1/20/2016 Chapter 3 Descriptive Study of Bivariate Data Ananda Manage, PhD Associate Professor of Statistics Department of Mathematics and Statistics Sam Houston State University Figure, Temperature and Ozone P 85 Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. 3.1 Summarization of Bivariate Categorical Data Table 1, Cross-Tabulated Frequency Counts, p. 86. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Table 2, Relative Frequencies for the Data of Table 1, p. 87. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Simpson’s Paradox Table 3, Relative Frequencies by Group, p. 87. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Does this appear to be a gender bias? Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . 1 1/20/2016 Table 5, Admission Rates by Department, p. 88. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. 3.3 Scatter Diagram of Bivariate Measurement Data Figure 1, Scatter diagram of applicants' scores, p. 94. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Figure 2, Scatter diagrams; A = Lake Apopka; B = Lake Woodruff, p. 95, 96. Data, p. 95. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . 2 1/20/2016 3.4 The Correlation Coefficient Box, Sample Correlation Coefficient, p. 98. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Box, Calculation Formula for the Sample Correlation Coefficient, p. 99. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Table 8, Calculation of r, p. 99. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Table 9, Alternate Calculation of r, p. 100. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. 3 1/20/2016 An observed correlation between two variables may be spurious. That is, it may be caused by the influence of a third variable. Figure 4, r is not appropriate – samples from two populations, p. 101. Correlation and Causation:- Spurious correlation can be caused by lurking variables Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Figure 6, Scatter diagram pattern has strong relation to year, p. 102. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Figure 8, Scatter diagrams (Problem 3.21), p. 105. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Box, Lurking Variables, p. 103. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . Figure 9, Scatter diagram (Problem 3.22), p. 105. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. . 4 1/20/2016 2.5 Linear Regression Table 11, Data of Concentration x and Drying Time y (in minutes) and the Basic Calculations, p. 109. Figure 10, The line yˆ ˆ ˆ , p. 108 0 1 Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. 28 Coefficient of Determination r2 Fitted Line The coefficient of determination r2 measures the goodness of fit of the regression equation to the data. We interpret r2 as the proportion of the variability in y that is accounted for by the linear relationship between y and x. The values that r2 can take are 0 ≤ r2 ≤ 1. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Regression Example STAT/MATH 3379 Weight of animals can be estimated using their heart girth. Following data set is from a sample of six cows. Weights and heart girth are given in pounds and inches respectively. Model Summary Model 1 Cow Weight Heart Girth 1 641 214 2 633 215 3 651 216 4 666 217 5 688 219 6 680 221 R .898a R Square .806 Adjusted R Std. Error of the Square Estimate .758 10.76225 a. Predictors: (Constant), HeartGirth Coefficientsa Standardized (a) (b) (c) (d) (e) Construct a scatterplot. Find the correlation coefficient. Find the regression line. Interpret coefficient b. Interpret coefficient of determination. Unstandardized Coefficients Model 1 B Std. Error (Constant) -974.049 400.543 HeartGirth 7.529 1.846 Coefficients Beta t .898 Sig. -2.432 .072 4.079 .015 a. Dependent Variable: Weight Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Statistics, 7/E by Johnson and Bhattacharyya Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. 5