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Unformatted text preview: Student Name : Student Number : Faculty of Science FINAL EXAMINATION Mathematics 203
Principles of Statistics I
Thursday April 20th 9 :0012 :00, 2006 Answer directly on the test (use front and back if necessary). You must justify all your answers to receive full credit. This means showing your
work and / or giving explanations. Nongraphing calculators are allowed. One 8.5” X 11” two—sided sheet of notes is allowed. Language dictionaries are allowed. There are 20 pages to this exam with 2 additional pages of tables. Total number of points for the exam is 100 with a bonus question that is worth up to 5 extra points. Examiner : Genevieve Lefebvre Associate Examiner : Professor R. Steele MATH 203 Final Examination April, 2006 Question 1. (15 points total) Are the following statements True (T) or False (F)? 1. In hypothesis testing, the type I error is the probability that we reject the null hypothesis when it is true. ...... 2. The Student—t distribution is more variable than the standard normal distribution for all ﬁnite values of degrees of freedom. ...... 3. It is desirable that the standard deviation of an estimator of a population parameter does not depend on the'sample size. ...... 4. In all cases, we cannot assume that X is normally distributed when the sample size is small (i.e. when n < 30). ...... 5. The p—value is not the probability that the null hypothesis is true. ...... MATH 203 Final Examination April, 2006 Question 2. (10 points total) You toss a fair coin three times and observe the result. 1) Enumerate all sample points associated with this experiment and their respective proba— bility. (4 points) ii) Let X be the random variable deﬁned as : {# of heads — # of tails}. Find the probability distribution of X. (4 points) MATH 203 Final Examination April, 2006 iii) Find the variance of X. (2 points) MATH 203 Final Examination April, 2006 Question 3. (10 points total) Starting at 7 :00 am, buses pass by the St—Laurent/Villeray stop every 15 minutes. The
refore, buses pass at 7:00 am, 7 :15 am, 7 :30 am and so on... A passenger arrives at this stop
between 7:00 am and 7:30 am, Where his exact time of arrival is uniformly distributed on this interval. i) Find the passenger’s expected time of arrival to the bus stop. (3 points) ii) Find the probability that the passenger’s arrival time is between 7:00 and 7 :15 am. (3 points) MATH 203 Final Examination April, 2006 iii) Find the probability that the passenger waits less than 5 minutes before a bus passes. (4 points) MATH 203 Final Examination April, 2006 Question 4. (10 points total) We suppose that the height (in cm) of a 25 year old male is a normal random variable with u = 175 and 02 = 36. i) Find the probability that a 25 year old male has a height of exactly 175 cm. (3 points) ii) Find the probability that a 25 year old male has a height greater than 180 cm. (3 points) MATH 203 Final Examination April, 2006 iii) Among the men. having a height greater than 180 cm, What percentage have a height greater than 185 cm. (4 points) MATH 203 Final Examination April, 2006 Question 5. (11 points total) Assume that among juveniles, the distribution of the score on a verbal IQ test has a mean of ,u = 107 and a standard deviation 0 = 15. i) Suppose that you take a random sample of n = 84 juveniles. Find the probability that the sample mean verbal IQ score will be 104 or less. (5 points) MATH 203 Final Examination April, 2006 ii) Over the past 20 years, the Journal of Abnormal Psychology has published numerous
studies on the relationship between juvenile delinquency and poor verbal abilities. In one
cited study, the researchers found that a random sample of n = 84 juvenile delinquents had
a mean verbal IQ of (E = 104. Considering your answer to part i), do you think that the population mean and standard deviation for delinquent juveniles are the same as those for all juveniles? Explain. (2 points) iii) Would :E = 104 be more likely to be observed from a population having a mean ,u = 106 and standard deviation 0 = 8 ? Explain. (4 points) 10 MATH 203 Final Examination April, 2006 Question 6. (8 points total) Scientists have linked a catastrophic decline in the number of frogs inhabiting the world
to ultraviolet radiation. Researchers at Oregon State University device the following expe
riment : eggs from the endangered Paciﬁc tree frog species are selected and divided into
two groups of size n1 2 70 and n2 = 80. In the ﬁrst group, the eggs are shielded with
ultraviolet—blocking sun shades, while in the second group the eggs are not. The number of eggs successfully hatched in each group is the following : Number of eggs that hatched for SunShaded Eggs : 34 out of 70
Number of eggs that hatched for Unshaded Eggs : 31 out of 80 Provide evidence of whether or not there is a statistically signiﬁcant difference in the pro— portion of eggs that hatched under the two conditions (use a = 0.10). 11 MATH 203 Final Examination April, 2006 (continued) 12 MATH 203 Final Examination April, 2006 Question 7. (12 points total) The company Ynos surveyed the community of online computer game players to deter—
mine the mean number of hours per week played on such games. Based on a random sample
of 75 online players, Ynos observed a sample mean and standard deviation of 5; = 15.80 and s = 7.00 hours, respectively. 1) Contruct a 98% conﬁdence interval for a. (5 points) ii) Interpret the result from i). (2 points) 13 MATH 203 Final Examination April, 2006 iii) Find the sample size required to estimate ,u within 0.5 with a conﬁdence of 98%, or in
other words, what is the sample size required to shrink the width of your 98% conﬁdence interval to 1. (5 points) 14 MATH 203 Final Examination April, 2006 Question 8. (12 points total) A survey is done to estimate the proportion of Quebec adult residents in favor of the
privatization of the SAQ (Societe des alcools du Quebec). A random sample of size n = 586
is taken among the Quebec adult residents. It is found that among these people, 277 are in favor of the privatization of the SAQ. i) Give a point estimate for the proportion of Quebec residents in favor of the privatization of the SAQ. (2 points) ii) Give a conservative 95% conﬁdence interval for the proportion of Quebec residents in favor of the privatization of the SAQ and interpret the result. (5 points) 15 MATH 203 Final Examination April, 2006 iii) Does this sample provide evidence that the population prefers one option to the other? (use oz 2 0.05). (5 points) 16 MATH 203 Final Examination April, 2006 Question 9. (12 points total) A psychologist wants to investigate the selfperception of the undergraduate students on
their scholar results, and in particular wants to test the hypothesis that the students are too selfconﬁdent. To do so he sets up the following experiment : He contacts a professor from a law school to get permission to come in class the day
of the ﬁnal exam and ask the students to guess their score on the exam. The students are
assured about the conﬁdentiality of the process so that only the psychologist knows what is
the score they guess. When the professor is done with the marking, she gives the results to the psychologist who
reports the following statistics on the 35 students who wrote the exam : 1. The mean and standard deviation of the “guessed” scores are respectively 71.73 and 8.95.
2. The mean and standard deviation of the (real) scores are respectively 67.67 and 9.73. 3. The mean and standard deviation of the differences between the “guessed” and the (real) score is 4.06 and 3.18. The psychologist has forgotten most of the statistics courses he took 10 years ago and asks you to perform a test to investigate his hypothesis. i) Perform an appropriate test for the hypothesis that the students are too self—conﬁdent about their scholar results at the level a = 0.10. (6 points) 17 MATH 203 Final Examination April, 2006 (continued) ii) Find the observed level of signiﬁcance (p—value) and interpret the result. (4 points) 18 MATH 203 Final Examination April, 2006 iii) Do you think that the previous results are likely to be valid and generalizable to the whole population of undergrad students? Explain clearly and brieﬂy. (2 points) 19 MATH 203 Final Examination April, 2006 Bonus question (5 points total) Ten people are being given a blood test to see if they have a certain disease. However,
rather than testing each person individually, it is decided to mix the 10 blood samples and
test that sample instead. If the test is negative, only one test is necessary for the 10 people.
However, if the test is positive, each person will be tested individually, and overall 11 tests
are performed on this group. We suppose that the probability that a person has the disease
is 0.1 and that the disease affects people independently from each of the other. We also suppose that the test for the group will be positive as soon as one person is sick. Find the expected number of tests necessary for the 10 people. Is this strategy good compared to the strategy of testing each individual separately? Explain. 20 59
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17
79
24 49
52
33
89
21 30
15
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l0 Critical Values of t f0) V ' hop: .1.qu 132's _ 1.010 . 3.995 [601 .t._ot)us 1 3.078 6.314 12.706 31.821 63.657 318.31 636.62 2 1.886 2.920 4.303 6.965 9.925 22.326 31.598 3 1.638 2.353 3.182 " 4.541 5.841 10.213 12.924 4 1.533 2.132 2.776 3.747 4.604 7.173 8.610 5 1.476 2.015 2.571 3.365 4.032 5.893 6.869 6 1.440 1.943 2.447 3.143 3.707 5.208 5.959 7 1.415 1.895 2.365 2.998 3.499 4.785 5.408 8 1.397 1.860 2.306 2.896 3.355 4.501 5.041 9 1.383 1.833 2.262 2.821 3.250 4.297 4.781 10 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 1.323 1.721 2.080 2.518 _.2.831 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 1.319 1.714 _2.069 2.500 2.807 3.485 3.767
24 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.450 3.725
26 1.315 1.706 2.056 2.479 2.779 3.435 3.707
27 1.314 1.703 2.052 2.473 2.771 3.421 3.690
28 1.313 1.701 2.048 2.467 2.763 3.408 3.674
29 1.311 1.699 2.045 2.462 2.756 3.396 3.659
30 1.310 1.697 2.042 2.457 2.750 3.385 3.646
40 1.303 1.684 2.021 2.423 2.704 3.307 3.551
60 1.296 1.671 2.000 2.390 2.660 3.232 3.460
120 . 1.289 1.658 1.980 2.358 2.617 3.160 3.373
1.282 1.645 1.960 2.326 2.576 3.090 3.291 Source: This table is reproduced with the ki
The Biometrika Tables for Statisticians, V01. nd permission of the Trustees of Biometrika from E. S. Pearson a 1, 3d ed., Biometrika, 1966. nd H. O. Hartley (eds), % Normal Curve Areas
. _'—__——“—‘'—  :.':e'r.::ia_: l z .00 .01  .02 .03 .04 .05 .06. .07 .08 .09
.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486, .2517 .2549
.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 i
.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 
10 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621
, 11 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830
" 12 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015
' 13 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177
;[ 14 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319
i 15 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1‘ 16 .4452 _ 4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
17 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
18 .4641 .4649 .4656 .4664 .4671 .4678. .4686 .4693 .4699 .4706
19 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767
20 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817
21 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857
22 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 _ .4887 .4890
23 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916
24 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936
25 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952
26 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964
27 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974
28 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981
29 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986
30 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990 —————_____—___—__________ Source: Abridged from Table I of A. Hald, Statistical Tables and Formulas (New York: Wiley), 1952. Reproduced by permission of A. Hald. ...
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