# VCU STAT 210 - Lecture16 (59 pages)

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## Lecture16

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- Pages:
- 59
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 16 Regression Line October 2 2017 Test 3 Friday October 6 Covers chapter 5 pages 99 138 Combination of multiple choice questions and short answer questions and problems Formulas provided please bring calculator and writing instrument Practice Problems Pages 128 through 134 Relevant 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 9 Additional Reading and Examples Read pages 120 and 121 Top Hat Motivating Example Students 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 Relationships We want to use the independent or explanatory variable X to predict the dependent or response variable Y Top Hat Describing Relationships To describe the relationship between two variables we must describe the direction form and strength of the relationship A scatterplot and the correlation coefficient are two statistical tools that can be used to help describe the relationship C Regression Line Now 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 Line Y 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 27 slope Sxy Sxx 690 682 5 1 011 This 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 27 intercept y slope x 21 2 1 011 14 5 21 2 14 66 6 54 This implies that if 0 ads are run X 0 then the dealer is predicted to sell 6 54 cars Motivating Example A 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 Example A 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 studying Y grade on test Motivating Example X time spent studying Y grade on a test Regression line intercept slope Predicted grade on test 36 10 time spent studying Top Hat 2 Prediction We 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 age At age 4 we predict weight 6 10 4 6 40 46 pounds Extrapolation When 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 Extrapolation When 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 28 Y 6 54 1 011 X x 10 y 6 54 1 011 10 6 54 10 11 16 65 Example 28 Y 6 54 1 011 X x 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 28 Y 6 54 1 011 X x 20 y 6 54 1 011 20 6 54 20 22 26 76 Example 28 Y 6 54 1 011 X X 20 Y 6 54 1 011 20 6 54 20 22 26 76 If the dealer runs 20 ads he can expect to sell between 26 and 27 cars Example 28 Y 6 54 1 011 X x 200 y 6 54 1 011 200 6 54 202 2 208 74 Example 28 Y 6 54 1 011 X x 200 y 6 54 1 011 200 6 54 202 2 208 74 Run 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 28 Number of cars sold 40 36 32 28 24 20 16 12 Intercept 8 6 54 4 0 5 10 15 20 25 30 35 Number of ads run Residual The difference between an observed dependent variable Y value and a predicted dependent variable value residual y y This is the vertical deviation of a data point from the regression line Residual Y X Example In 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 Plot The residuals can be used to analyze the quality and usefulness of the regression line Residual Plot 1 Compute the residual for each observation Residual Plot 1 Compute the residual for each observation 2 Create a scatterplot with the independent variable X on the horizontal axis and the residuals on the vertical axis This is called a residual plot Ideal Residual Plot Points are randomly scattered around 0 with no obvious pattern Patterned Residual Plot A residual pattern that reveals a pattern like that below indicates that a linear relationship may not exist but instead the relationship may be quadratic Example 29 Ads run 6 20 0 14 25 16 28 18 10 8 Cars sold y 6 54 1 011 x 15 31 10 16 28 20 40 25 12 15 12 606 26 760 6 540 20 694 31 815 22 716 34 848 24 738 16 650 14 628 Residual 2 394 4 240 3 460 4 694 3 815 2 716 5 152 …

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