Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Unformatted text preview: 1. (25 pts.) The scatterplot on the front of the exam indicates a linear pattern between highway miles per gallon and reciprocal weight (in reciprocal tons) of a vehicle. Here 1s some Minitab output for these two variables. Descriptive Statistics Varlable —EEM va MPG m— RecirocaIWeiht 0.58165 0.11816 Correlation Correlation of HWY MPG and Reciprocal Weight = 0.836 a. Write the least squares equation fitting highway mpg to a cars reciprocal weight. (Use the variables “highway mpg” and “reciprocal weight” rather than generic x and y.) HIWUW ”Wk 3 29673 + 0.334 (ILE‘IVWWW’ 2119/4 b. Predict the highway mpg for the 2004 Subaru Forester w ' h has a reciprocal weight of 0.647249 tons“. [we 1» 0.6%? 21'? W W / Maw/r} M196— 9’.“0 c. If the Subaru Forester actually gets 28 miles per gallon, find the associated residual. 2?, 26.7; [email protected] ll A H/mw moor — NW WW (1. What is the typical size of the 409 residuals when fitting highway mpg linearly in reciprocal weight? Give a numerical value. . 1 These data are from the December 2003 issue of Kiplinger's Personal Finance. 1 s l— (.3991 5.101 E 230mg 2. (20 pts.) Two objects of unknown weights w1 and w2 are weighed on an error-prone pan balance in the following way: 0 Object I is weighed by itself and the measurement is 13 grams, 0 Object 2 is weighed by itself and the measurement is 4 grams, 0 The difference of the weights (the weight of object 1 minus the weight of object 2) is measured by placing the objects indifferent pans, and the result is 6 grams, 0 The sum of the weights is measured as 20 grams Find the least squares estimates of w1 and w2. 4 , m = 0 ' ”I ,, 6" = zfzre (g M; l~\ U1. ) a” 0+ l l 4* 3. (17 pts.) Rosner (1999)2 contains a study on how respiratory function —- in terms of the volume of air expelled in the first second of a forceful breath (fev), is related to smoking. The study involved 654 young people under the age of 20. Variables other than fev measured in the study: Age (in years) . Height (in inches) Gender (1 for male, 0 for female) .9 lforasmoker 0f. . The plot below, along with further graphical analysis, suggests as a possible model ln(fev) = ,60 + A ln(height) + fl2(age) + ,63 (gender) + ,64(smoke ?) + e 3.8 3.9 4.0 4.1 4.2 4.3 |n(height) Below (and on the next page) is the Minitab output corresponding the fit of this model Regression Analysis The regression. equation is ln(fev) = — 9.84 + 2.56 ln(height) + 0.0233 age + 0.0365 gender - 0.0407 smoke? Predictor Coef ' StDev T P Constant 4.8387 0.3868 -25.44 0.000 1n(height) 2.5574 0.1006 25.43 0.000 age 0.023318 0.003356 6.95 0.000 gender 0.03648 0.01167 3 . 12 0.002 smoke? ~0.04070 0.02095 —1.94 0.052 2 Rosner, B. (1999), Fundamentals of Biostatistics, 5th ed., Pacific Grove, CA: Duxbury. S = 0.1456 R-Sq = 8;@Q§ R-Sq(adj) = 80.9% Analysis of Variance Source DP SS HS P P Regression 4 58.762 14.691 692.70 0.000 Residual Error 649 13.764 0.021 . Total 653 72.526 a. State the null and alternative hypotheses for the ‘age’ row in the output on the previous page. (The two hypotheses will involve beta value(s).) b. State the null and alternative hypotheses corresponding to the ‘Analysis of Variance’ section above. (The two h theses will involve beta value(s).) c. Using the fitted model, predict the fev for a 12 year~old, 60 inch tall, female, non- smoker. ' A690 = ’- ‘U’f- + 7.54 A(40) 4— 0.0233049 1“ b.03€5(a) - 0.0mm) ‘ c. .72.“: 4. (18 pts.) Multiple Choice: a. When choosing to estimate the ,6 vector for the model Y=Xfl+e using ,3” = (X 'X )'l X ’ Y we are choosing the value of ,3 to minimize (circle one) . . . b. (AB)' =B'A’ is ii. sometimes true iii. never true c. AB = BA is / i. always true ii. sometim iii. never true d. When adding a predictor variable to a least squares model, the resulting value of R2 i. is alwa s at least as big as the old v ue ii. is always no more than t e o . v ue iii. neither of the above are necessarily true e. The importance of a predictor variable in a least squares model can .- -- rmined directly from the magnitude of its coefficient in the fitted model. True 0 ® circle your choice)? f. If the correlation coefficient between the response variable and a possible predictor variable is close to zeri I en that predictor variable must not be useful in helping predict the response. True 0 tircle your choice)? 5. (20 pts.) Short Answer: a. If day of the week (Sunday, Monday, . . ., Saturday) is to incorporated into a least squares model, then how many “dummy” (or “0-1”) variables must be used? 7,, >@ b. Tell me the name of a modified version of R2 that may be reasonably used to compare least squares models having possibly different numbers of predictor variables. 0450*; B?— c. One condition for forcing the additive constant in a least squares model to be zero is that the origin “make (physical) sense”. What is the other condition mentioned by your instructor? Orv"; flaulé ba "1;," m. "he..." 42:. 4042 Vain/2b d. What type of plot may be used to see if a predictor variable has been adequately incorporated into a least squares model? H‘s/4&4 ( P/a'f' / e. 3x2+5y2+6z2+2xy+4xz+10yz+x+9y+llz+17=fl'Q/3+,B‘c+d for what Q (symmetric), c, and d if x fi=y? Z Biz ' n55 ,azi/A/v 2.5; H _% II II . '- «pt- \.W, an ...
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