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Least-squares regression curves
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8.68. Fit a least-squares parabola, y a bx cx2, to the data in Table 8-34. Table 8-34 x y 0 2.4 1 2.1 2 3.2 3 5.6 4 9.3 5 14.6 6 21.9
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8.69. Table 8-35 gives the stopping distance d (feet) of an automobile traveling at speed v (miles per hour) at the instant danger is sighted. (a) Graph d against v. (b) Fit a least-squares parabola of the form d a bv cv2 to the data. (c) Estimate d when v 45 miles per hour and 80 miles per hour. Table 8-35 Speed, v (miles per hour) Stopping Distance, d (feet) 20 54 30 90 40 138 50 206 60 292 70 396
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8.70. The number y of bacteria per unit volume present in a culture after x hours is given in Table 8-36. (a) Graph the data on semilogarithmic graph paper, with the logarithmic scale used for y and the arithmetic scale for x. (b) Fit a least-squares curve having the form y abx to the data, and explain why this particular equation should yield good results, (c) Compare the values of y obtained from this equation with the actual values. (d) Estimate the value of y when x 7. Table 8-36 Number of Hours (x) Number of Bacteria per Unit Volume (y) 0 32 1 47 2 65 3 92 4 132 5 190 6 275
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Multiple regression
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8.71. Table 8-37 shows the corresponding values of three variables x, y, and z. (a) Find the linear least-squares regression equation of z on x and y. (b) Estimate z when x 10 and y 6.
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CHAPTER 8 Curve Fitting, Regression, and Correlation
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Table 8-37 x y z 3 16 90 5 10 72 6 7 54 8 4 42 12 3 30 14 2 12
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Standard error of estimate and linear correlation coefficient
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8.72. Find (a) sy.x, (b) sx.y for the data in Problem 8.67. 8.73. Compute (a) the total variation in y, (b) the unexplained variation in y, (c) the explained variation in y for the data of Problem 8.67. 8.74. Use the results of Problem 8.73 to find the correlation coefficient between the two sets of quiz grades of Problem 8.67. 8.75. Find the covariance for the data of Problem 8.67 (a) directly, (b) by using the formula sxy of Problem 8.74. rsxsy and the result
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8.76. Table 8-38 shows the ages x and systolic blood pressures y of 12 women. (a) Find the correlation coefficient between x and y. (b) Determine the least-squares regression line of y on x. (c) Estimate the blood pressure of a woman whose age is 45 years. Table 8-38 Age (x) 56 42 72 36 63 47 55 49 38 42 68 60
Blood Pressure (y) 147 125 160 118 149 128 150 145 115 140 152 155 8.77. Find the correlation coefficients for the data of (a) Problem 8.64, (b) Problem 8.66. 8.78. The correlation coefficient between two variables x and y is, r 0.60. If sx y 20, find the equations of the regression lines of (a) y on x, (b) x on y. # 8.79. Compute (a) sy.x, (b) sx.y for the data of Problem 8.78. 8.80. If sy.x 3 and sy 5, find r. 1.50, sy 2.00, x # 10 and
8.81. If the correlation coefficient between x and y is 0.50, what percentage of the total variation remains unexplained by the regression equation 8.82. (a) Compute the correlation coefficient between the corresponding values of x and y given in Table 8-39. (b) Multiply each x value in the table by 2 and add 6. Multiply each y value in the table by 3 and subtract 15. Find the correlation coefficient between the two new sets of values, explaining why you do or do not obtain the same result as in part (a). Table 8-39 x y 2 18 4 12 5 10 6 8 8 7 11 5