548 CHAPTER 10 Correlation and Regression Coefficient of Determination In Section 10-1 we saw that the linear correlation coefficient r can be used to find the proportion of the total variation in y that can be explained by the linear correlation. This statement was made in Section 10-1: The value of r2 is the proportion of the variation in y that is explained by the linear relationship between x and y. This statement about the explained variation is formalized with the following definition. DEFINITION The coefficient of determination is the proportion of the variation in y that is explained by the regression line. It is computed as r 2 = explained variation total variation We can compute r2 by using the formula given in the preceding definition (along with Formula 10-7), or we can simply square the linear correlation coefficient r. Go with squaring r. If we use the nine pairs of jackpot>tickets data from Table 10-1, we find that the linear correlation coefficient is r = 0.947. Find the coefficient of determination. Also, find the percentage of the total variation in y (tickets) that can be explained by the linear correlation between the jackpot amount and number of tickets sold. CP YOUR TURN. Do Exercise 5 “Times of Taxi Rides and Tips.” EXAMPLE 2 Jackpot , Tickets Data: Finding the Coefficient of Determination SOLUTION With r = 0.947 the coefficient of determination is r2 = 0.897. INTERPRETATION Because r2 is the proportion of total variation that can be explained, we conclude that 89.7% of the total variation in tickets sold can be explained by the amount of the jackpot, and the other 10.3% cannot be explained by the jackpot. The other 10.3% might be explained by some other factors and>or random variation. FORMULA 10-7 1total variation2 = 1explained variation2 + 1unexplained variation2 Σ1y - y22 = Σ1yn - y22 + Σ1y - yn22 squares of the total deviation values, the explained variation is the sum of the squares of the explained deviation values, and the unexplained variation is the sum of the squares of the unexplained deviation values.
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