Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences
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10
20
90
80
70
60
50
40
30
GFR
Boxplot of GFR
ANSWERS TO ODD-NUMBERED EXERCISES A-113
17. Some examples include difference, diversity, departure, discrepancy, deviation, and
entropy
19.
21. Answers will vary: It is not uncommon for students to score higher on exams as a
semester progresses; therefore, the exam scores are likely to be left skewed, making
the median, which is less affected by skew, to be the better choice.
23. Answers will vary: Using Sturges's rule, where
. An estimate of sample standard deviation can be found by dividing the sample
range by 4. Therefore, so that . Using this formula, then and
suggesting that (d) or (e) may be appropriate.
25. Answers will vary: Imagine you are examining protein intake among college stu-
dents. In general most students are likely to consume the average daily protein intake,
but among this population, there is likely to be a fair number of athletes who consume
large amounts of protein owing to the demands of their sport. In that case, the data are
likely to be positively skewed, and the median better represents the central tendency of
the data.
27. Variable N Mean Median TrMean StDev SE Mean
S/R 216 0.3197 0.2440 0.2959 0.2486 0.0169
Variable Minimum Maximum Q1 Q3
S/R 0.0269 1.1600 0.1090 0.4367
IQR = .3277, Range = 1.1331, IQR>R = .2892
w =
160
9.23
= 17.33
R = 160R L 4ss L
R
4
9.23
k = 1 + 3.3221log
10
300) Mw =
R
k
,
x = 3.95, Median = 3, s = 3.605, s
2
= 12.998
0.0
1.2
1.0
0.8
0.6
0.4
0.2
S/R ratio
Boxplot of S/R ratio
*
*
*
*
*
Stem-and-leaf of C1 N = 107
Leaf Unit = 1.0
145
5 5 0004
12 5 5556899
18 6 013444
31 6 5555666777888
(28) 7 0000011122222222333333344444
48 7 666666666677888999
A-114
ANSWERS TO ODD-NUMBERED EXERCISES
40
30
20
10
0
50 75 100
Nutritional Status
Frequency Polygon of Status
Frequency
125 150
0
10
40
30
20
Frequency
60 80
Nutritional Status
Histogram of Status
100 120
29. (a) Variable N Mean Median TrMean StDev SE Mean
nutri 107 75.40 73.80 74.77 13.64 1.32
Variable Minimum Maximum Q1 Q3
nutri 45.60 130.00 67.50 80.60
Variance = 186.0496, Range = 84.4, IQR = 13.1, IQR>R = .1552
ANSWERS TO ODD-NUMBERED EXERCISES A-115
30 8 000002234444
18 8 56889
13 9 01223
8 9 679
51000
3109
211
211
2123
112
1130
40
130
120
100
80
70
60
50
110
90
Status
Boxplot of status
*
*
*
*
(d)
(e) (f)
Chapter 3
3.4.1. (a) .6631 (b) marginal (d) .0332 (e) joint (f) .0493
(g) conditional (h) .3701 (i) addition rule
3.4.3. (a) male and split drugs, .3418 (b) male or split drugs or both, .8747
(c) male given split drugs, .6134 (d) male, .6592
3.4.5. .95
3.4.7. .301
3.5.1. (a) A subject having the symptom (S) and not having the disease.
(b) A subject not having S but having the disease. (c) .96
(d) .9848 (e) .0595 (f) .99996 (g) .00628, .999996, .3895, .9996,
.8753, .9955 (h) predictive value increases as the hypothetical disease rate
increases
3.5.3. .9999977
Review Exercises
3. (a) .2143 (b) .5519 (c) .1536 (d) .4575 (e) .5065
5. (a) .1349 (b) .6111 (c) .3333 (d) .5873 (e) .3571
(f) .6667 (g) 0 (h) .3269
1>107 = .0093102>107 = .9533
102.68; 103>107 = .9626; 75.4 ; 3113.642; 34.48, 116.32; 105>107 = .9813
75.4 ; 13.64; 61.76, 89.04; 79>107 = .7383; 75.4 ; 2113.642; 48.12,
7. (a) 1. .2200 2. .5000 3. .0555 4. .1100 5. .5900
(b) 1. .3000 2. .3900 3. .3900 4. .1667 5. .0700 6. .6000
9. (a) .0432 (b) .0256 (c) .0247 (d) .9639 (e) .5713
(f) .9639 (g) .9810
11. .0060
13. .0625
15. mothers under the age of 24
17. null set, as events are mutually exclusive
19. (a) plasma lipoprotein between 10–15 or greater than or equal to 30.
(b) plasma lipoprotein between 10–15 and greater than or equal to 30.
(c) plasma lipoprotein between 10–15 and less than or equal to 20.
(d) plasma lipoprotein between 10–15 or less than or equal to 20.
21. (a) .7456 (b) .3300
23. .0125
Chapter 4
A-116 ANSWERS TO ODD-NUMBERED EXERCISES
4.2.1 (a)
Relative Cumulative
Number of Substances Used Frequency Frequency Frequency
0 144 .19 .19
1 342 .44 .63
2 142 .18 .81
3 72 .09 .90
4 39 .05 .95
5 20 .03 .98
6 6 .01 .99
7 9 .01 1.00
8 2 .003 1.003
9 1 .001 1.004
Total 777 1.004
(b)
0.5
0.4
0.3
0.2
0.1
0.0
Number of Substances Used
Frequency
0123456789
ANSWERS TO ODD-NUMBERED EXERCISES A-117
4.2.3.
4.3.1. (a) .1484 (b) .8915 (c) .1085 (d) .8080
4.3.3. (a) .2536, (b) .3461 (c) .7330 (d) .9067 (e) .0008
4.3.5.
4.3.7. (a) .5314 (b) .3740 (c) .0946 (d) .9931 (e) .0946
(f) .0069
4.3.9.
Number of
Successes, x Probability, f (x)
0
1
2
3
Total 1
4.4.1. (a) .156 (b) .215 (c) .629 (d) .320
4.4.3. (a) .105 (b) .032 (c) .007 (d) .440
4.4.5. (a) .086 (b) .946 (c) .463 (d) .664 (e) .026
4.6.1. .4236
4.6.3. .2912
4.6.5. .0099
4.6.7. .95
4.6.9. .901
4.6.11.
4.6.13. 1.77
4.6.15. 1.32
-2.54
3!
3! 0!
1.22
0
1.82
3
= .512
3!
2! 1!
1.22
1
1.82
2
= .384
3!
1! 2!
1.22
2
1.82
1
= .096
3!
0! 3!
1.22
3
1.82
0
= .008
mean = 4.8, variance = 3.264
x = 1.58, s
2
= 2.15, s = 1.47
0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1234
Number of Substances Used
Frequency
56789
4.7.1. (a) .6321 (b) .4443 (c) .0401 (d) .3064
4.7.3. (a) .1357 (b) .2389 (c) .6401 (d) .0721 (e) .1575
4.7.5. (a) .3413 (b) .1056 (c) .0062 (d) .3830
4.7.7. (a) .0630 (b) .0166 (c) .7719
Review Exercises
15. (a) .0212 (b) .0949 (c) .0135 (d) .7124
17. (a) .034 (b) .467 (c) .923 (d) .010 (e) .105
19. (a) .4967 (b) .5033 (c) .1678 (d) .0104 (e) .8218
21. (a) .0668 (b) .6247 (c) .6826
23. (a) .0013 (b) .0668 (c) .8931
25. 57.1
27. (a) 64.75 (b) 118.45 (c) 130.15 (d) 131.8
29. 14.90
31. 10.6
33. (a) Bernoulli assuming there is an equal probability of both genders
(b) Not Bernoulli—more than two possible outcomes
(c) Not Bernoulli—weight is not a binary variable
Chapter 5
5.3.1. 204, 6.2225
5.3.3. (a) .1841 (b) .7980 (c) .0668
5.3.5. (a) .0020 (b) .1736 (c) .9777 (d) .4041
5.3.7. (a) .9876 (b) .0668 (c) .0668 (d) .6902
5.3.9.
Sample
6, 8, 10 8.00
6, 8, 12 8.67
6, 8, 14 9.33
6, 10, 12 9.33
6, 10, 14 10.00
6, 12, 14 10.67
8, 10, 12 10.00
8, 10, 14 10.67
8, 12, 14 11.33
10, 12, 14 12.00
5.4.1. .3897
5.4.3. .0038
5.4.5. .0139
5.5.1. .1131
5.5.3. .0808
5.5.5. (a) .1539 (b) .3409 (c) .5230
5.6.1. .1056
5.6.3. .7938
m
x
= 10, s
x
= 1.333
x
A-118 ANSWERS TO ODD-NUMBERED EXERCISES
ANSWERS TO ODD-NUMBERED EXERCISES A-119
Review Exercises
11. .0003
13. .0262
15. .1335
17. .1093
19. .1292
21. 252
23. .53, .0476
25. At least approximately normally distributed
27. .9942
29. (a) No (b) Ye s (c) No (d) Ye s (e) Ye s (f ) Yes
Chapter 6
6.2.1. (a) 88, 92 (b) 87, 93 (c) 86, 94
6.2.3. (a) 7.63, 8.87 (b) 7.51, 8.99 (c) 7.28, 9.22
6.2.5. 1603.688, 1891.563; 1576.125, 1919.125; 1521.875, 1973.375
6.3.1. (a) 2.1448 (b) 2.8073 (c) 1.8946 (d) 2.0452
6.3.3. (a) 1.549, .387 (b) 2.64, 4.36; .49, .91 (c) Nitric oxide diffusion
rates are normally distributed in the population from which the sample was
drawn. (f) narrower (g) wider
6.3.5. 66.2, 76.8; 65.1, 77.9; 62.7, 80.3
6.4.1.
6.4.3. 17.70; 19.95; 24.40
6.4.5. 64.09, 479.91; 19.19, 524.81; 621.49
6.4.7. 2.1, 4.5; 1.8, 4.8; 1.3, 5.3
6.4.9.
6.5.1. .1028, .1642
6.5.3. .4415, .6615
6.6.1. .0268, .1076
6.6.3. .2667
6.7.1. 27, 16
6.7.3. 19
6.8.1. 683, 1068
6.8.3. 385, 289
6.9.1. 6.334, 44.63
6.9.3. 793.92, 1370.41
6.9.5. 1.17, 2.09
6.9.7.
6.10.1. .44, 17.37
6.10.3. .49, 2.95
6.10.5. .9, 3.52
6.10.7. 5.13, 60.30
Review Exercises
13.
15. pN = .30; .19, .41
x
= 79.87, s
2
= 28.1238, s = 5.3; 76.93, 82.81
170.98503 … s
2
… 630.65006
-.0843,
-32.58, -25.42; -33.33, -24.67, -34.87, -23.13
-77.49,
-12.60,-8.15,-5.90,
-549.82, -340.17; -571.28, -318.72; -615.52, -274.48
17.
19.
21.
23. 2.82
25. 362.73, 507.27
27. .44, .74
29. .188
31. Level of confidence decreases. The interval would have no width. The level of
confidence would be zero.
33. z, 8.1, 8.1
35. All drivers ages 55 and older. Drivers 55 and older participating in the vision
study.
37. .2865, .3529 (Use z since )
Chapter 7
7.2.1. Reject because
7.2.3. Fail to reject because
7.2.5. Yes, reject
7.2.7. No, fail to reject
7.2.9. Yes, reject
7.2.11. reject
7.2.13. fail to reject
7.2.15. reject
7.2.17. fail to reject
7.2.19. Reject since
7.3.1. Reject because
7.3.3. Reject because
7.3.5. Reject because
7.3.7. Reject
7.3.9. Reject
7.3.11. Reject
7.4.1. Reject because
7.4.3. Reject
7.4.5. Reject since
7.5.1. Reject since
7.5.3. Reject because
7.5.5. Reject because
7.6.1. Reject because
7.6.3. Fail to reject because
7.7.1. Do not reject since (two-sided test).
7.7.3. Do not reject (two-sided test)
7.7.5. Do not reject
7.7.7.
7.8.1. Fail to reject because
7.8.3. No,
7.8.5. Reject
7.8.7. V.R. = 2.1417, p 7 .10
V.R. = 4, .02 6 p 6 .05H
0
.
V.R. = 1.83, p 7 .10
V.R. = 1.226 6 1.74 p 7 .10
x
2
= 22.036, .10 7 p 7 .05
p 7 .10H
0
.x
2
= 28.8.
p 7 .05H
0
.x
2
= 6.75.
5.142 6 20.723 6 34.267, p 7 .01H
0
,
z = 1.70 6 1.96, p = .088H
0
-2.86 6-2.58, p = .0042H
0
z =-2.21, p = .0136H
0
-1.77 6-1.645, p = .0384H
0
-1.62 7-1.645. p = .0526H
0
-4.4580 6-2.4469, p 6 .01H
0
-3.1553 6-1.8125, .005 6 p 6 .01H
0
.
3.17 7 2.624, p 6 .005H
0
p 6 .01H
0
.t =-3.3567.
p = 211 - .99972= .0006H
0
.z = 3.39.
p 6 21.0052= .010H
0
.s
2
p
= 5421.25, t =-6.66.
z = 3.39 7 1.96. p = 21.00032H
0
-9.60 6-2.6591, p 6 21.0052H
0
-10.9 6-2.388, p 6 .005H
0
z =-4.00. p 6 .0001H
0
p = 21.04752= .095H
0
.z = 1.67,
p 6 .0001H
0
.z =-4.18,
p 7 .2H
0
.t = .1271,
H
0
, p 6 .0001z = 4,
H
0
, z = 3.08. p = .0010
H
0
. t =-1.5 7-1.709. .05 6 p 6 .10
H
0
, z =-5.73 6-1.645. p 6 .0001
.76 6 1.333. p 7 .10H
0
-2.57 6-2.33, p = .0051 6 .01H
0
n 7 30
-.416,
-2.18,
x
= 19.23, s
2
= 20.2268; 16.01, 22.45
pN = .90; .87, .93
pN
1
= .20; pN
2
= .54, .26, .42
A-120
ANSWERS TO ODD-NUMBERED EXERCISES
ANSWERS TO ODD-NUMBERED EXERCISES A-121
7.9.1.
Value of
Alternative Power Function
Value of
516 0.9500 0.0500
521 0.8461 0.1539
528 0.5596 0.4404
533 0.3156 0.6844
539 0.1093 0.8907
544 0.0314 0.9686
547 0.0129 0.9871
7.9.3.
1 BBM
515 520 525 530 535 540 545 550
1.0
0.8
0.6
0.4
0.2
0.0
Alternative values of mu
Power
Value of
Alternative Power Function
Value of
4.25 0.9900 0.0100
4.50 0.8599 0.1401
4.75 0.4325 0.5675
5.00 0.0778 0.9222
5.25 0.0038 0.9962
1 BBM
4.2 4.4 4.6 4.8 5.0 5.2
1.0
0.8
0.6
0.4
0.2
0.0
Alternative values of mu
Power
7.10.1. Select a sample of size 548 and compute If
reject If do not reject
7.10.3. Select a sample of size 103 and compute If
reject If do not reject
Review Exercises
19. Reject since
21. Fail to reject the null because
23. Reject since
25. Fail to reject because
27.
29.
31. Reject since
Answers to Exercises 41–55 obtained by MINITAB
41. 95.0% C.I.
(456.8, 875.9)
tpvalue
7.09 0.0000
Test of vs.
43. 95.0% C.I.
(0.224, 0.796)
tpvalue
3.65 0.0010
Test of vs.
45. Leg press: 95.0% C.I. Arm abductor: 95.0% C.I.
(32.22, 56.45) (3.717, 7.217)
tpvalue tpvalue
7.85 0.0000 6.70 0.0000
Test of vs. Test of vs.
Hip flexor: 95.0% C.I. Arm abductor: 95.0% C.I.
(3.079, 6.388) (4.597, 7.670)
tpvalue tpvalue
6.14 0.0000 8.56 0.0000
Test of vs. Test of vs.
Hip extensor: 95.0% C.I.
(6.031, 10.236)
tpvalue
8.30 0.0000
Test of vs.
47. 95.0% C.I.
tpvalue
1-71.9, -26.52
m not = 0m = 0
m not = 0m = 0m not = 0m = 0
m not = 0m = 0m not = 0m = 0
m not = 0m = 0
m not = 0m = 0
-2.286 6-1.7530, .025 7 p 7 .01H
0
d = 11.49, s
2
d
= 256.679, s
d
= 16.02, t = 2.485, .025 7 p 7 .01
t = 3.873, p 6 .005
1.10 6 1.645, p = .1357H
0
12.79 7 2.58. p 6 .0001H
0
z = 1.48 6 1.96. p = .1388
29.49 7 2.33. p 6 .0001H
0
H
0
.x
6 4.66,
H
0
.
x Ú 4.66,x
.
n = 103; C = 4.66.
H
0
.x 6 518.25H
0
.x Ú 518.25,
x
.
n = 548; C = 518.25.
A-122
ANSWERS TO ODD-NUMBERED EXERCISES