Get answer: Finding the Best Model. In Exercises 516, construct a scatterplot and

Identifying a Model and R 2 A data set is obtained as follows: Squares of different sizes are used, and x represents the length (cm) of a side and y represents the corresponding area ( cm 2 ) of the square. What formula best describes the relationship between x and y? Which of the five models describes this relationship? What should be the value of R 2 ?

Super Bowl and R 2 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 Super Bowl from 1980. Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of R 2 for the different models: linear: 0.0185 quadratic: 0.103 logarithmic: 0.000557 exponential: 0.0253 power: 0.00264. 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?

Interpreting R 2 In Exercise 2, the quadratic model results in R 2 = 0.103 . Identify the percentage of the variation in Super Bowl points that can be explained by the quadratic 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 quadratic model?

Interpreting a Graph The accompanying graph plots the numbers of points scored in each Super Bowl from 1980 (coded as year 1) to the last Super Bowl at the time of this writing. The graph also shows the graph of the quadratic equation that best fits the data. What feature of the graph justifies the value of R 2 = 0.103 for the quadratic model?

Finding the Best Model. In Exercises 516, 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.Dropped Object The table lists the distance d (meters) above the ground for an object dropped in a vacuum from a height of 300 m. The time t (in sec) is the time after the object has been released. t 1 2 3 4 5 d 295.1 280.5 256.1 222.0 178.1

Finding the Best Model. In Exercises 516, 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.Landscape Material The table lists the cost y (in dollars) of purchasing a volume of red landscaping mulch from the Red Barn Farm and Nursery in Dutchess County, New York. The mulch is sold by the yard, which is actually a cubic yard, and the variable x is the length (yd) of each side of a cubic yard. x 1 1.5 2 2.5 3 Cost 35 118.13 280 546.88 945

Finding the Best Model. In Exercises 516, 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.CD Value The table lists the value y (in dollars) of $100 deposited in a certificate of deposit at Bank of New York. Year 1 2 3 4 5 20 Value 103 106.09 109.27 112.55 115.93 180.61

Finding the Best Model. In Exercises 516, 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.Sound Intensity The table lists intensities of sounds as multiples of a basic reference sound. A scale similar to the decibel scale is used to measure the sound intensity. Sound Intensity 316 500 750 2000 5000 Scale Value 25.0 27.0 28.75 33.0 37.0

Finding the Best Model. In Exercises 516, 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.Harry Potter Listed below are the amounts (millions of dollars) grossed during the first three weeks of the movie Harry Potter and the Half Blood Prince. Use the best model to predict the gross amount on the 22nd day of the movie's release. Is the result close to the actual amount of $2.2 million? Is there any characteristic of the data not revealed by the model? (Hint: The movie opened on a Wednesday.) 58.2 22.0 26.8 29.2 21.8 9.9 9.5 7.5 6.9 9.3 11.2 8.9 4.0 4.0 4.6 3.8 5.4 6.8 5.8 2.5 2.4

Finding the Best Model. In Exercises 516, 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.Deaths from Motor Vehicle Crashes Listed below are the numbers of deaths in the United States resulting from motor vehicle crashes. Use the best model to find the projected number of such deaths for the year 2020. Year 1975 1980 1985 1990 1995 2000 2005 2010 Deaths 44,525 51,091 43,825 44,599 41,817 41,945 43,443 32,708

Finding the Best Model. In Exercises 516, 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.. Richter Scale The table lists different amounts (metric tons) of the explosive TNT and the corresponding value on the Richter scale used for explosions of the TNT. TNT 2 10 15 50 100 500 Richter Scale 3.4 3.9 4.0 4.4 4.6 5.0

Finding the Best Model. In Exercises 516, 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.v

Finding the Best Model. In Exercises 516, 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.Bacteria Growth In a carefully controlled experiment, bacteria are allowed to grow for a week. The number of bacteria are recorded at the end of each day with these results: 20, 40, 80, 160, 320, 640, 1280.

Finding the Best Model. In Exercises 516, 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.. Sunspot Numbers Listed below in order by row are annual sunspot numbers beginning with 1980. Is the best model a good model? Carefully examine the scatterplot and identify the pattern of the points. Which of the models fits that pattern? 154.6 140.5 115.9 66.6 45.9 17.9 13.4 29.2 100.2 157.6 142.6 145.7 94.3 54.6 29.9 17.5 8.6 21.5 64.3 93.3 119.6 123.3 123.3 65.9 40.4 29.8 15.2 7.5 2.9 3.1 16.5

Finding the Best Model. In Exercises 516, 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.Stock Market Listed below in order by row are the annual high values of the Dow Jones Industrial Average for each year beginning with 1990. Use only the first 10 values (for 19901999) to find the best model and then predict the value for the year 2010. Is the predicted value close to the actual value of 11,655? 3000 3169 3413 3794 3978 5216 6561 8259 9374 11,568 11,401 11,350 10,635 10,454 10,855 10,941 12,464 14,198 13,338 10,606

Finding the Best Model. In Exercises 516, 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.. Stock Market Repeat the preceding exercise using all of the data values

Moore's Law In 1965, Intel cofounder Gordon Moore initiated what has since become known as Moore's law: the number of transistors per square inch on integrated circuits will double approximately every 18 months. In the table below, the first row lists different years and the second row lists the number of transistors (in thousands) for different years. 1971 1974 1978 1982 1985 1989 1993 1997 2000 2002 2003 2007 2011 2.3 5 29 120 275 1180 3100 7500 42,000 220,000 410,000 789,000 2,600,000 a. Assuming that Moore's law is correct and transistors per square inch double every 18 months, which mathematical model best describes this law: linear, quadratic, logarithmic, exponential, power? What specific function describes Moore's law? b. Which mathematical model best fits the listed sample data? c. Compare the results from parts (a) and (b). Does Moore's law appear to be working reasonably well?

Sum of Squares Criterion In addition to the value of R 2 , another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 103 that a residual is the difference between an observed y value and the value of y predicted from the model, which is denoted as y ^ . Better models have smaller sums of squares. Refer to the data in Table 107. a. Find ( y y ^ ) 2 , the sum of squares of the residuals resulting from the linear model. b. Find the sum of squares of residuals resulting from the quadratic model. c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.