570 CHAPTER 10 Correlation and Regression 17. 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. 10-5 Beyond the Basics 16. Global Warming Listed below are mean annual temperatures (°C) of the earth for each decade, beginning with the decade of the 1880s. Find the best model and then predict the value for 2090–2099. Comment on the result. 13.819 13.692 13.741 13.788 13.906 14.016 14.052 13.983 13.938 14.014 14.264 14.396 14.636 Chapter Quick Quiz Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author. Cost of Dinner (dollars) 46.60 33.46 50.68 87.92 98.84 63.60 107.34 49.88 Tip (dollars) 7.50 5.50 5.00 8.08 17.00 12.00 16.00 7.00 1. Scatterplot Construct a scatterplot and comment on the pattern of points. 2. Conclusion The linear correlation coefficient r is found to be 0.846, the P-value is 0.008, and the critical values for a 0.05 significance level are {0.707. What should you conclude? 3.Fixed Percentage If a restaurant were to change its tipping policy so that a constant tip of 20% of the bill is added to the cost of the dinner, what would be the value of the linear correlation coefficient for the paired amounts of dinners>tips? 1971 1974 1978 1982 1985 1989 1993 1997 2000 2002 2003 2007 2011 2018 2.3 5 29 120 275 1180 3100 7500 42,000 220,000 410,000 789,000 2,600,000 19,200,000 a. Ignoring the listed data and 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? 18.Sum of Squares Criterion In addition to the value of R2, another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is y - yn (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7 on page 565. a. Find Σ1y - yn22, 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.
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