# VCU STAT 210 - Lecture16(3) (2) (53 pages)

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## Lecture16(3) (2)

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## Lecture16(3) (2)

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Pages:
53
School:
Virginia Commonwealth University
Course:
Stat 210 - Basic Practice of Statistics
##### Basic Practice of Statistics Documents
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STAT 210 Lecture 16 Regression Line October 3 2016 Test 2 Results Mean 82 0 Max 100 Score 90 s 80 s 70 s 60 s 60 Median 86 Min 37 n 142 Frequency Relative Frequency 53 37 32 38 26 76 27 19 01 13 9 15 11 7 75 Practice Problems Pages 130 through 137 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 through 121 Test 3 Thursday October 6 Questions 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 Clicker Describing Relationships We want to use the independent or explanatory variable X to predict the dependent or response variable Y Clicker 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 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 Estimates slope Sxy r sy Sxx sx intercept 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 Calculator See 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 Example Watching 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 Example The scatterplot on the next slide depicts the relationship for a random sample of 23 people The regression line is Y 0 8 3 75 X Interpret what the intercept and slope values imply about the relationship Clicker2 Motivating Example N u m b e r o f C o m m e rc ia ls Y 0 8 3 75 X 45 40 35 30 25 20 15 10 5 0 0 2 4 6 Hours of Television 8 10 12 Motivating Example The scatterplot on the next slide depicts the relationship for a random sample of 23 people The regression line is Y 0 8 3 75 X Intercept 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 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 35 year old is extrapolation weight 6 10 35 356 pounds Extrapolation 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 when x 20 ads were run y 31 cars were sold In example 28 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 Ideal Residual Plot Points are randomly scattered around 0 with no obvious pattern 0 X Patterned Residual Plot A residual plot that reveals a pattern like that below indicates that a linear relationship may not exist but instead the relationship may be quadratic 0 Unequal Variability 0 Example 29 6 4 2 X 0 0 2 4 6 5 10 …

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