The correlation coefficient measures the strength of the linear association between two variables. … the correlation coefficient and the standard error of the estimate of inversely related. This is because the larger the correlation coefficient, the smaller the standard of error, or scatter. As the strength of the linear relationship between two variables increases (correlation coefficient), the scatter decreases (the standard error of the estimate decreases). We also noted that the square of the correlation coefficient is the coefficient of determination. IN SUM, regression analysis provides two statistics to evaluate the predictive ability of a regression equation: • The standard error of the estimate • The coefficient of determination 13.8—INTERVAL ESTIMATES OF PREDICTION The standard error of the estimate and the coefficient of determination are two statistics that provide an overall evaluation of the ability of a regression equation to predict a dependent variable. Assumptions Underlying Linear Regression • For each value of X, there are corresponding Y values. These Y values follow the normal distribution. • The means of these normal distributions lie on the regression line. • The standard deviations of these normal distributions are all the same. The best estimate we have of this common standard deviation is the standard error of estimate. • The Y values are statistically independent. This means that in selecting a sample, a particular X does not depend on any other value of X. whattttt?????? Recall from Chapter 7 that if the values follow a normal distribution, then the mean plus or minus one standard deviation will encompass 68 percent of the observations, the mean plus or minus two standard deviations will encompass 95 percent of the observations, and the mean plus or minus three standard deviations will encompass virtually all of the observations. CONSTRUCTING CONFIDENCE AND PREDICTION INTERVALS A confidence interval refers to all cases with a given value of X. A prediction interval refers to a particular case for a given value of X. the prediction interval will always be wider because of the extra 1 under the radical in the second equation. 13.9 TRANSFORMING DATA
The correlation coefficient describes the strength of the linear relationship between two variables. It could be that two variables are closely related, but their relationship is not linear. BOOK CHAPTER SUMMARY • A scatter diagram is a graphic tool to portray the relationship between two variables. o The dependent variable is scaled on the Y axis and is the variable being estimated. o The independent variable is scaled on the X axis and is the variable used as the predictor. • The correlation coefficient measures the strength of the linear association between two variables. o Both variables must be at least the interval scale of measurement o The correlation coefficient can range form –1.00 to +1.00 o If the correlation coefficient between two variables is 0, there is no association between them.
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