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1.7 Derive, with a proof, the slope coefficient that would have been obtained in Exercise 1.5 if weight and height had been measured in metric units. (Note: one pound is 454 grams, and one inch is 2.54 cm.) 1 EXERCISE 1.7

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2 To simplify the algebra, we will write weight, in pounds, as Y and height, in inches, as X. The slope coefficient using these variables is shown.

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3 EXERCISE 1.7 We then define Y' = 0.454Y as weight measured in kilos and X' = 2.54X as height measured in cm.

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4 EXERCISE 1.7 This is the expression for the revised estimator using X' and Y'.

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5 EXERCISE 1.7 We substitute for X' and Y'.

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6 EXERCISE 1.7 2.54 is a factor of the first term in the numerator, so it can be taken out. Similarly 0.454 can be taken out of the second term. 2.54 is a factor in the squared term in the denominator, so it can be taken out as a square.

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7 EXERCISE 1.7 Thus b 2 ' is equal to 1.79 times the expression for the original slope coefficient.

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8 EXERCISE 1.7 The original slope coefficient was 5.19.

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9 Hence the revised one, using metric units, will be 0.93. EXERCISE 1.7

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10 We will run the regression and verify that this is correct. First we construct W1 and H1, the variables measured in metric units, using the Stata generate command. ‘g’ is short for generate. EXERCISE 1.7. g W1=WEIGHT85*0.454. g H1=HEIGHT*2.54. reg W1 H1 Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 355.97 Model | 53819.2324 1 53819.2324 Prob > F = 0.0000 Residual | 81340.044 538 151.189673 R-squared = 0.3982 -------------+------------------------------ Adj R-squared = 0.3971 Total | 135159.276 539 250.759325 Root MSE = 12.296 ------------------------------------------------------------------------------ W1 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- H1 |.9281928.0491961 18.87 0.000.8315528 1.024833 _cons | -88.38539 8.472958 -10.43 0.000 -105.0295 -71.74125 ------------------------------------------------------------------------------

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. g W1=WEIGHT85*0.454. g H1=HEIGHT*2.54. reg W1 H1 Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 355.97 Model | 53819.2324 1 53819.2324 Prob > F = 0.0000 Residual | 81340.044 538 151.189673 R-squared = 0.3982 -------------+------------------------------ Adj R-squared = 0.3971 Total | 135159.276 539 250.759325 Root MSE = 12.296 ------------------------------------------------------------------------------ W1 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- H1 |.9281928.0491961 18.87 0.000.8315528 1.024833 _cons | -88.38539 8.472958 -10.43 0.000 -105.0295 -71.74125 ------------------------------------------------------------------------------ 11 The slope coefficient is 0.93, confirming the analysis above. EXERCISE 1.7

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Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use. 24.06.06

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