VCU STAT 210 - Lecture15 (40 pages)

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Lecture15



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Lecture15

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Pages:
40
School:
Virginia Commonwealth University
Course:
Stat 210 - Basic Practice of Statistics
Basic Practice of Statistics Documents
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STAT 210 Lecture 15 Describing Relationships Between Variables September 29 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 130 through 137 Relevant problems V 3 c V 4 c V 7 b V 9 d and V 10 c Recommended problems V 3 c V 7 b V 9 d and V 10 c Additional Reading and Examples Read pages 120 and 121 Top Hat 2 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 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 Example 26 x y 6 20 0 14 25 16 28 18 10 8 15 31 10 16 28 20 40 25 12 15 145 Sx x2 36 400 0 196 625 256 784 324 100 64 212 Sy y2 225 961 100 256 784 400 1600 625 144 225 2785 S x2 xy 90 620 0 224 700 320 1120 450 120 120 5320 S y2 3764 S xy Correlation Coefficient Sxx S x2 S x 2 n Syy S y2 S y 2 n Sxy S xy S x S y n TI 83 84 Calculator 1 First turn on diagnostics Hit 2ND and 0 bringing up the Catalog Scroll down to DiagnosticOn and hit Enter twice You only need to do this the first time 2 Enter the data into two lists say L1 and L2 3 Hit STAT then CALC and choose option 8 LinReg a bx 4 Enter the list containing the X data say L1 then comma then the list containing the Y data say L2 Hit Enter and r is the correlation coefficient Anscombe Data Page 111 As directed in class compute the correlation coefficient for the set of data you are assigned Determine the value to two decimal places You can and are encouraged to work together Data Set 1 x y Data Set 2 x y Data Set 3 x y Data Set 4 x y 10 8 13 9 11 14 6 4 12 7 5 10 8 13 9 11 14 6 4 12 7 5 10 8 13 9 11 14 6 4 12 7 5 8 8 8 8 8 8 8 19 8 8 8 8 04 6 95 7 58 8 81 8 33 9 96 7 24 4 26 10 84 4 82 5 68 9 14 8 14 8 74 8 77 9 26 8 10 6 13 3 10 9 13 7 26 4 74 7 46 6 77 12 74 7 11 7 81 8 84 6 08 5 39 8 15 6 42 5 73 6 58 5 76 7 71 8 84 8 47 7 04 5 25 12 50 5 56 7 91 6 89 Data Set 1 y Anscombe data set1 12 10 8 6 4 2 0 0 5 10 x 15 Data Set 2 Data Set 3 Data Set 4 Top Hat 2 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 Regression Line Y X Regression Line Y X Regression Line Y X Equation of a Line Y intercept slope X The intercept is the predicted value of Y when x 0 Hence when 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 Hence if x increases by 1 unit the predicted y increases or decreases by slope units Equation of a Line Y intercept slope X where the intercept is the predicted value of Y when X 0 and the slope is the amount that Y changes increases or decreases when X is increased by one unit Example weight in pounds 6 10 age in years intercept 6 when a child is 0 years old the child is predicted to weigh 6 pounds slope 10 if a child s age increases by one year then his or her weight is predicted to increase by 10 pounds weight increases 10 pounds each year Prediction Equation Once we determine the intercept and the slope we can use the line Y intercept slope X to predict values of Y given values of X The prediction equation is Y intercept slope X Residual y the observed value of the dependent variable y the predicted value of the dependent variable y y is called a residual and our goal is to make the residuals as small as possible Method of Least Squares Determine values of the intercept and slope such that the sum of the squared residuals is minimized minimize S Y Y 2 which is equivalent to minimize S Y intercept slope X 2 Least Squares Estimates slope Sxy r sy Sxx sx intercept y slope x where Sxx and Sxy are defined on page 102 and r is the correlation between X and Y Again x and y are the means of the X and Y data respectively and s x and sy are the standard deviations of the X and Y data respectively Example 27 From example 26 Sxx 682 5 Syy 825 6 x S x 145 10 14 5 n y S y 212 10 21 2 n Sxy 690 Example 27 slope Sxy Sxx Example 27 slope Sxy Sxx 690 682 5 1 011 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 Example 27 intercept y slope x 21 2 1 011 14 5 21 2 14 66 6 54 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 Example 27 Regression line Y intercept slope X 6 54 1 011 X TI 83 84 Calculator See pages 129 and 130 for instructions on using a calculator to compute the regression line intercept and slope 1 Enter the data into two lists say L1 and L2 2 Hit STAT then CALC and choose option 8 LinReg a bx 3 Enter the list containing the X data say L1 then comma then the list containing the Y data say L2 Hit Enter r is the correlation coefficient a intercept 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 …


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