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Unformatted text preview: 063:161 Introduction to Biostatistics
Fall 1992, Exam 2
October 8, 1992 I. True/False Questions (2 points each) NOTE: If the statement is true, mark response A
If the statement is false, mark response B 1. If the random variableX has a Poisson distribution, its mean and variance are equal;
. The mean of a normal distribution must be greater than or equal to zero. . For any binomial distribution, the variance is always less than the mean. $03M . For every probability distribution, approximately 95% of the probability mass falls within
two standard deviations of the mean. 5. The normal approximation to the binomial distribution is valid whenever the binomial pa
rameter n is greater than 20'. 6. ‘ The number of possible outcomes for a Poisson random variable is inﬁnite.
7. A normal distribution with mean H and variance 02 is symmetric about zero 8. Under certain circumstances, the binomial distribution can be approximated by the Poisson
distribution. 9. If X is a continuous random variable, then Pr(a < X < b) < Pr(a S X g b), for any two
numbers a and b (a < b). 10. If X has a normal distribution with parameters ,u and 02, then
Pr(X <p—a)=Pr(X >u+a) II. MultipleChoice Questions (3 points each) 11. Suppose that the random variable X has a binomial distribution with parameters n = 1000
and p = 0.4. The preferred method of approximating this distribution is by:
A a normal distribution with mean 400 and variance 240
B a normal distribution with mean 400 and variance x/ZTO
C a Poisson distribution with mean 400
D a Poisson distribution with mean 240
E none of the above 12. Suppose that the random variable X has a binomial distribution with parameters n = 200
and p = 0.01. The preferred method of approximating this distribution is by:
A a normal distribution with mean 2 and variance 1.98
B a normal distribution with mean 2 and variance «ll—95
. C a Poisson distribution with mean 2 D a Poisson distribution with mean 1.98
E none of the above The next six questions refer to an experiment in which a fair coin is tossed three times. Let X
be the random variable denoting the number of heads. 13. X is: A a continuous random variable
B 'a discrete tandem variable
C neither continuous or discrete 14. The expected value of X is:
A 0.75
B 1.0
C 1.5
D 2.0 E none of the above 15. The variance of X is:
A 0.75
B 1.0 _
C 1.5
D 2.0
E none of the above , 16. Pr(X 2 2) =
A 0.0072
B 0.125
C 0.250
D 0.375
E 0.5 17. Pr(X < 1) = p
A 0.125
B 0.375
C 0.500
D 0.625
E 0.875 18. The probability distribution function of X is given by:  _ 1/3 0 g a: g 3
A fa) — {0 otherwise _ e050.5”‘/x! a: = 0,1,2,3
B ﬂat) _ {0 otherwise 1 —0 5a:2 ——— ' 0 < a: < 3
C f(x)={\/_21re   0 otherwise 1 3 ‘ — = 0, 1,2,3
D...={.<.> . 0 otherwise E none of the above WWW mfmm_mumwwﬂx
w...“ . ,— III. MultipleChoice Questions (4 points each) V The next two questions refer to a random variable X with. probability mass function given by: 0.2 if 2:0
0.1 if 2:1 ' f(x) = 0.3 if 2:2
? if 3:3
0.0 otherwise 19. f(3)=
, A 0.0
B 0.2 C 0.3 D 0.4 E 0.5 20. E(X) 2 ~ A 0.475
B 0.633
C 1.5
D 1.9
E 2.0 The next two questions refer to the function: ' _ a: '0 _<_ a: g c
f(x) _ {0 otherwise 21. What value of c makes f(a:) a valid probabilitydensity function?
A 1
B x/5
C 2
D 00
E none of the above 22. If the random variable X has probability distribution function given by f(z). ﬁnd Pr(X < 1)
A 0 ' 
B 0.25 '
C 0.5
D 0.75
E 1 _ ....,.._M,..____.__._____._______T_____.___.__._.. The next three questions refer to the random variable X, which has a normal distribution with
parameters a  5 and a2  —.4 Evaluate the following probabilities. 23. Pr(4 g X g 7) =
A 0.1498
B 0.2902
C 0.5328
D 0.8502
E 0.4672 24. Pr(X > 3) =
A 0.1587
B 0.3085
0 0.3413
D 0.6915
E' 0.8413 ' A 0.3085
B 0.4013
C 0.5910
D 0.5987
E 0.6915 Suppose that the probability that a randomly selected baby will be albino 15 1/20, 000: 0. 00005.
Use this information to answer the next two questions. 26. If 80,000 babies are born in a given year, what is the approximate probability that at most
one will be albino? A 0.018
30067
C 0.073
D 0.092
E 0.106 1 27. If 80,000 babies are born in a given year, what is the approximate probability that at least
two will be albino? A 0.146
B 0.841
C 0.853
D 0.894
E 0.908 28. According to national statistics, 25 cases of leukemia are expected to occur in a particular community in one year. What is the approximate probability of observing 30 or more cases, under the assumption that the rate predicted by the national statistics is correct?
A 0.136 ’ B 0.159
C 0.184
D 0.421
E 0.816 The next four questions refer to the following situation. Suppose that the probability that a
newly~diagnosed patient with lung. cancer survives for two years is 0.6. In a sample of 150
newlydiagnosed cases, what is the approximate probability that: 29. exactly 90 patients survive for two years:
A, 0.000 I
B 0.004
C 0.008
D 0.032
E 0.064 30. at least 80, but no more than 90, patients survive for two years:
A 0.11 B 0.12
C 0.45
D 0.49
E 0.57 31. at least 85 patients survive for two years:
A 0.20
B 0.44
C 0.56
D 0.77
E 0.82 32. at most 80 patients survive for two years:
A 0.0401
B 0.0475
G 0.0571
D 0.0778
E 0.3897 ——————————T————— mmmm_m_.mwﬂwm. 0...... ..
_.. .....M.Mﬂ«W_—m _——_.._—.._____—___——.._____———____—_——..__——_..—u_______~__—___ ______________ .9025 .8100 .7225 .6400 .5625 .4900 .4225 I .3600 .3025 .2500
' .0950 .1800 .2550 .3200 .3750 .4200 .4550 .4800 .4950 . .5000
.0025 .0100 .0225 .0400 .0625 .0900 .1225 .1600  .2025 .2500 .8574 .7290 .6141 .5120 .4219 .3430 .2746 .2160 .1664‘ .1250
.1354 .2430 .3251 .3840 .4219 .4410 .4436 .4320 . .4084 .3750
.0071 .0270 .0574 .0960 .1406 .1890 .2389 .2880 .3341 .3750
.0001 .0010 .0034 .0080 .0156 .0270. .0429 .0640 .091 1 .1250 .8145 .6561 .5220 .4096 .3164 .2401 .1785 .1296 .0915 .0625
.1715 .2916 .3685 .4096 .4219 .4116 .3845 .3456 .2995 .2500
.0135 .0486 _ .0975 .1536 .2109 .2646 .3105 .3456 .3675 .3750
.0005 .0036 .01 15 .0256 .0469 .0756 .1 1 15 .1536 .2005 .2500 i
.0000 .0001 .0005 .0016 .0039 .0081 .0150 .0256 .0410 .0625 .7738 .5905  .4437 .3277 .2373 .1681 .1 160 .0778 .0503 .0313
.2036 .3280 .3915 .4096 .3955 .3602 .3124 .2592 .2059 .1563
.0214 .0729 .1382 .2048 .2637 .3087 .3364 .3456 .3369 .3125
.0011 .0081 L0244 .0512 .0879 .1323 .1811 .2304  .2757 .3125
.0000 .0004 .0022 .0064 .0146 .0283 .0488 .0768 .1 128 .1563
.0000 .0000 .0001 .0003 .0010 .0024 .0053 .0102 .0185 .0313 0
l
2
0
1
2
3
0
1
2
3
4
0
1
2
3
4
5
6 0 .7351 .5314 .3771 .2621 .1780 .1176 .0754 .0467 .0277 .0156
1
2
3
4
5
6
0
l
2
3
4
5
6
7
0
1
2
3 .2321 .3543 .3993 .3932 .3560 .3025 .2437 .1866 .1359 .0938
.0305 .0984 .1762 .2458 .2966 .3241 .3280 .31 10 .2780 .2344
.0021 .0146 .0415 .0819 .1318 .1852 .2355 .2765 .3032 .3125
.0001 .0012 .0055 .0154 .0330 .0595 .0951.1382 .1861 .2344'
.0000 .0001 .0004 .001 5 .0044 .0102 .0205 .0369 ' .0609 .09 38
.0000 .0000 .0000 .0001 .0002 .0007 .0018 .0041 .0083 .0156 .6983 .4783 .3206 .2097 .1335 .0824 .0490 .0280 .0152 .0078 .
.2573 .3720 .3960 .3670 .3115 .2471 .1848 .1306 .0872 .0547 1‘
.0406 .1240 .2097 .2753 .3115 .3177 .2985 .2613 .2140 .1641
.0036 .0230 .0617 .1 147 .1730 .2269 .2679 .2903 .2918 .2734
.0002 .0026 .0109 .0287 .0577 .0972 .1442 .1935 .2388 .2734
.0000 .0002 .0012 .0043 .01 15 .0250 .0466 .0774 .1 172 .1641
.0000 , .0000 .0001 .0004 .0013 .0036 .0084 .0172 .0320 .0547
.0000 .0000 ~ .0000 .0000 .0001 .0002 .0006 .0016 .0037 .0078 .6634 .4305 .2725 .1678 .1001 .0576 .0319 .0168 .0084 .0039
.2793 .3826 , .3847 .3355 .2670 .1977 .137 3 .0896 .0548 .0313
.0515 .1488 .2376 .2936 .3115 .2965 .2587 .2090 .1569 .1094
.2786 .0054 .0331 .0839 .1468 .2076 .2541 532 APPENDIX ONE / TABLES a ,— k
TABLE 2 Exact Poisson Probabilities Pr(X = k) = E '7‘ ___—____——_——_——__—__——_————__——_——_m———_———_—__——_—____—_____—_ ____________ It 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
————_________.___—_._____________
.6065 1.3679 .2231 .1353 .0821 .0498 .0302 .0183 .0111 .0067
.3033 .3679 .3347 .2707 .2052 .1494 .1057 .0733 .0500 .0337
.0758 .1839 .2510 .2707 .2565 .2240 .1850 .1465 .1125 .0842
.0126 .0613 .1255 .1804 .2138 .2240 .2158 .1954 .1687 .1404 g i
.0016 .0153 .0471 .0902 .1336 .1680 .1888 .1954 .1898 .1755 ‘
.0002 .0031 .0141 .0361 .0668 .1008 .1322 .1563 .1708 .1755 "'
. . .0035 .0120 .0278 .0504 20771 .1042 .1281 .1462
.0000 .0001 .0008 .0034 .0099 .0216 .0385 .0595 .0824 .1044
.0000 .0000 .0001 .0009 .0031 .0081 .0169 .0298 .0463 .0653
.0000 .0000 .0000 .0002 .0009 .0027 .0066 .01 32 .0232 .0363 7 5 1 SwmuauAuN—c
é éééééé
éééééé
§§§§§§
§§§§§§
§§§§§§
éééééé
éééééé
éééééé
§§§§§§
§§§§§§ it 5.5 6.0 6.5 7.0 7.51 8.0 , 8.5 9.0 9.5 10.0 f 0 .0041 .0025 .0015 .0009 .0006 .0003 .0002 .0001 .0001 .0000 _ ‘2
1 _.0225 .0149 .0098 .0064 .0041 , .0027 .0017 .0011 .0007 .0005 , ‘
2 .0618 .0446 .0318 .0223 .0156 .0107 .0074 .0050 .0034 .0023 3 .1133 .0892 .0688 .0521 .0389 .0286 .0208 .0150 .0107 .0076 4 .1558 .1339 .1118 .0912 .0729 .0573 .0443 .0337 .0254 .0189 5 .1714 .1606 .1454 .1277 .1094 .0916 .0752 .0607 .0483 .0378 6 .1571 .1606 .1575 .1490 .1367 .1221 .1066 .0911 .0764 .0631 7 , .1234 .1377 .1462 .1490 .1465 .1396 .1294 .1171 .1037 .0901 8 .0849 .1033 .1188 .1304 .1373 .1396 .1375 .1318 .1232 .1126 9 .0519 .0688 .0858 .1014 .1144 .1241 .1299 .1318 .1300 .1251 10 .0285 .0413 .0558 .0710 .0858 .0993 .1104 .1186 .1235 .1251 11 .0143 .0225 .0330 .0452 .0585 .0722 .0853 .0970 .1067 .1137 12 .0065 .0113 .0179 .0263 .0366 ”.0481 .0604 .0728 .0844 .0948 13 .0028 .0052 .0089 .0142 .0211 .0296 .0395 .0504 .0617 .0729 14 .0011 .0022 .0041 .0071 .0113 .0169 .0240 .0324 .0419 .0521 i
i .0013 .0033 .0057 .0090 .0136 .0194 .0265 .0347 g;
.0007 .0014 .0025 .0045 .0072 .0109 .0157 .0217 g.
.0003 .0006 .001; .0021 .0036 .0058 .0088 .0123 1*
.0001 .0002 .0005 .0009 .0017 .0029 .0046 .0071 "
.0001 .0002 .0004 .0003 .0014 0023 .0037 .0001 .0002 .0003 .0006 .0011 '.0019 33%
E33 E
E.
E E
i
i
E THE NORMAL DISTRIBUTION / TABLE 3 535 TABLE 3 The Normal Distribuﬂoh A(x) = 91):) =. Pr(XS x) .4039
.4108
.4177
.4245
‘ . . . . . . .4313
I _ . ._ . .  . .4381
, I .4448 I .4515
.4581
.4647
.4713 536 APPENDIX ONE / TABLES TABLE 3 (Continued x )v at c: .‘—m_v<,. 4. X L56
L57
1.58
L59
L60
L61
. L62
:5 L63
.. _L64
L65
L66
L67
L68
L69
170
L71
L72
L73
L74
L75
L76
L77
L78
L79
L80
L81
L82
L83
L84
L85
L86
L87
L88
L89
L90
L91
L92
L93
194
L95
.L96
L97
L98
L99
200
201
202 ....._____._..._.___.______..________.__.T__.____.__._____._____ IAILE 3 (Clmwmwuetb All .9406
.9418
.9429
.9441
.9452
.9463
.9474
.9484
.9495
.9505
.9515
.9525
.9535
.9545
.9554
.9564
.9573
.9582
.9591
.9599
.9608
.9616
.9625‘
.9633
.9641
.9649
.9656
.9664 .9671 .9678 .9686
.9693
.9699
.9706
.9713
.9719
.9726
.9732
.9738
.9744
.9750
.9756
.9761
.9767
.9772
.9778
.9783 51. .0594
.0582
.0571
.0559
.0548
.0537
.0526
.0516
.0505
.0495
.0485
.0475
.0465
.0455
.0446
.0436
.0427
.0418
.0409
.0401 ..0392
.0384
.0375
.0367
.0359
.0351
.0344
.0336 .0329 .0322
.0314 .0307
.0301
.0294
.0287
.0281
.0274
.0268
.0262
.0256
.0250
.0244
.0239
.0233
.0228
.0222
.0217 ¢=I .4406
.4418
.4429
.4441
.4452
.4463
.4474
.4484
.4495
.4505
.4515
.4525
.4535
.4545
.4554
.4564
.4573
.4582.
.4591
.4599
.4608
.4616
.4625
.4633
.4641
.4649
.4656
.4664 »
.4671
.4678
.4686
.4693
.4699
.4706
.4713
.4719
.4726
.4732
.4738
.4744
.4750
.4756
.4761
.4767 .4772
.4778
.4783 D§ .8812
.8836
.8859
.8882
.8904
.8926
.8948
.8969
.8990
.9011
.9031
.9051
.9070 9090
.9109
.9127
.9146
.9164 .9181
.9199 .9216
.9233 .9249
.9265 ..9281
.9297 .9312
.9327
.9342
.9357
.9371
.9385
.9399
.9412
.9426
.9439
.9451
.9464
.9476
.9488 .9500 ’.9512 .9523
.9534
.9545
.9556
.9566 THE NORMAL DISTRIBUTION / TABLE 3 537 ...
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