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Unformatted text preview: ’DO NUT 1' . “n‘ve's‘w University of Waterloo
Waterlc     \ g Flnal Examination = ,_
9 FALL 2000 (Term) (Year) Student Name Student ID Number Course Abbreviation and Number STAT 331/361 Course Title Section(s) Held With Course(s) Section(s) of Held With Courses(s) Instructor Bovas Abraham Date of Exam December 11, 2000 Time Period: Start Time: 7:00 End Time: 10:00 i Duration of Exam 3 hours Number of Exam Pages 12
(including this cover sheet) Exam Type Closed Book Additional Materials Allowed Calculators Instructions 1. Answer each question in the space provided
2. Check to see that you have all the 12 pages including the tables Marking Scheme: Page 1 of 12 Statistics 331/361
Final Examination, Fall 2000 [10] 1. Circle true or false. Note that true means always true; false means can be false. i) In a simple linear regression of y on 3:, it was found that the F statistic for testing ,81 = 0 was nonsigniﬁcant. This implies that there is no relation between y and
m. True False ii) Addition of a variable to a linear regression equation never decreases R2.
True False iii) All automatic methods (i.e. forward, backward, and stepwise) lead to the same
set of selected independent variables. True False
iv) Consider the model yi = [30 + q, 2': 1, 2,)..., 10 such that E(€.;) = 0,
Var(ei) = 02 and Cov(ei, ej) = 0. The least squares estimate of 0'2 is
i 2(92‘ — ﬂu)?
True False v) Consider the model in (iv). The leverage for the ﬁrst observation is 1.
True False vi) Observations with large studentized residuals are inﬂuential.
True False vii) Addition of a variable to linear regression never increases the residual sum of
squares. ' True False viii) The model 3; = ,60 + $6” + 6 is a linear model in the sense used in class.
True False ix) The residuals from a regression ﬁt are independent.
True False x) The residuals from a regression fit have constant variance.
True False Page 2 of 12 2. The model y = [30 + ﬁlm + e was ﬁt to a set of data containing 22 observations and the
following table was obtained a
Regression
Residual Lack of ﬁt
Pure Error [5] (a) Complet the table and test H : ﬁl = 0. What are your conclusions? [2] (b) Perform a Lack of ﬁt test for this model and state your conclusions. [4] (0) Suppose that a new model
Y=ﬂo+ﬁ1m+ﬂ2m2+e is ﬁt to the same data and the regression sum of squares is 423. Perform a lack
of ﬁt test for this model and comment. ‘ [4] (d) Test the signiﬁcance of the regression model in What percent of the variation
in y is explained by the model? Page 3 of 12 3. The following, (X ’X )‘1, 3 and residual sum of squares, were obtained from the re
gression of plant dry weight (grams) on percent soil organic matter (:31) and kilograms
of supplemental nitrogen per 1000 m2 (3:2) from n = 13 experimental ﬁelds. The re
gression model included an intercept. [B] = (52, 10, 3) Residual sum of squares = 30 2 5 —3
(X'X)1= —5 1 —1
—3 —1 5 [1] (a) Write down the prediction (ﬁtted) model. [4] (b) Interpret ﬁg and ﬁl ﬁat
1615
[1] (c) How many degrees of freedom does the residual sum of squares have? Degrees of freedom
[1] (d) Compute 32. [1] (e) Compute the estimated variance of 81. Var(Bl) Page 4 of 12 [1] (f) Compute the estimated covariance of Bl and 82. 007461, 182) [4] (g) Test the hypothesis that ,62 = 0 using a 95% conﬁdence interval C.I. [6] (h) Suppose previous experience has led you to believe that a 2 percentage point
increase in organic matter is equivalent to a 1 kilogram/ 1000 m2 increase in sup
plement nitrogen. That is, you want to test the hypothesis H : 2/61 = ﬁg (i.e.
Zﬂl —— ﬁg = 0). Use a ttest to test this hypothesis. Conclusion: Page 5 of 12 r 4. The plasma lipid levels of total cholesterol (y), the weight (221), and age (332) were
measured in a sample of 10 patients with hyperlipoproteinemia. Assuming linear re
lationships, three separate regressions were carried: y on m1, 3; on $2, y on $1 and x2
simultaneously. The following results were obtained. you .771 yon I132 yon £121,112 [4] (a) Test for a signiﬁcant linear relationship between y and 2:1, given that 2:2 is already
in the model. Conclusion: [4] (b) Test for a signiﬁcant linear relationship between y and 2:2, given that (1:1 is already
in the model. Conclusion: [3] (0) Test for a signiﬁcant multiple linear regression of y on 331 and 3:2 simultaneously. Conclusion: [3] (d) Compare your results in a, b and c. Do these results make sense? Explain your
answer brieﬂy. Page 6 of 12 5. In an experiment with 10 observations, there were four potential explanatory variables
x1, x2, x3, and :34. All possible regressions were ﬁt to the data and the results are
summarized below. Note: all models contain an intercept. Variables in Model Residual Sum of Squares None (intercept only) 2715.76
x1 1265.69 x2 906.33 x3 1939.40 x4 883.87 x1 x2 57.90 x1 x3 1227.07 x1 x4 74.76
x2 X3 415.44 x2 x4 868.88 x3 x4 175.74
x1 x2 x3 48.11
x1 x2 x4 ' 47.97
x1 x3 x4 50.84
x2 x3 x4 73.81
x1 x2 x3 x4 . 47.86 [3] (a) Which variable enters the model ﬁrst in an automatic forward selection procedure?
Justify your choice. (use a signiﬁcance level = .1 for inclusion) Selected Variable ___________ [4] (b) Which variable will be dropped ﬁrst from the model in an automatic backward selection procedure? Justify you choice. (use a signiﬁcance level = .05 for exclu—
sion) Selected Variable Page 7 of 12 [5] (0) Compare the value of the 0,, statistic for the model containing 171 and $2 with
that for the model containing x1, x2, x3 and :04. Which of the two models would
you recommend and why? 013(131 1'2)
Op($1 {132 (1)3 1174) Recommended Model . [5] (a) Fitting of the regression model to a set of data led to the following plots. What
are your assessments. Page 8 of 12 7. In the regression model
y =Xﬁ+e, e N N(O,02,I) the least square estimate of ﬂ is 3 = (X ’X )‘lX ’ y, fl is the vector of ﬁtted values and
e is the vector of residuals. [5] (a) Obtain the distribution of [5] (b) Obtain the distribution of e. [6] (c) Are ,3 and e statistically independent. Why or Why not? (Show your work). Page 10 of 12 (a) In the regression model 3; = X ,6 + e; 6 N N (0, 021), the X matrix is given by 1 1
+1 —1
1 +1
+1
—1 —1
+1 —1
—1 +1
+1 +1 HHHp—AHHHH
H (2) What are the degrees of freedom for the pure error sum of squares. Degrees of freedom (1) (ii) What are the degrees of freedom for the residual sum of squares? Degrees of freedom (6) (iii) Are 31 and 32 statistically independent? Show your work. Yes or No Page 9 of 12 Student’s t Distn'bution (upper tail probabilities) Prpbability of a Numerically Larger Value of t 0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0005 
4._______.¥._2‘4.. . . p—a
OQWQQ “#UJNH 1 . ~ 6.314 12706 31.821 636.619
2 .816 1.061 1.386 1.886 2.920 4.303. 6.965 9.925 31.598
3 .765 .978 1250 1.638 2353 3.182 4541 5.841 12.941
4 .741 .941 1.190 1533 2.132 2776 3.747 4.604 8.610
5 . . 2.015 2571 3.365 6.859
6 . 1.943 2.447 3.143 5959
: 7 .711 .896 1.119 1.415 1.895 2.365 2.998 3.499 5.405
8 .706 .889 1.108 1.397 1.860 2.306 2896 3.355 5.041
9 .703 ' .883 1.100 1.383 1.833 2.262 2.821 3250 4.781
10 . . 1.812 2228 2.764 4587
V 11 . . 1.796 2201 2.718 4.437 11
12 .695 .873 1.083 1.356 1.782 2179 2.681 3.055 4.318 12
13 .694 .870 1.079 1550 1.771 2160 ‘ 2650 3.012 4.221 13
14 .692 .868 1.076 1.345 1.761 2145 2.624 2.977 4.140 14
15 .691 .866 1.074 1541 1.753 2131 2602 2947 4.073 15 
16 .690 .865 1.071 1.337 1.746 2.120 2.583 2921 4.015 16
17 .689 .863 1.069 1.333 1.740 2110 2567 2898 3.965 17
18 .688 .862 1.067 1.330 1.734 2101 2552 2.878 3922 18
19 .688 .861 1.066 1.328 1.729 2093 2539 2861 3.883 19
20 .687 .860 1.064 1.325 1.725 2086 2528 2845 3.850 20
21 . . 1.721 2080 2.518 3.819 21
22 .686 .858 1.061 1.321 1.717 2074 2508 2819 3.792 22
23 .685 .858 1.060 1.319 1.714 2069 2500 2.807 3.767 23
24 .685 .857 1.059 1.318 1.711 2064 2492 2797 3.745 24
25 . 1.708 2.060 2.485 3.725 25
I 26 . . _1.706 2056 2479 3.707 26
27 .684 .855 1.057 1.314 1.703 2.052 2473 2771 3.690 27
28 .683 .855 1.056 1.313 1.7011 2048 2467 2763  3.674 28
29 .683 .854 1.055 1.311 1.699 2045 2.462 2756 3.659 29
30 .683 .854 1.055 1.310 1.697 2.042 2457 2750 3.646 30
40 .681 .851 1.050 1.303 1.684 2021 2.423 2704 3551 40
60 .679 .848 1.046 1.296 1.671 2.000 2390 2.660 3.460 60
120 . . 1.658 1.980 2.358 3.373 120
.. 1.960 2.326 3291 .. 1.645 Page 11 of 12 m : denominator
depees of Random 3&83 88855 Somqa 9&QNH 1 IWIU V ' The FbiStrIbuu'on AREAUNDEkan TOTHERIGIHOFTHETABW‘IED VALUESIS 0.1 n  numerator degrees of freedom 4 5 >6 8 12 . 24 39.86 49.50 53.59 55.83 57.24 58.20 59.44 60.70 62.00 63.33 8.53
5.54
4.54
4.06 3.78
3.59
3.46
3.36
3.28 3.18
3.07
2.97
2.92
2.88 2.84
2.79 2.75
2.71 9.00
5.46
4.32
3.78 ' 3.46 3.26
3.1 1
3.01
2.92 2.81
2.70
2.59
2.53
2.49 2.44 '239 2.35
2.30 9.16
5.39
4.19
3.62 3.29
3.07
2.92
2.8 1
2.73 2.61
2.49
2.38
2.32
2.28 2.23
2.18 ' 2.13 2.08 AREAUNDER Fun To THE RIGHT OFTHETABULA'IED VALUES 15 0.05 m : denominator degrees of freedom 8§S$ 83855 somqm ubuup 161.4
 1851
10.13
7.71
6.61 5.99
5.59
5.32
5.12
4.96 4.75 2 199.5
19.00
9.55
6.94
5.79 5.14
4.74
4.46
4.26
4.10 3.88
3.68
3.49
3.38
3.32 3.23
3.15
3.07
2.99 215.7
19.16
9.28 659 5.41 4.76 I _4.35
4.07
3.86
3.71 3.49
3.29
3.10
2.99 2.92 2.84 ' 2.76
2.68
2.60 9.24 9.29 9.33 9.37 9.41 9.45
5.34 5.31 5.28 5.25 5.22 ' 5.18
4.11 4.05 4.01 3.95 3.90 3.83
3.52 3.45 3.40 334 327 3.19
3.18 3.11 3.05 2.98 2.90 2.82
2.96 2.88 2.83 2.75 2.67 2.58
2.81 2.73 2.67 2.59 250 2.40
2.69 2.61 2.55 2.47 238 2.28
2.61 2.52 2.46 2.38 2.28 2.18
2.48 2.39 2.33 2.24 _ 2.15 2.04
2.36 2.27 2.21 2.12 2.02 1.90
2.25 2.16 2.09 2.00 1.89 1.77
2.18 2.09 2.02 1.93 ' 1.82 1.69
2.14 2.05 1.98 1.88 1.77 1.64
2.09 2.00 1.93 1.83 1.71 1.57
2.04 1.95 1.87 1.77 1.66 1.51
1.99 1.90 1.82 1.72 1.60 1.45
1.94 1.85 1.77 1.67 ’ 1.55 1.38
n  numcmlor degxees of [modem
4 5 6 8 12 24
224.6 230.2 234.0 238.9 243.9 249.1
19.25 19.30 19.33 19.37 19.41 19.45
9.12 9.01 8.94 8.84 8.74 8.64
6.39 6.26 6.16 , 6.04 5.91 5.77
5.19 5.05 4.95 4.82 4.68 4.53
4.53 4.39 4.28 4.15 4.00 3.84
4.12 3.97 3.87 3.73 3.57 3.41
3.84 ' 3.69 3.58 ' 3.44 3.28 3.12
3.63 3.48 3.37 3.23 3.07 2.90
3.48 3.33 3.22 3.07 2.91 2.74
3.26 3.11 3.00 2.85 2.69 ‘ 2.50
3.06 2.90 2.79 2.64 2.48 2.29
2.87 2.71 2.60 2.45 2.28 2.08
 2.76 2.60 2.49 2.34 2.16 1.96
2.69 2.53 2.42 2.27 2.09 1.89
2.61 2.45 2.34 2.18 2.00 1.79
252 2.37 2.25 2.10 1.92 1.70
2.45 2.29 2.17 2.02 1.83 1.61
2.37 2.21 2.10 1.94 1.75 1.52 9.49
5.13
3.76
3.10 2.72 2.47
2.29
2.16
2.06 1.90
1.76
1.61
1.52
1.46 1.38
1.29
1.19
1.00 254.3
19.50
8.53
5.63
4.36 3.67
3.23
2.93
2.71
2.54 2.30
2.07
1.84 . 1.71 1.62 1.51
1.39
1.25
1.00 Page 12 of 12 ...
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