568 CHAPTER 10 Correlation and Regression 2. Super Bowl and R2 Let x represent years coded as 1, 2, 3, . . . for years starting in 1980, and let y represent the numbers of points scored in each annual Super Bowl beginning in 1980. Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of R2 for the different models: linear: 0.008; quadratic: 0.023; logarithmic: 0.0004; exponential: 0.027; power: 0.007. Based on these results, which model is best? Is the best model a good model? What do the results suggest about predicting the number of points scored in a future Super Bowl game? 3. Interpreting R2 In Exercise 2, the exponential model results in R2 = 0.027. Identify the percentage of the variation in Super Bowl points that can be explained by the exponential model relating the variable of year and the variable of points scored. (Hint: See Example 2.) What does the result suggest about the usefulness of the exponential model? 4. Interpreting a Graph The accompanying graph plots the numbers of points scored in each Super Bowl from the first Super Bowl in 1967 (coded as year 1) to the last Super Bowl at the time of this writing. The graph of the quadratic equation that best fits the data is also shown in red. What feature of the graph justifies the value of R2 = 0.205 for the quadratic model? Finding the Best Model. In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models. 5. Landing on the Moon When the Apollo spacecraft landed on the Moon, the rocket engine would typically cut off at about 1.3 meters above the surface so that hot gases and dust and other surface materials would not cause damage. The landing module was in freefall starting at about 1 meter above the surface. The table below lists the time t (seconds) after being dropped and the distance d (meters) travelled by an object dropped near the surface of the Moon. t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 d 0.008 0.032 0.072 0.128 0.200 0.288 0.392 0.512 6. Dirt Cheap The Cherry Hill Construction company in Branford, CT sells screened topsoil by the “yard,” which is actually a cubic yard. Let the variable x be the length (yd) of each side of a cube of screened topsoil. The table below lists the values of x along with the corresponding cost (dollars). x 1 2 3 4 5 6 7 8 Cost 22 176 594 1408 2750 4752 7646 11,264 7. CD Yields The table lists the value y (in dollars) of $1000 deposited in a certificate of deposit at Bank of New York (based on rates currently in effect). Year 1 2 3 4 5 6 10 Value 1010.00 1020.10 1030.30 1040.60 1051.01 1061.52 1104.62
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