Chapter 4 - 6851F_ch04_64_98 13/09/2002 05*08 PM Page 64 4...

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64 4 4.1 (a) Since 2.54 cm 5 1 inch, inches are changed to centimeters by multiplying by 2.54. Letting x 5 height in inches and y 5 height in centimeters, the transformation is y 5 2.54 x, which is monotonic increasing. (b) If x 5 typing speed in words per minute, then 5 typing speed in words per second and number of seconds needed to type a word. The transformation is , which is monotonic decreasing (we can ignore the case x , 0 here). (c) Letting x 5 diame- ter and y 5 circumference, the transformation is y 5 p x, which is monotonic increasing. (d) Letting x 5 the time required to play the piece, the transformation is y 5 ( x 2 5) 2 , which is not monotonic. 4.2 Monotonic increasing: intervals 0 to and to 2 p . Monotonic decreasing: interval to . 4.3 Weight 5 c 1 (height) 3 and strength 5 c 2 (height) 2 ; therefore, strength 5 c (weight) 2/3 , where c is a constant. 4.4 A graph of the power law y 5 x 2/3 (see below) shows that strength does not increase linearly with body weight, as would be the case if a person 1 million times as heavy as an ant could lift 1 million times more than the ant. Rather, strength increases more slowly. For example, if weight is multiplied by 1000, strength will increase by a factor of (1000) 2/3 5 100. 3 p 2 p 2 3 p 2 p 2 y 5 60 x y 5 60 x 5 1 60 x 4.5 Let y 5 average heart rate and x 5 body weight. Kleiber’s law says that total energy con- sumed is proportional to the three-fourths power of body weight, that is, Energy 5 c 1 x 3/4 . But total energy consumed is also proportional to the product of the volume of blood pumped by the heart and the heart rate, that is, Energy 5 c 2 (volume) y . Furthermore, the volume of blood pumped by the heart is proportional to body weight, that is, Volume 5 c 3 x . Putting these three equations together yields c 1 x 3/4 5 c 2 (volume) y 5 c 2 ( c 3 x ) y . Solving for y, we obtain . y 5 c 1 x 3 > 4 c 2 c 3 x 5 cx 2 1 > 4 6851F_ch04_64_98 13/09/2002 05*08 PM Page 64
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More on Two-Variable Data 65 4.6 (a) (b) The ratios are 226,260/63,042 5 3.59, 907,075/226,260 5 4.01, and 2,826,095/907,075 5 3.12. (c) log y yields 4.7996, 5.3546, 5.9576, and 6.4512. Here is the plot of log y vs. x. 3,000,000 1,500,000 0 Acres 1978.20 1978.90 1979.60 1980.30 1981.00 Year 6.0 5.0 log y 1978.20 1978.90 1979.60 1980.30 1981.00 Year (d) Defining list L 3 on the TI-83 to be log(L 2 ) and then performing linear regression on (L 1 , L 3 ) where L 1 holds the year (x) and L 3 holds the log acreage yield the least squares line log . For comparison, Minitab reports The regression equation is log y 52 1095 1 0.556 Year Predictor Coef Stdev t-ratio p Constant 2 1094.51 29.26 2 37.41 0.001 Year 0.55577 0.01478 37.60 0.001 s 5 0.03305 R–sq 5 99.9% R–sq(adj) 5 99.8% ˆ y 1094.507 1 .55577 x 6851F_ch04_64_98 13/09/02 12:52 Page 65
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66 Chapter 4 (e) 0.035 0.000 C7 1978.20 1978.90 1979.60 1980.30 1981.00 C6 B B B B A A A 5 C7 vs. C6 B 5 RESI2 vs. Year The residual plot of the transformed data shows no clear pattern, so the line is a reasonable model for these points. (f) log , so , and so With the regression equation for log y on x installed as Y 1 on the TI-83, define Y 2 5 10 (Y 1 ).
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Chapter 4 - 6851F_ch04_64_98 13/09/2002 05*08 PM Page 64 4...

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