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Unformatted text preview: UNIVERSITY OF TORONTO AT MISSISSAUGA
APRIL EXAMINATIONS 2008
ECO 220Y5Y
Duration~ 2 hours You may be charged with an academic oﬂence for possessing the following items during
the writing of an exam unless otherwise specified: any unauthorized aids, including but
not limited to calculators, cell phones, pagers, wrism'atch calculators, personal digital
assistants (PDAs), iPODS, MP3 players, or any other device. If any of these items are in
your possession in the area of your desk, please turn them off and put them with your
belongings at the ﬁ'ont of the room before the examination begins. A penalty MY BE
imposed if any of these items are kept with you during the writing of your exam. Please note, students are NOT allowed to petition to RE~ WRIT E a ﬁnal examination. Aids: Non—Programmable calculator, l aid sheet Note: Do six questions making sure that you do at least one question from each
section. All questions are of equal value (20 marks). I I I I I l l O U O OIIIIOOOOCIIDIDlII. I I I O I I I I l I I o l l l O I O I l l O I l l l I I O l l I I I i l l I O I I I l II
Student Number ....................................................... ..
Section .................... ............................................ .. Questions Attemgted. Page 1 of 14 Part 1.
I. Give the formula for three measures of location and three measures of dispersion.
Explain what they do and discuss any advantages or disadvantages they may have. Page 2 of14 2 The following table reports the probabilities of reaching academic attainment levels 1
through 5 (5 being higher than 1) for cohorts of girls and boys born in the 1930’s and the
1970’s. Calculate the expected values and variances for girls and boys in the cohorts and
comment on the change in status of male and female academic attainments. —
_
0.0723 0.1288 0.1472 0.3313 0.3204
0.4442 0.1358 0.1052 0.1520 0.1628
0.4426 0.1756 0.1062 0.1801 0.0955 Page 3 of 14 3. X ~N(9,9) and Y = 6 + 503. What is the expected value and variance of Y?
What is P(Y>10)? What is the value of Y* where P(Y < Y*) = 0.45?
What is the distribution of Y — X. Page 4 of 14 Part 2. 4) Your company makes sunscreen oil weight. When the active ingredient is present in
doses less than 0.025 milligrams, the oil is completely ineffective. a)[12 marks] Outline the way you would formulate a statistical test for the presence of the
ingredient based upon a sample of oil taken from a batch and tested to destruction to see
if the rest of the batch are good enough for use. Discuss the type of errors the test could
make and explain which of the errors you would make type 1. b)[8 marks] A sample of 49 bottles yielded an average presence of 0.025l milligrams
with a sample variance of 0.000001. Suppose you are willing to accept a 5% chance of
making a type one error. What would you conclude? Page 5 of 14 5. You have to estimate the parameter 6 based on a sample of size n and your research assistant has provided you with three estimators 61, 62, and 63 . It can be shown that 61=
6+V, 82 = 8 + (v+1)/n, and 83 = 6 + v/n, where v is a random variable with a zero mean and constant variance 62. Discuss the merits of the three estimators in terms of their
unbiasedness, consistency and efﬁciency. Which estimator would you choose? Page 6 of 14 6. Explain what the maximum likelihood estimation technique is. Derive the maximum
likelihood estimator of 02 for a random sample x;, i = 1,. . .,n drawn from a distribution f(xi)= l I‘ e 262 27m 2 Page 7 of 14 Part 3. 7. In a recent summer exam 4 students arrived late of whom three were left handed, in the
class of 50 students (all of whom turned up to the exam) 5 were leﬁ handed. Test the
hypothesis that being late for an exam and being left handed are independent? Page 8 of 14 8. In a random sample of 200 births at a hospital, 40 took place between midnight and 6
in the morning. Test the hypothesis that births at the hospital are independent of the time of day (Le. they are uniformly distributed) set the type one error at 5%. An additional
sample of 400 births yielded 80 more births between midnight and 6 am, using the
combined example what would you conclude now. Why is it that the same proportion of births yields a different conclusion? Page 9 of i4 9) Prove that in the analysis of variance problem the Total Sum of Squares is equal to the
Error Sum of Squares plus the Treatment Sum of Squares and demonstrate that their
respective degrees of freedom obey the same relationship. Page 10 of 14 Part 4. Questions 10 and 1 1 reiate to the foilowing table which reports the results of the
regression equation y 2 u+B1x+B2x2+e linking y, a child’s educational attainment to x, it’s
parent’s educational attainment {Standard normal ratio’s (coefﬁcient/Standard error) in
brackets} for Canadian Children. BoysFathers GirlsMothers  Born in 50’s Born in 70’s Born in 50’s
(20.7241) (29.0204) (23.7804) (32.0526)
(10.5623) (3.7405) (10.0323) (6.7228)
(6.7923) (0.3932) (5.7141) —3.1192) Page 11 0f14 10. There is a theory which says that the relationship between a child’s attainment and
their parent’s attainment will become more Iinear (ﬁg—>0) for younger cohorts as
opportunities become more equal. Thus the coefﬁcient should be closer to 0 for more
recent cohorts than it is for older cohorts. Clearly stating the hypothesis you are testing
test this hypothesis for boys and girls in Canada setting the type one error at 5%. Page 12 of 14 11. Educationai mobility is measured by the marginal effect of a parents educational
attainment on a child’s educational attainment (0 corresponds to complete mobility and 1
corresponds to complete immobility). Assuming that average parental attainment (both
genders) was 2 in the 1930’s and 2.5 in the 1970’s calculate the mobiiity for girls and
boys in the two cohorts and comment on the trends in mobility. Page 13 of 14 12. For the simple regression model Y1 = a+BXi+gig i = 1,. . .,n, R2 is given by the
formula: I! im~hkzﬁ R2 : i=1 i=1
Zm—EZ
i=1 State what R2 is and what it is used for, derive its relationship to the covariance of X and
Y and explain why caution should be used in its inteipretation. Page 14 of 14 Table 3.4 Standard Normal Distribution The cumuIative standardized normal distribution FL?) is deﬁned by
Exnmpfe: PL‘. 4 L75) = F635) ' 0.959$J\n '1 P(: 3 [.75) 1" 0.049] z .00 .01 .02 .05 .04 .05 .05 .07 .05 _ .09
.0 .5000 .5040 .5050 .5120 .5150 .5199 .5239 .5279 .5519 .5559
.1 .5595 .5455 .5475 .5517 .5557 .5595 .5555 .5575 .5714 .5755
.2 .5793 .5552 .5571 .5910 .5945 .5957 .5025 .5054 .5105 .5141
, .3 .5179 .5217 .5255 .5295 .5521 .5555 .5405 .5445 .5450 .5517
.4 .5554 .5591 .5525 .5554 .5700 .5725 .5772 .5505 .5544 .5579
.5 .5915 .5950 .5955 .7019 .7054 .7055 .7125 .7157 .7190 .7224
.5 .7257 .7291 .7524 ' .7357 .7559 .7422 .7454 .7495 .7517 .7549
.7 .7550 .7511 .7542 '.7575 .7704 .7754 .7754 .7794 .7522 .7552
.5 .7551 .7910 .7939 .7957 .7995 .5025 .5051 .5075 .5105 .5155
.9 .9159 .5155 .5212 .5255 .5254 .5259 .5515 .5540 .5555 .9559
1.0 .5415 .5435 .5451 .5495 .5505 .5521 .5554 .5577 .5599 .9521
1.1 .5545 .5555 .9555 .5705 .5729 .5749 .5770 .5790 .9510 .5530
1.2 .5549 .5559 .5555 .9907 .9925 .5944 .5952 .5950 .5997 .9015
1.5 .9052 .9049 .9055  .9052 .9099 .9115 .9151 .9147 .9152 .9177
1.4 .9192 .9207 .9222 .9255 .9251 .9255 .9279 .9292 .9305 .9519
1.5 .9552 .9245 .9557 .9570 .9592 .9394 .9405 .9415 .9429 .9441
1.5 .9452 .9453 .9474 .9454 .9495 .9505 .9515 .9525 .9555 .9545
1.7 .9554 .9554 .9575 .9552 .9591 .9599 .9505 .9515 .9525 .9555
1.5 .9541 .9549 .9555 .9554 .9571 .9575 .9595 .9595 .9599. .9705
1.9 .9715 .9719 .9725 .9752 .9755 .9744 .9750 .9755 .9751 .9757
2.0 .9772 .97. .9755 .9755 .9795 .9795 .9505 ' .9505 .9512 .9517
2.1 .9521 .9525 .9950 .9534 .9555 .9942  .9545 .9550 .9554 .9557 A
2.2 .9551 .9554 .9555 .9571 .9575 .9575 .9551 .9554 .9557 .9590
2.3 .9595 .9595 , .9595 .9901 .9904 .9905 .9909 .991 1 .9915 .9915
2.4 .9915 .9920 .9922 .9925 .9927 .9929 .9951 .9952 .9954 .9935
5.5 .9955 .9940 .9941 .9945 .9945 .9945 .9945 .9949 .9951 .9952
2.5 .9953 .9955 .9955 .9957 .9959 .9950. _..9951 .9952 .9955 .9954
2.7 .9955 .9955 .9057 .9955 .9959 .9975 .9971 .9972 .9975 .9974
2.5 .9974 .9975 .9975 .9977 .9977 .9975 .9979 .9979 .9950 .9951
5.9 .0951 .9952 .9952 .9905 .9954 .9954 .9955 .9955 .9955 .9955
5.0 .9957 .9957 .9957 .9955 .9955 .9959 .9959 .9959 .9990 .9990
5.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9995 .9995
5.2 .9995 .9993 .9994 .9994 _ .9994 .9994 .9994 .9995 .9995 .9995
5. .9995 _ .9995 .9995 .9995 .9995 .9995 .9995 .9995 .9995 .9997
3.4 .9997 . .9997 .997 .9997 .9997 .9997 .9997 .9997 .9997 .9995 Appendix 3: Random Numbers and Probability Functions Table 3.5 The cumulative t distribution is deﬁned by t Distribution* = PU!) < 2.2523 = 0.973 I I.
F!” = I a I. r”
— _ a  q I '
(1r.i ) : HOP'7 01 — 0.1123
Exmnpl‘r: 11 = H). v = 9.
’9 0.1.
F111 .75 .90 .95 .975 .99 .995 .9995
v 151 1.251 1.101 1.051 1.0251 [.011 [.0051 1.00051
1 1.000 3.075 5.314 12.705 31.021 53.557 535.519
2 .515 1.555 2.920 4.303 5.955 9.925 31.595
3 .755 1.535 2.353 3.102 4.541 5.541 12.941
4 .741 1.533  2.132 2.77 3.747 4.504 5.510
5 .727 1.475 2.015 2.571 3.355 4.032 5.059
5 .715 1.440 1.943 2.447 3.143 3.707 5.959
7 .711 1.415 1.595 2.355 2.995 3.499 5.405
5 .705 1.397 1.550 2.305 2.595 3.355 5.041
9 .703 1.35:1 1.533 2.252 2.521 3.250 4.751
10 .700 1.372 1.512 2.220 2.754 3.159 4.557
11 .597 1.353 1.795 2.201 2.715 3.105 4.437
12 .595 1.355 1.752 2.179 2.551 3.055 4.315
13 .594 1.350 1.771 2.150 2.550 3.012 4.22
14 .592 1.345 1.751 2.145 2.524 2.977 4.140
15 .591 1.341 1.753 2.131 2.502 2.947 4.073
15 _ .590 1.337 1.745 2.12 2.553 2.921 4.015
17 .559 1.333 1.740 2.110 2.557 2.590 3.955
15 .555 1.330 1.734 2.101 2.552 2.975 3.922
19 .555 1.325 1.72 2.093 2.539 2.551 3.053
20 .557 1.325 1.72 2.055 2.525 2.545 3.550
21 .555 1.323 1.72 2.0110 2.515 2.531 3.019
22 .555 1.321 . 1.717 2.074 2.5011 2.019 3.792
23 .555 1.319 1.714 2.059 2.500 2.007 3.757
24 .555 1.315 1.711 2.054 2.492 2.797 3.745
25 .554 1.310 1.705 2.050 2.405 2.707 3.725
25 .554 1.315 1.705 2.055 2.479 2.77 3.707
7 .504 1.314 1.703 3.052 2.47.1 2.771 3.590
20 .553 1.313 1.701 2.045 2.457 2.753 3.574
29 .553 1.311 1.599 2.045 2.452 2.755 3.559
30 .553 1.310 1.597 2.042 2.457 2.750 3.545
40 .501 1.303 1.554 2.021~ 2.423 2.704 3.551
60 .579 1.295 1.571 2.000 2.390 2.550 3.450
120 .57. 1.209 1.555 1.950 2.355 2.517 3.373
=12.) .574 1.202 1.545 1.950 2.325 2.575 3.291 ' This table is abrldnad from rhu 'Slausliual Tables" 01' R. A. Fisher and Frank Yale:
published bv Oliver .2 80011. 1.111.. Edinburgh 3nd London. 1933. El 15 here published with
Ihe kmd permiSslon 01' 1h: aulhors .1011 their publishers. Table 8.7 ChiSquare Distribution
‘ The~ cumulative chisquare distribution is deﬁned by
1 xiv * Evaxde [(1121 r. 0115.9)=0.950 F 2 Ix
0‘} 0 2””[01—217211 51:01:11.1!“ Png < 16.9 for (if. = 9 141x? <01“) = F1159) = 0.950 '   x§d_r_
[6.9 _
th’) .005 .010 .025 .050 .100 .900 .950 .975 .990 .995
v [51 (.995) 1.990] 1.9751 1.950} (.9001, 1.1001 1.050] 1.0251 1.0101 1.0051
1 .0393 .01157 .0902 .0=393 0.150 2.71 3.04 5.02 5.53 7.00
2 .0100 .0201 .0500 .103 .211 4.51 5.99 7.30 9.21 10.5
3 .0717 .115 .210 .352 .554 5.25 7.01 9.35 11.3 12.0
4 .207 .297 .404 .711 1.05 7.70 9.49 11.1 13.3 14.9
5 .41 . .554 .031 1.15 1.51 9.24 11.1 12.0 15.1 15.7
5 0 .072 1.24 1.54 2.20 10.5 12.5 14.4 15.0 10.5
7 . I 1.24 1.59 2.17 2.03 12.0 14.1 15.0 10.5 20.3
0 1.34 1.55 2.15 2.73 3.49 13.4 15.5 17.5 20.1 22.0
9 1.73 2.09 2.70 3.33 4.17 14.7 15.9 19.0 21.7 23.0
10 2.15 2.55 3.25 . 3.94 4.57 15.0 10.3 20.5 23.2 25.2
11 2.50 3.05 3.02 4.57 5.50 17.3 19.7 21.9 24.7 25.3
12 3.07 3.57 4.40 5.23 5.30 15.5 21.0 23.3 25.2 20.3
13 3.57 4.11 5.01 5.09 7.04 19.0 . 22.4 24.7 27.7 29.5
14 4.07 4.05 5.53 5.57 7.79 21.1 23.7 25.1 29.1 31.3
15 4.50 5.23 0.25 7.25 0.55 22.3 25.0 27.5 30.5 32.0
10 5.14 5.01 5.91 7.95 9.31 23.5 25.3 20.0 32.0 34.3
17 5.70 5.41 7.50 0.57 10.1 24.0 27.5 30.2 33.4 35.7
10 5.25 7.01 0.23 9.39 10.9 25.0 20.9 31.5 34.0 37.2
19 5.04 7.53 0.91 10.1 11.7 27.2 30.1 32.9 35.2 30.5
20 7.43 0.20 9.59 10.9 12.4 23.4 31.4 34.2 37.5 40.0
21  0.03 0.90 10.3 _ 11.5 13.2 29.5 32.7 35.5 30.9 41.4
22 0.54 9.54 11.0 12.3 14.0 30.0 33.9 35.0 40.2 42.0
23 9.25 10.2 11.7 13.1 14.0 32.0 35.2 30.1 41.0 44.2
24 9.09 10.9 12.4 13.0 15.7 33.2 30.4 39.4 43.0 45.5
25 10.5 11.5 13.1 14.5 15.5 34.4 37.7 40.5 44.3 45.9
25 11.2 ' _ 12.2 13.5 15.4 17.3 35.5 30.9 41.9 45.5 40.3
27 11.0 12.9 14.5 15.2 10.1 ‘ 35.7 40.1 43.2 47.0 49.5
30 12.5 13.5 15.3 15.9 10.9 37.9 41.3 44.5 40.3 51.0.
29 13.1 1‘4.3 15.0 17.7 19.0 39.1 42.5 45.7 49.5 52.3
10 13.0 15.0 15.0 10.5 20.5 40.3 43.0 47.0 50.9 53.7
r.  2575  2.325 — 1.950 — 1.045 1.‘252 + 1.202 + 1.545 4 1.000 4 2.325 + 2.575 NOTE: For 9 > 30 (1.9.. for more than 30 degrees of freedom] take 2 {2 ’ 2......—
3 . _. —— _ 2 I ._ 1
x v[1 9p + 2. 9V] or x 15;, + v:21 1}!  according to the 'degree of accuracy required. 2. is the standardized normal deﬂate
corresponding 10 the a level of significance. and is shown in the bottom line of the tabie.
This table ls abridged from “Tables of percentage points of tha Incomplete beta function
and of the chisquare distribution." Biometrikc. Vol. 32 {1941}. Reprinted with permission
of its author. Catherine M. Thompson. and the editor of Biometrikc. UL; APPENUIKD aaure: '2.an vm Percentage Points of the Fdiscribution. a = .05 "fF } NUMERATOR DEGREES OF FREEDOM DENOMINATUR DEGREES 0F FREEDOM 2.51 2.46 2.40 2.42 2.45 2.39 2.42 2.37 2.40 2.34 2.37 2.32 . 2.35 230
2. 2.34 2.28
2.59 2.22 2.77
2.57 2.31 2.2%
2.56 2.29 2.2:
2.55 2.22 2.22
3.53 2.27 2.2:
2.45 2.13 2.12
2.37 2.10 2.04
2.29 2.02 L96
3.31 1.94 1.33 Source: From M. Merriagwn and C. M.Thompson. "Tables of Perccmnge Poinlsnf the Inverted Bclu (FzDiszriburion." Biamrrrika. 1913. 3173—88. Reproduced by permission of the Br'mnerrika Trustees. APPEND‘IXB Tables l023 TABLE VI II Continued NUMERATOR DEGREES 01“ FREEDOM DENOMINA’I‘OR DEGREES 0F FREEDOM
J ' d
J 20 ....35  2.1 2.32 181
22 2.30 1.73
13 2.27 1.76
24 2.25 173
25 2.24 1.71
26 2.2: L69
:7 2.20 167 ...
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 Spring '11
 Wolfson
 Normal Distribution, Variance, ........., Biometrikc

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