret what the intercept and slope values imply about the relationship.Clicker2Motivating Example0 2 4 6 8 10 12051015202530354045Hours of TelevisionN u m b e r o f C o m m e r c ia lsY = -0.8 + 3.75*XMotivating ExampleThe scatterplot on the next slide depicts the relationship for a random sample of 23 people. The regression line isY = -0.8 + 3.75*XIntercept: If the number of hours of television watched equals 0, then the predicted number of commercials watched is -0.8.Slope: If the number of hours of television watched increases by 1 hour, then the predicted number of commercials watched increases by 3.75.PredictionWe 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 35 year old is extrapolation: weight = 6 + 10(35) = 356 pounds.ExtrapolationExample: 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 = 16.65 ^^Example 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 = 26.76^^Example 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 = 208.74^^Example 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
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