Bivariate Normal Data Analysis: LPGA 2008 Season Overview

Explore the analysis of bivariate normal data focusing on LPGA driving distance and fairway percent from the 2008 season. Learn how to compute confidence ellipses, estimated means, variance-covariance matrix, eigenvalues, eigenvectors, and plot insightful visualizations. Understand the method, set up the ellipse, rotate it, and add major and minor axes to analyze the data effectively.

• Bivariate Normal Data
• LPGA
• Confidence Ellipse
• Data Analysis
• 2008 Season

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1. Confidence Ellipse for Bivariate Normal Data LPGA Driving Distance and Fairway Percent 2008 Season

2. Overview of Method 1. Obtain Bivariate Data on n Units 2. Compute the Estimated Means and Variance-Covariance Matrix 3. Obtain the Eigenvalues and Eigenvectors of the Estimated Variance- Covariance Matrix 4. Choose the Confidence Coefficient for the Ellipsoid and obtain the critical value corresponding to it for the Chi-Square Distribution with df=2 5. Form a grid of angles from 0 to 2 radians (equivalently, 0 to 360 degrees) 6. Obtain the Base of the Ellipse scaling the cosine of the angle for X and the sine of the angle for Y 7. Rotate the ellipse 8. Center the Ellipse 9. Add Major and Minor Axes

3. Plot of Fairway % vs Drive Distance with Marginal Histograms

4. Plot of Fairway % vs Drive Distance with Marginal Densities

5. Sample Means, Variance-Covariance Matrix, and Eigenvalues/Eigenvectors 249.4 64.6 253.8 62.7 156 235.2 78.3 254.3 67.2 x x = x x x x 11 12 21 M 22 M 246.7108 67.5777 1 n = = = = = X x X'1 n M M n 155,1 x 155,2 x 156,1 156,2 89.0538 22.9350 22.9350 33.3357 1 1 n = S X' I J X n n 1 n S x x 'x Computing eigenvalues and eigenvectors of : x = = = = = S I Sx 1 1 x Sx x e e 0 1 2 1 2 2 2 1 2 x 'x 1 1 2 2 0.94129 0.33761 ) .05 = 0.33761 0.94129 0.9413 0.3376 0.3376 0.9413 = = = = = = e e P e e 97.27994 25.10957 1 2 1 2 1 2 ( = 2 2 Confidence Coefficient: 1- 0.95 5.991

6. Setting up and rotating the ellipse : 0 1 0 ( ) 1 /360 2 2 360 D = = = Obtain a set of angles from 0 to 360 degrees or 0 to 2 radians: or Note that R and EXCEL use Radians R ( ) 359 360 ( ) 2 359 /360 2 = 2 2 2 Obtain the base o ( ) 1 B = f the ellipsis with squared statistical distance c ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = = 2 2 cos 97.27994(5.991)cos 24.14134cos sin 25.10597(5.991)sin 12.26417sin c B c 1 2 2 ( ) ( ) 1 /360 ( ) ( ) 1 /360 24.14134cos 0 24.14134cos 2 12.26417sin 0 12.2641 7sin 2 24.14134 24.13766 0.00000 0.21404 ( ) ( ) = = B ( ) ( ) ( ) ( ) 24.14134cos 2 359 /360 12.26417sin 2 359 /360 24.13766 24.14134 0.21404 0.00000 ( ) ( ) 24.14134cos 2 12.26417sin 2 223.9870 223.9182 24.14134 24.13766 0.00000 0.21404 22.7239 22.7927 8.1504 7.9477 0.94129 0.33761 0.33761 0.94129 = = B P' R = BP' Rotate the base of the el lipsis by multiplying by : 22.6482 22.7239 24.13766 24.14134 0.21404 0.00000 8. 8.1504 3507 246.7108 246.7108 67.5777 67.5777 75.7281 75 .5254 = = + = x Y R x Y Add the the "Center Point" to each observation: Plot the second column of versus the first 246.7108 246.7108 67.5777 67.5777 224.0627 223.9870 75.9284 75.7281

7. Adding the Major and Minor Axes and Data Points Obtain the square root of the variance-covariance matrix: 0 0 = 97.27994 0 0 1 = = = 1/2 1/2 S P P' 25.10597 2 0.33761 0.94129 0.94129 0.33761 0.33761 0.94129 97.2799 0 0.94129 0.33761 9.2840 3.3299 1.6916 4.7164 = 1/2 S 0 25.0916 2 Create the following 2 matrices where + = ( 11 Place points at X ( 12 Place points at X is the squared statistical distance for the ellipse: c X + + + 1/2 11 2 1/2 12 2 1/2 21 2 1/2 22 2 223 269.4347 .9870 242.5703 250.8514 75.7281 56.0336 59.4273 79.1218 x S c x S c x S c x S c 1 1 2 2 = = = * * X Y 1/2 11 2 1/2 12 2 1/2 21 2 1/2 22 2 x S c x S c x S c x S c 1 1 2 2 ) ) ( ( ) * * * 21 * 21 Y Y , and , and join with a line (Major axis ) and join with a line (Minor axis) ) 11 * * * 22 * 22 Y X Y , and , 12 Add Data Points.

8. Fairway Percent (X2) versus Drive Distance (X1) with 95% Confidence Ellipse 85 80 75 70 Fairway Percent Series1 65 Series2 Series3 Series4 60 55 50 45 220 225 230 235 240 245 250 255 260 265 270 275 Drive Distance