1.Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let x be the weight of the car (in hundreds of pounds), and let y be the miles per gallon (mpg).
In the least-squares line 1? = 5 + 4x, what is the marginal change in 1? for each unit change in x? E Does prison really deter violent crime? Let x represent percent change in the rate of violent crime and y represent percent change in the rate of imprisonment in the general U.S. population. For 7 recent years, the following data have been obtained. X 6.0 5.9 4.2 5.2 6.2 6.5 11.1 y -1.5 -4.0 -7.4 -4.0 3.6 -0.1 -4.4 Complete parts (a) through (e), given Ex = 45.1, Ey = -17.8, Ex2 = 319.39, Ey? = 121.34, Exy = -111.65, and r & 0.0648. (a) Draw a scatter diagram displaying the data.Ex = (b) Verify the given sums Ex, Zy, Ex , Eye, Exy, and the value of the sample correlation coefficient r. (Round your value for r to four decimal places.) Ey = Ex2 = Ey= = Exy = X = (c) Find x, and y. Then find the equation of the least-squares line y = a + bx. (Round your answer to four decimal places.) Y = V = (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. 10 y 10 5 a 0 OA - 51 b -5 O 2 4 6 8 10 X 12 14 O 2 4 6 8 X 10 12 14 y 10 5 0 C -5 O 2 4 U X 6 8 10 14 O 2 4 6 8 X 12 14(e) Find the value of the coefficient of determination . What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for 2 to four decimal places Round your answers for the percentages to two decimal place.) 12 = explained % unexplained (f) Considering the values of r and r2, does it make sense to use the least-squares line for prediction? Explain your answer. O The correlation between the variables is so low that it makes sense to use the least-squares line for prediction. O The correlation between the variables is so high that it makes sense to use the least-squares line for prediction. O The correlation between the variables is so high that it does not make sense to use the least-squares line for prediction. O The correlation between the variables is so low that it does not make sense to use the least-squares line for prediction.We use the form y = a + bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost- free days per year in a state. A Minitab printout provides the following information. Predictor Coef SE Coef Constant 315.54 28. 31 11. 24 0 . 002 Elevation -28. 599 3. 511 -8.79 0. 003 S = 11.8603 R-Sq = 94.0$ Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation y = a + bx. (a) Use the printout to write the least-squares equation. + X (b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.) (c) The printout gives the value of the coefficient of determination /2. What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.) (d) What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? % What percentage is unexplained? %An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs). X 13 34 52 28 50 25 L 3 Complete parts (a) through (e), given Ex = 202, Ey = 27, Ex? = 7938, Ey? = 159, Exy = 1071, and r & 0.7844. (a) Draw a scatter diagram displaying the data.(b) Verify the given sums Ex, Ey, Ex , Ey , Exy, and the value of the sample correlation coefficient r. (Round your value for r to four decimal places.) Ex = Ey = Ex2 = Ey2 : Exy = (c) Find x, and y. Then find the equation of the least-squares line y = a + bx. (Round your answers to four decimal places.) X = V : (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. a 10 b 10 C 10 10 5 5. 5 5 10 20 30 40 50 60 AO 10 20 30 40 50 60 10 AO 20 30 40 50 60 AO 10 20 30 40 50 60 (e) Find the value of the coefficient of determination . What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for / to four decimal places. Round your answers for the percentages to two decimal place.) explained % unexplained (f) For a neighborhood with x = 31 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.) hundred jobsFuming because you are stuck in traffic? Roadway congestion is a costly item, both in time wasted and fuel wasted. Let x represent the average annual hours per person spent in traffic delays and let y represent the average annual gallons of fuel wasted per per in traffic delays. A random sample of eight cities showed the following data. x (hr) 29 22 38 19 y (gal) 49 3 20 32 55 15 34 39 25 9 (a) Draw a scatter diagram for the data. 65 45 Fill 40 35 30 25 No Solution 20 15 aPHO 10 15 20 25 30 35 40 WebAssign. Graphing Tool Verify that Ex = 153, Ex2 = 3805, Zy = 246, Zy2 = 9842, and Exy = 6076. Compute r. (Round your answer to four decimal places.)The data in part (a) represent average annual hours lost per person and average annual gallons of fuel wasted per person in traffic delays. Suppose that instead of using average data for different cities, you selected one person at random from each city and measured the annual number of hours lost x for that person and the annual gallons of fuel wasted y for the same person. x (hr) 20 4 20 44 16 26 2 37 y (gal) 64 8 14 52 24 35 4 70 (b) Compute x and y for both sets of data pairs and compare the averages. (Round your answers to four decimal places.) X Data 1 Data 2 Compute the sample standard deviations s, and s, for both sets of data pairs and compare the standard deviations. (Round your answers to four decimal places.) SX Sy Data 1 Data 2 In which set are the standard deviations for x and y larger? O The standard deviations for x and y are larger for the first set of data. O The standard deviations for x and y are larger for the second set of data. O The standard deviations for x and y are the same for both sets of data. Look at the defining formula for r. Why do smaller standard deviations s, and sy tend to increase the value of r? O Dividing by smaller numbers results in a larger value. O Multiplying by smaller numbers results in a larger value. O Multiplying by smaller numbers results in a smaller value. O Dividing by smaller numbers results in a smaller value.(c) Make a scatter diagram for the second set of data pairs. Graph Layers After you add an object to the graph you can use Graph Layers to view and edit its properties. 80 O Fill V . 6 8 8 8 8 8 6 8 8 No Solution aPHO 10 15 20 25 30 35 40 45 50 WebAssign. Graphing Tool Verify that Ex = 169, Ex2 = 5057, Zy = 271, Zy2 = 13,777, and Exy = 7772. Compute r. (Round your answer to four decimal places.) (d) Compare r from part (a) with r from part (b). Do the data for averages have a higher correlation coefficient than the data for individual measurements? O No, the data for averages do not have a higher correlation coefficient than the data for individual measurements. O Yes, the data for averages have a higher correlation coefficient than the data for individual measurements. List some reasons why you think hours lost per individual and fuel wasted per individual might vary more than the same quantities averaged over all the people in a city.Do heavier cars reallyr use more gasoline? Suppose a car is chosen at random. Let x be the weight of the car {in hundreds of pounds), and let 1/ be the miles per gallon (mpg). x 29 44 33 47 23 40 34 52 y 30 19 24 13 29 17 21 14 Complete parts (a) through (e), given Ex = 302, E]; = 16?, Ex2 = 12,064, 21/2 = 37'73, Exy = 5898, and r a 0.9312. {a) Draw a scatter diagram displaying the data. Ex (b) Verify the given sums Ex, Ey, Ex2, Ey2, Exy, and the value of the sample correlation coefficient r. (Round your value for r to four decimal places.) Ey = Ex2 = Ey2 Exy x (c) Find x, and y. Then find the equation of the least-squares line y = a + bx. (Round your answers to four decimal places.) V = V = (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. 50 50 40 50 . K 40 a 30 50 40 30 20 40 b 30 20 10 C 20 30 10 10 20 10 20 30 X O-10 40 50 60 10 10 20 30 @ 0-10 40 50 60 10 20 30 40 X DO-10, 50 60 10 20 30 X @ O-10 40 50 60 Round your answers for the percentages to two decimal place.) (e) Find the value of the coefficient of determination . What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for / to four decimal places explained % unexplained % mpg (f) Suppose a car weighs x = 35 (hundred pounds). What does the least-squares line forecast for y = miles per gallon? (Round your answer to two decimal places.)The following Minitab display gives information regarding the relationship between the body weight of a child (in kilograms) and the metabolic rate of the child (in 100 kcal/ 24 hr). Predictor Coef SE Coef P Constant 0. 8435 0. 4148 2.06 0 . 84 Weight 0. 38324 0. 02978 13.52 0.000 S = 0. 517508 R-Sq = 97.86 (a) Write out the least-squares equation. y = X (b) For each 1 kilogram increase in weight, how much does the metabolic rate of a child increase? (Use 5 decimal places.) (c) What is the value of the correlation coefficient r? (Use 3 decimal places.)
