10-2 Regression 533 Making Predictions Regression equations are often useful for predicting the value of one variable, given some specific value of the other variable. When making predictions, we should consider the following: 1. Bad Model: If the regression equation does not appear to be useful for making predictions, don’t use the regression equation for making predictions. For bad models, the best predicted value of a variable is simply its sample mean. However, the sample mean is not a good predicted value because it is the predicted value for any value of the other variable. 2. Good Model: Use the regression equation for predictions only if the graph of the regression line on the scatterplot confirms that the regression line fits the points reasonably well. 3. Correlation: Use the regression equation for predictions only if the linear correlation coefficient r indicates that there is a linear correlation between the two variables (as described in Section 10-1). 4. Scope: Use the regression line for predictions only if the data do not go much beyond the scope of the available sample data. (Predicting too far beyond the scope of the available sample data is called extrapolation, and it could result in bad predictions.) Figure 10-5 summarizes a strategy for predicting values of a variable y when given some value of x. Figure 10-5 shows that if the regression equation is a good model, then we substitute the value of x into the regression equation to find the predicted value of y. However, if the regression equation is not a good model, the best predicted value of y is simply y, the mean of the y values. (But if y is the best predicted value, it isn’t very good because it is the predicted value of y for any value of x.) Remember, this strategy applies to linear patterns of points in a scatterplot. If the scatterplot shows a pattern that is nonlinear (not a straight-line) pattern, other methods apply. Is the regression equation a good model? • The regression line graphed in the scatterplot shows that the line fits the points well. • r indicates that there is a linear correlation. • The prediction is not much beyond the scope of the available sample data. Strategy for Predicting Values of y Yes. The regression equation is a good model. No. The regression equation is not a good model. Substitute the given value of x into the regression equation y 5 b0 1 b1x Regardless of the value of x, the best predicted value of y is the value of y (the mean of the y values). ˆ FIGURE 10-5 Recommended Strategy for Predicting Values of y n nPostponing Death Several studies addressed the ability of people to postpone their death until after an important event. For example, sociologist David Phillips analyzed death rates of Jewish men who died near Passover, and he found that the death rate dropped dramatically in the week before Passover, but rose the week after. Other researchers of cancer patients concluded that there is “no pattern to support the concept that ‘death takes a holiday.’” (See “Holidays, Birthdays, and Postponement of Cancer Death,” by Young and Hade, Journal of the American Medical Association, Vol. 292, No. 24.) Based on records of 1.3 million deaths, this more recent study found no relationship between the time of death and Christmas, Thanksgiving, or the person’s birthday. The findings were disputed by David Phillips, who said that the study focused on cancer patients, but they are least likely to have psychosomatic effects.
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