Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Chapter 10 Correlation and Regression 10-1 Review and Preview 10-2 Correlation 10-3 Regression 10-4 Prediction Intervals and Variation 10-5 Multiple Regression 10-6 Nonlinear Regression

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Key Concept In this section we present a method for constructing a prediction interval, which is an interval estimate of a predicted value of y. (Interval estimates of parameters are confidence intervals, but interval estimates of variables are called prediction intervals.)

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Requirements For each fixed value of x, the corresponding sample values of y are normally distributed about the regression line, with the same variance.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Formulas For a fixed and known x 0, the prediction interval for an individual y value is: with margin of error:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Formulas The standard error estimate is: (It is suggested to use technology to get prediction intervals.)

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example If we use the 40 pairs of shoe lengths and heights, construct a 95% prediction interval for the height, given that the shoe print length is 29.0 cm. Recall (found using technology, data in Appendix B):

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued The 95% prediction interval is 162 cm < y < 186 cm. This is a large range of values, so the single shoe print doesn’t give us very good information about a someone’s height.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Explained and Unexplained Variation Assume the following: There is sufficient evidence of a linear correlation. The equation of the line is ŷ = 3 + 2x The mean of the y-values is 9. One of the pairs of sample data is x = 5 and y = 19. The point (5,13) is on the regression line.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Explained and Unexplained Variation The figure shows (5,13) lies on the regression line, but (5,19) does not. We arrive at:

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. (total deviation) = (explained deviation) + (unexplained deviation) = + (total variation) = (explained variation) + (unexplained variation) Relationships = +

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Definition The coefficient of determination is the amount of the variation in y that is explained by the regression line. The value of r 2 is the proportion of the variation in y that is explained by the linear relationship between x and y.

Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example If we use the 40 paired shoe lengths and heights from the data in Appendix B, we find the linear correlation coefficient to be r = Then, the coefficient of determination is r 2 = = We conclude that 66.1% of the total variation in height can be explained by shoe print length, and the other 33.9% cannot be explained by shoe print length.