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Unformatted text preview: McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 203 Principle of Statistics 1 Examiner: Dr. Axel Hundemer Date: Friday December 7, 2007
Associate Examiner: Benjamin Rich Time: 9:00 AM to 12:00 PM $93.59!“ INSTRUCTIONS Please attempt to answer questions in the exam booklets provided.
This is a closed book exam. No notes, cribsheets or books are permitted.
Calculators are permitted. Use of a regular and or translation dictionary is‘permitted This examination is comprised of the cover page and 9 questions and 5 pages of tables. This exam has printed on lettersize paper and is
double—sided. MATH 203 Final Exam Page 1 12/7/07 1. 5‘ (7 marks) Answer the following ’True’ or ’False’ questions. No justiﬁcation is required and no partial credit will
be given.
(a) If a distribution has many outliers, then it is right—skewed. (b) According to Chebyshev’s Rule, for any data set at most 1/4 of the observations are more than 2 standard
deviations away from the mean. (c) The conditional probability P(AB) is always smaller than the unconditional probability P(A).
(d) The standard deviation of E decreases as the sample size increases.
(e) If 2 is a standard normal random variable and P(—a < z < a) = 1 — a, then a = 2:042. (f) The Central Limit Theorem only applies when the observed data :31, . . .,:rn are sampled from a normal
distribution. (g) In a hypothesis testing setup, a Type I error is committed when the null hypothesis is rejected, but in reality
the null hypothesis is true. . (10 marks) Let A, B and C be events with P(A) = 0.5, P(B) 7— 0.7 and P(C) = 0.4. Furthermore, it is known that A and B are independent and that A and C are mutually exclusive. (a) Compute P(A ('1 B) and P(A D C). (b) Compute P(A U B) and P(A U C). (c) Compute P(BA). (d) Are B and C mutually exclusive? Justify! . (12 marks) Suppose that 40% of the Canadian population has blood type 0+. A sample of 20 Canadians is selected at random. (a) What is the probability that exactly 9 of them have blood type 0+?
(b) What is the probability that at least 6 of them have blood type 0""?
(c) What is the probability that more than 4 but less than 11 of them have blood type 0+? (d) If the experiment is repeated many times, how many people in the sample, on average, should be expected
to have blood type 0+? (12 marks) The picture tubes for the Xenon 40—inch plasma television sets are manufactured in 2 plants. The
ﬁrst plant manufactures 35% 0f the picture tubes, whereas the second plant manufactures the remaining 65%.
The quality control department has determined that 2% of the picture tubes produced in the ﬁrst plant are
defective, whereas 3% of the tubes produced in the second plant are defective. (a) What is the probability that a randomly selected picture tube is defective? (b) If a randomly selected picture tube is found to be defective, what is the probability that it was manufactured
in the ﬁrst plant? (c) A picture tube is selected at random. Are the two events “the picture tube is defective” and “the picture
tube was manufactured at the ﬁrst plan ” independent? Justify! (10 marks) A medical researcher wishes to determine the percentage of females who take vitamins. He wishes
to be 99% conﬁdent that the estimate is within 2 percentage points of the true population proportion. A recent
study of 180 females showed that 25% took vitamins. (a) How large should the sample size be? (b) If no estimate of the sample proportion were available, how large should the sample be? —Please turn over— MATH 203 Final Exam Page 2 12/7/07 6. (13 marks) A study assessed fatigue in steel plant workers due to heat stress. A random sample of 58 casting
workers had a mean post—work heart rate of 75.7 bpm (beats per minute) with a standard deviation of 11.2 bpm. (a) Is there sufﬁcient evidence at a level of signiﬁcance of a = 0.01 to conclude that the mean postwork heart
rate of for casting workers exceeds the normal resting heart rate of 72 bpm? (1) State the hypotheses. (2) What type of test will you use? What is the suitable test statistic? Justify!
(3) Compute the value of the test statistic. (4) Find the p—value of the test. (5) State your conclusion. (b) Find a 99% conﬁdence interval for the true average postwork heart rate of casting workers. Interpret your
result. Does or doesn’t it contradict your results from part (a)? Explain! (12 marks) A recent study claimed that at least 15% of all eighth—grade students are overweight. In a sample of
80 students, 9 were found to be overweight. At or = 0.05, is there enough evidence to reject the claim? (a) State the hypotheses. (b) What type of test will you use? What is the suitable test statistic? Justify!
(c) Find the critical value(s) and the rejection region(s). (d) Compute the value of the test statistic. (e) State your conclusion. . (12 marks) A real estate agent compares the selling prices of homes in two suburbs of Montreal to see whether there is a difference in price. The results of the study are shown here. Is there evidence to reject the claim that
the average cost of a home in both locations is the same? Use a = 0.01. —— $163,255 $159,102
$5,602 $4,731
34 40 (a) State the hypotheses. (b) What type of test will you use? What is the suitable test statistic? Justify!
(c) Find the critical value(s) and the rejection region(s). ((1) Compute the value of the test statistic. (e) State your conclusion. (12 marks) Glaucoma is a leading cause of blindness. A study measured the corneal thickness of six patients who
have glaucoma in one eye but not in the other. The results of the study were (thickness measured in microns): Patient 1 2 3 4 5 6
Normal 484 478 492 444 436 476 Glaucoma 488 478 480 426 440 460 Does this provide sufficient evidence at oz = 0.1 to conclude that mean corneal thickness is greater in normal
eyes than in eyes with glaucoma? Assume that the difference in thickness between the normal eye and the eye
with glaucoma is approximately normally distributed. (a) State the hypotheses. (b) What type of test Will you use? What is the suitable test statistic? Justify!
(c) Find the critical value(s) and the rejection region(s). ((1) Compute the value of the test statistic. (e) State your conclusion. Appendix A Tables 885 TABLE II Binomial Probabilities .006) ._. ﬂ. _... .~.q..._,.._..__...— .__..___. 0w_____— .1: 0‘1"2—3""Ii"'5”'6'7'_8"'9 1o k
Tabulated values are Ep(x). (Computations are rounded at the third decimai place.) x=0 (continued) 886 Appendix A Tables TABLE II Continued d.n = 8
kunmunnnunm
0 .923 .663 .430 .168 .058 .017 .004 .001 .000 .000 .000 .000 .000
1 997 .000
2
3
4
5
6
7 en—9 Vuanun
.914 .630 .387 .134 .040 .010 .002 .000 .000 .000 .000 .000 WQQUIBUJNHG f.n = 10
Vim“unmum
.904 .599 .349 .107 .028 .006 .001 .000 .000 .000 .000 .000 996 .000 \OWQQUIBUJNH: ..  (continued) Appendix A Tables 887 TABLE II Continued HHH
NHexoooqatJIAmNrH: H
DJ 0
1
2
3
4
5
6
7
8
9
0 H HHHHHHH I'd
\oooqaxmhwnrt (continued) 894 Appendix A Tables TABLE IV Normal Curve Areas .0
.1
.2
.3
.4
.5
.6
.7
.8 Source: Abridged from Table I of A. Hald, Statistical Tables and Formulas (New York: Wiley), 1952. Reproduced by permission of A. Hald. 896 Appendix A Tables TABLE VI Critical Values of t # f0) Degrees of
Freedom [0005
1 636.62
2 31.598
3 12.924
4 8.610
5 6.869
6 5.959
7 5.408
8 5.041
9 4.781
10 4.587
11 4.437
12 4.318
13 4.221
14 4.140
15  4.073
16 4.015
17 3.965
18 3.922
19 3.883
20 3.850
21 3.819
22 3.792
23 3.767
24 3.745
25 3.725
26 3.707
27 3.690 '
28 3.674
29 3.659
30 3.646
40 3.551
60 3.460
120 3.373
00 3.291 Source: This table is reproduced with the/kind permission of the Trustees of Biometrika from E. S. Pearson and H. O. Hartley (eds),
The Biometrika Tables for Statisticians, Vol. 1, 3d ed., Biometrika, 1966. \ ...
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