After the oil crisis in 1973, the average fuel efficiency, E, of cars increased until

Table A.5 gives the gross world product, G, which measures global output of goods and services.3 If t is in years since 1950, the regression line for these data is G = 3.543 + 0.734t. (a) Plot the data and the regression line on the same axes. Does the line fit the data well? (b) Interpret the slope of the line in terms of gross world product. (c) Use the regression line to estimate gross world product in 2005 and in 2020. Comment on your confidence in the two predictions. Table A.5 G, in trillions of 1999 dollars Year 1950 1960 1970 1980 1990 2000 G 6.4 10.0 16.3 23.6 31.9 43.2

Table A.6 shows worldwide cigarette production as a function of t, the number of years since 1950.4 (a) Find the regression line for this data. (b) Use the regression line to estimate world cigarette production in the year 2010. (c) Interpret the slope of the line in terms of cigarette production. (d) Plot the data and the regression line on the same axes. Does the line fit the data well?

Table A.7 shows the US Gross National Product (GNP).5 (a) Plot GNP against years since 1970. Does a line fit the data well? (b) Find the regression line and graph it with the data. (c) Use the regression line to estimate the GNP in 1985 and in 2020. Which estimate do you have more confidence in? Why? Table A.7 GNP in 2003 dollars Year 1970 1980 1990 2000 GNP (billions) 1045 2824 5838 9856

The acidity of a solution is measured by its pH, with lower pH values indicating more acidity. A study of acid rain was undertaken in Colorado between 1975 and 1978, in which the acidity of rain was measured for 150 consecutive weeks.6 The data followed a generally linear pattern. If P is the pH and t is the number of weeks into the study, the regression line was determined to be P = 5.43 0.0053t. (a) Is the pH level increasing or decreasing over the period of the study? What does this tell you about the level of acidity in the rain? (b) According to the line, what was the pH at the beginning of the study? At the end of the study (t = 150)? (c) What is the slope of the regression line? Explain what this slope is telling you about the pH.

In a 1977 study7 of 21 of the best American female runners, researchers measured the average stride rate, S, at different speeds, v. The data are given in Table A.8. (a) Find the regression line for these data, using stride rate as the dependent variable. (b) Plot the regression line and the data on the same axes. Does the line fit the data well? (c) Use the regression line to predict the stride rate when the speed is 18 ft/sec and when the speed is 10 ft/sec. Which prediction do you have more confidence in? Why? Table A.8 Stride rate, S, in steps/sec, and speed, v, in ft/sec v 15.86 16.88 17.50 18.62 19.97 21.06 22.11 S 3.05 3.12 3.17 3.25 3.36 3.46 3.55

Table A.9 shows the atmospheric concentration of carbon dioxide, CO2 (in parts per million, ppm), at the Mauna Loa Observatory in Hawaii.8 (a) Find the average rate of change of the concentration of carbon dioxide between 1980 and 2000. Give units and interpret your answer in terms of carbon dioxide. (b) Plot the data, and find the regression line for carbon dioxide concentration against years since 1980. Use the regression line to predict the concentration of carbon dioxide in the atmosphere in the year 2020. Table A.9 Year 1980 1985 1990 1995 2000 CO2 349.6 346.7 354.4 360.8 368.9

In Problem 6, carbon dioxide concentration was modeled as a linear function of time. However, if we include data for carbon dioxide concentration from as far back as 1900, the data appear to be more exponential than linear. (They looked linear in Problem 6 because we were only looking at a small piece of the graph.) If C is the CO2 concentration in ppm and t is in years since 1900, an exponential regression function to fit the data is C = 272.27(1.0026)t . (a) What is the annual percent growth rate during this period? Interpret this rate in terms of CO2 concentration. (b) What CO2 concentration is given by the model for 1900? For 1980? Compare the 1980 estimate to the actual value in Table A.9.

(a) Fit an exponential function to the population data in Table A.10. Plot the data and the exponential function on the same axes. (b) At approximately what percentage rate was the population growing between 1960 and 2000? (c) If the population continues to grow at the same percentage rate, what population is projected for 2020?

Table A.11 shows the public debt, D, of the US9 in billions of dollars, t years after 1998. (a) Plot the public debt against the number of years since 1998. (b) Does the data look more linear or more exponential? (c) Fit an exponential function to the data and graph it with the data. (d) What annual percentage growth rate does the exponential model show? (e) Do you expect this model to give accurate predictions beyond 2004? Explain. Table A.11 t 0 1 2 3 4 5 6 D 5526 5656 5674 5808 6228 6783 7379

A company collects the data in Table A.12. Find the regression line and interpret its slope. Sketch the data and the line. What is the correlation coefficient? Why is the value you get reasonable? Table A.12 Cost to produce various quantities of a product q (quantity in units) 25 50 75 100 125 C (cost in dollars) 500 625 689 742 893

Match the r values with scatter plots in Figure A.11. r = 0.98, r= 0.5, r= 0.25, r = 0, r= 0.7, r= 1. 2 4 6 8 10 2 4 6 8 10 x (a) y 2 4 6 8 10 2 4 6 8 10 x (b) y 2 4 6 8 10 2 4 6 8 10 x (c) y 2 4 6 8 10 2 4 6 8 10 x (d) y 2 4 6 8 10 2 4 6 8 10 x (e) y 2 4 6 8 10 2 4 6 8 10 x (f) y Figure A.11

Table A.13 shows the number of cars, N, in millions in the US10 t years after 1940. (a) Plot the data, with number of passenger cars as the dependent variable. (b) Does a linear or exponential model appear to fit the data better? (c) Use a linear model first: Find the regression line for these data. Graph it with the data. Use the regression line to predict the number of passenger cars in the year 2010 (t = 70). (d) Interpret the slope of the regression line found in part (c) in terms of passenger cars. (e) Now use an exponential model: Find the exponential regression function for these data. Graph it with the data. Use the exponential function to predict the number of passenger cars in the year 2010 (t = 70). Compare your prediction with the prediction obtained from the linear model. (f) What annual percent growth rate in number of US passenger cars does your exponential model show? Table A.13 Number of passenger cars, in millions t 0 10 20 30 40 50 60 N 27.5 40.3 61.7 89.2 121.6 133.7 133.6

Table A.14 gives the population of the world in billions. (a) Plot these data. Does a linear or exponential model seem to fit the data best? (b) Find an exponential regression function. (c) What annual percent growth rate does the exponential function show? (d) Predict the population of the world in the year 2020 and in the year 2050. Comment on the relative confidence you have in these two estimates. Table A.14 World population in billions Year (since 1950) 0 10 20 30 40 50 58 Population (bn) 2.6 3.0 3.7 4.5 5.3 6.1 6.7

In 1969, all field goal attempts in the National Football League and American Football League were analyzed. See Table A.15. (The data has been summarized: all attempts between 10 and 19 yards from the goal post are listed as 14.5 yards out, etc. ) (a) Graph the data, with success rate as the dependent variable. Discuss whether a linear or an exponential model fits best. (b) Find the linear regression function; graph it with the data. Interpret the slope of the regression line in terms of football. (c) Find the exponential regression function; graph it with the data. What success rate does this function predict from a distance of 50 yards? (d) Using the graphs in parts (b) and (c), decide which model seems to fit the data best. Table A.15 Successful fraction of field goal attempts Distance from goal, x yards 14.5 24.5 34.5 44.5 52.0 Fraction successful, Y 0.90 0.75 0.54 0.29 0.15

Table A.16 shows the number of Japanese cars imported into the US.11 (a) Plot the number of Japanese cars imported against the number of years since 1964. (b) Does the data look more linear or more exponential? (c) Fit an exponential function to the data and graph it with the data. (d) What annual percentage growth rate does the exponential model show? (e) Do you expect this model to give accurate predictions beyond 1971? Explain. Table A.16 Imported Japanese cars, 19641971 Year since 1964 0 1 2 3 4 5 6 7 Cars (thousands) 16 24 56 70 170 260 381 704

Figure A.12 shows oil production in the Middle East.12 If you were to model this function with a polynomial, what degree would you choose? Would the leading coefficient be positive or negative? 1970 1980 1990 0 500 1000 1500 million tons of oil year Figure A.12

After the oil crisis in 1973, the average fuel efficiency, E, of cars increased until the early 1990s, when it started to decrease again. (a) Plot the data 13 in Table A.17, using t in years since 1975. If you were to fit a quadratic polynomial to the data, what would be the sign of the leading coefficient? (b) Fit a quadratic polynomial and plot it with the data. Table A.17 Year 1975 1980 1985 1990 1995 2000 E, mpg 13.1 19.2 21.3 21.5 21.1 20.7

Table A.18 gives the area of rain forest destroyed for agriculture and development.14 (a) Plot these data. (b) Are the data increasing or decreasing? Concave up or concave down? In each case, interpret your answer in terms of rain forest. (c) Use a calculator or computer to fit a logarithmic function to this data. Plot this function on the axes in part (a). (d) Use the curve you found in part (c) to predict the area of rain forest destroyed in 2010. Table A.18 Destruction of rain forest x (year) 1960 1970 1980 1988 y (million hectares) 2.21 3.79 4.92 5.77

World solar power, S, inmegawatts; t in years since 1990 t 0 1 2 3 4 5 6 7 S 47 55 58 60 69 78 89 126 t 8 9 10 11 12 13 14 15 S 153 201 277 386 547 748 1194 1782

Nuclear warheads, N, in thousands; t in years since 1960 t 0 5 10 15 20 25 30 35 40 45 N 22 38 39 48 55 65 59 40 33 28

Carbon dioxide, C, in ppm; t in years since 1970 t 0 5 10 15 20 25 30 35 C 326 331 339 346 354 361 369 380

For each graph in Figure A.13, decide whether the best fit for the data appears to be a linear function, an exponential function, or a polynomial.