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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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- Spring '04
- JOHNSON
- Math, Statistics, Probability