A.17 β Curves for t Tests

Answers to Selected

Odd-Numbered Exercises



Chapter 1

1. a. Los Angeles Times, Oberlin Tribune, Gainesville Sun,

Washington Post

b. Duke Energy, Clorox, Seagate, Neiman Marcus

c. Vince Correa, Catherine Miller, Michael Cutler, Ken Lee

d. 2.97, 3.56, 2.20, 2.97

3. a. How likely is it that more than half of the sampled computers will need or have needed warranty service? What is

the expected number among the 100 that need warranty

service? How likely is it that the number needing warranty

service will exceed the expected number by more than 10?

b. Suppose that 15 of the 100 sampled needed warranty

service. How confident can we be that the proportion of all

such computers needing warranty service is between .08

and .22? Does the sample provide compelling evidence for

concluding that more than 10% of all such computers need

warranty service?

5. a. No. All students taking a large statistics course who participate in an SI program of this sort.

b. Randomization protects against various biases and helps

ensure that those in the SI group are as similar as possible to

the students in the control group.

c. There would be no firm basis for assessing the effectiveness of SI (nothing to which the SI scores could reasonably

be compared).

7. One could generate a simple random sample of all singlefamily homes in the city, or a stratified random sample by

taking a simple random sample from each of the 10 district

neighborhoods. From each of the selected homes, values of

all desired variables would be determined. This would be an

enumerative study because there exists a finite, identifiable

population of objects from which to sample.



9. a. Possibly measurement error, recording error, differences

in environmental conditions at the time of measurement, etc.

b. No. There is no sampling frame.



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11. 6L 430

6H 769689

7L 42014202

7H

8L 011211410342

8H 9595578

9L 30

9H 58

The gap in the data—no scores in the high 70’s.

13. a. 12 2

leaf: ones digit

12 445

12 6667777

12 889999

13 00011111111

13 222222222233333333333333

13 44444444444444444455555555555555555555

13 6666666666667777777777

13 888888888888999999

14 0000001111

14 2333333

14 444

14 77

symmetry

b. Close to bell-shaped, center Ϸ 135, not insignificant dispersion, no gaps or outliers.



691



692



Answers to Selected Odd-Numbered Exercises



15. Crunchy



Creamy

2 2

644 3 0069

77220 4 00145

stem: tens

6320 5 003666

leaf: ones

222 6 258

55 7

0 8

Both sets of scores are rather spread out. There appear to be

no outliers. The distribution of crunchy scores appears to be

shifted to the right (toward larger values) of that for creamy

scores by something on the order of 10.



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17. a. # Nonconforming

0

1

2

3

4

5

6

7

8



Frequency



Rel. freq.



7

12

13

14

6

3

3

1

1

60



.117

.200

.217

.233

.100

.050

.050

.017

.017

1.001



b. .917, .867, 1 Ϫ .867 ϭ .133

c. The histogram has a substantial positive skew. It is centered somewhere between 2 and 3 and spreads out quite a bit

about its center.

19. a. .99 (99%), .71 (71%)

b. .64 (64%), .44 (44%)

c. Strictly speaking, the histogram is not unimodal, but is

close to being so with a moderate positive skew. A much

larger sample size would likely give a smoother picture.

21. a. y

0

1

2

3

4

5



23. a.



b.



Freq.



Rel. freq.



17

.362

22

.468

6

.128

1

.021

0

.000

1

.021

47

1.000

.362, .638



Class

0–Ͻ100

100–Ͻ200

200–Ͻ300

300–Ͻ400

400–Ͻ500

500–Ͻ600

600–Ͻ700

700–Ͻ800

800–Ͻ900



Freq.

21

32

26

12

4

3

1

0

1

100



b.



z

0

1

2

3

4

5

6

7

8



Freq.



13

.277

11

.234

3

.064

7

.149

5

.106

3

.064

3

.064

0

.000

2

.043

47

1.001

.894, .830



Rel. freq.

.21

.32

.26

.12

.04

.03

.01

.00

.01

1.00



Rel. freq.



Class



Freq.



Rel. freq.



Density



0–Ͻ50

50–Ͻ100

100–Ͻ150

150–Ͻ200

200–Ͻ300

300–Ͻ400

400–Ͻ500

500–Ͻ600

600–Ͻ900



8

13

11

21

26

12

4

3

2

100



.08

.13

.11

.21

.26

.12

.04

.03

.02

1.00



.0016

.0026

.0022

.0042

.0026

.0012

.0004

.0003

.00007



c. .79

25.



Class



Freq.



Class



10–Ͻ20

20–Ͻ30

30–Ͻ40

40–Ͻ50

50–Ͻ60

60–Ͻ70

70–Ͻ80



8

14

8

4

3

2

1

40



1.1–Ͻ1.2

1.2–Ͻ1.3

1.3–Ͻ1.4

1.4–Ͻ1.5

1.5–Ͻ1.6

1.6–Ͻ1.7

1.7–Ͻ1.8

1.8–Ͻ1.9



Freq.

2

6

7

9

6

4

5

1

40



Original: positively skewed;

Transformed: much more symmetric, not far from bell-shaped.

27. a. The observation 50 falls on a class boundary.

b.

Class

Freq.

Rel. freq.

0–Ͻ50

50–Ͻ100

100–Ͻ150

150–Ͻ200

200–Ͻ300

300–Ͻ400

400–Ͻ500

500–Ͻ600



9

.18

19

.38

11

.22

4

.08

4

.08

2

.04

0

.00

1

.02

50

1.00

A representative (central) value is either a bit below or a bit

above 100, depending on how one measures center. There is a

great deal of variability in lifetimes, especially in values at the

upper end of the data. There are several candidates for outliers.

c.



Class

2.25–Ͻ2.75

2.75–Ͻ3.25

3.25–Ͻ3.75

3.75–Ͻ4.25

4.25–Ͻ4.75

4.75–Ͻ5.25

5.25–Ͻ5.75

5.75–Ͻ6.25



Freq.



Rel. freq.



2

.04

2

.04

3

.06

8

.16

18

.36

10

.20

4

.08

3

.06

50

1.00

There is much more symmetry in the distribution of the

ln(x) values than in the x values themselves, and less variability. There are no longer gaps or obvious outliers.

d. .38, .14



Answers to Selected Odd-Numbered Exercises



29. Complaint

J

F

B

M

C

N

O



31.



Freq.



Rel. freq.



10

9

7

4

3

6

21

60



.1667

.1500

.1167

.0667

.0500

.1000

.3500

1.0001



693



59. a. ED: .4, .10, 2.75, 2.65;

Non-Ed: 1.60, .30, 7.90, 7.60

b. ED: 8.9 and 9.2 are mild outliers, and 11.7 and 21.0 are

extreme outliers.

There are not outliers in the non-ED sample.

c. Four outliers for ED, none for non-ED. Substantial positive skewness in both samples; less variability in ED

(smaller fs), and non-ED observations tend to be somewhat

larger than ED observations.



Class



Freq.



Cum. freq.



Cum. rel. freq.



0–Ͻ4

4–Ͻ8

8–Ͻ12

12–Ͻ16

16–Ͻ20

20–Ͻ24

24–Ͻ28



2

14

11

8

4

0

1



2

16

27

35

39

39

40



.050

.400

.675

.875

.975

.975

1.000



33. a. ෆx ϭ 192.57, ~

x ϭ 189.0

b. New ෆx ϭ 189.71; ~

x unchanged

c. 191.0, 7.14%

d. 122.6

~

35. a. xෆ ϭ 12.55, x ϭ 12.50, ෆxtr(12.5) ϭ 12.40. Deletion of the

largest observation (18.0) causes ~

x and ෆxtr to be a bit smaller

than ෆx.

b. By at most 4.0

c. No; multiply the values of xෆ and ~

x

by the conversion factor 1/2.2.

37. ෆxtr(10) ϭ 11.46

x ϭ 1.009

b. .383

39. a. ෆx ϭ 1.0297, ~

41. a. .7

b. Also .7

c. 13

43. ~

x ϭ 68.0, ෆxtr(20) ϭ 66.2, ෆxtr(30) ϭ 67.5

45. a. ෆx ϭ 115.58; the deviations are .82, .32, Ϫ.98, Ϫ.38, .22

b. .482, .694

c. .482

d. .482

47. ෆx ϭ 116.2, s ϭ 25.75. The magnitude of s indicates a substantial amount of variation about the center (a “representative” deviation of roughly 25).



61. Outliers, both mild and extreme, only at 6 A.M. Distributions

at other times are quite symmetric. Variability increases

somewhat until 2 P.M. and then decreases slightly, and the

same is true of “typical” gasoline-vapor coefficient values.

63.



6

34

7

17

8

4589

9

1

10

12667789

11

122499

12

2

13

1

~

ෆx ϭ 9.96, x ϭ 10.6, s ϭ 1.7594, fs ϭ 2.3, no outliers, negative skew



65. a. Representative value ϭ 90. Reasonably symmetric, unimodal, somewhat bell-shaped, fair amount of variability.

b. .9231, .9053

c. .48

67. a. M: ෆx ϭ 3.64, ~

x ϭ 3.70, s ϭ .269, fs ϭ .40

F: ෆx ϭ 3.28, ~

x ϭ 3.15, s ϭ .478, fs ϭ .50

Female values are typically somewhat smaller than male

values, and show somewhat more variability. An M boxplot

shows negative skew whereas an F boxplot shows positive

skew.

b. F: xෆtr(10) ϭ 3.24 M: xෆtr(10) ϭ 3.652 Ϸ 3.65

b. 189.14, 1.87

69. a. ෆy ϭ axෆ ϩ b, sy2 ϭ a2sx2



55. a. 33

b. No

c. Slight positive skewness in the middle half, but rather

symmetric overall. The extent of variability appears substantial.

d. At most 32



71. a. The mean, median, and trimmed mean are virtually identical, suggesting a substantial amount of symmetry in the

data; the fact that the quartiles are roughly the same distance

from the median and that the smallest and largest observations are roughly equidistant from the center provides additional support for symmetry. The standard deviation is quite

small relative to the mean and median.

b. See the comments of (a). In addition, using 1.5(Q3 Ϫ Q1)

as a yardstick, the two largest and three smallest observations are mild outliers.

73. ෆx ϭ .9255, s ϭ .0809, ~

x ϭ .93, small amount of variability,

slight bit of skewness

75. a. The “five-number summaries” ( ~

x, the two fourths, and



57. a. Yes. 125.8 is an extreme outlier and 250.2 is a mild

outlier.

b. In addition to the presence of outliers, there is positive

skewness both in the middle 50% of the data and, excepting

the outliers, overall. Except for the two outliers, there appears

to be a relatively small amount of variability in the data.



the smallest and largest observations) are identical and

there are no outliers, so the three individual boxplots are

identical.

b. Differences in variability, nature of gaps, and existence

of clusters for the three samples.

c. No. Detail is lost.



49. a. 56.80, 197.8040



b. .5016, .708



51. a. 1264.766, 35.564



b. .351, .593



53. a. 2.74, 3.88

d. At most .40



b. 1.14

e. 1.19



c. Unchanged



694



77. a.



Answers to Selected Odd-Numbered Exercises



0

1

2

3

4

5

6

7

8

9

10

HI



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2355566777888

0000135555

00257

0033

0057

stem: ones

044

leaf: tenths

05

8

0

3

22.0, 24.5



b.



Class



Freq.



Rel. freq.



Density



0–Ͻ2

2–Ͻ4

4–Ͻ6

6–Ͻ10

10–Ͻ20

20–Ͻ30



23

9

7

4

1

2



.500

.196

.152

.087

.022

.043



.250

.098

.076

.022

.002

.004



79. a. ෆxnϩ1 ϭ (nxෆn ϩ xnϩ1)/(n ϩ 1)

c. 12.53, .532

81. A substantial positive skew (assuming unimodality)

83. a. All points fall on a 45° line. Points fall below a 45° line.

b. Points fall well below a 45° line, indicating a substantial

positive skew.



Chapter 2

1. a. S ϭ {1324, 3124, 1342, 3142, 1423, 1432, 4123, 4132,

2314, 2341, 3214, 3241, 2413, 2431, 4213, 4231}

b. A ϭ {1324, 1342, 1423, 1432}

c. B ϭ {2314, 2341, 3214, 3241, 2413, 2431, 4213, 4231}

d. A ʜ B ϭ {1324, 1342, 1423, 1432, 2314, 2341, 3214,

3241, 2413, 2431, 4213, 4231},

A ʝ B contains no outcomes (A and B are disjoint),

AЈ ϭ {3124, 3142, 4123, 4132, 2314, 2341, 3214, 3241,

2413, 2431, 4213, 4231}

3. a. A ϭ {SSF, SFS, FSS}

b. B ϭ {SSF, SFS, FSS, SSS}

c. C ϭ {SFS, SSF, SSS}

d. CЈ ϭ {FFF, FSF, FFS, FSS, SFF},

A ʜ C ϭ {SSF, SFS, FSS, SSS},

A ʝ C ϭ {SSF, SFS},

B ʜ C ϭ {SSF, SFS, FSS, SSS} ϭ B,

B ʝ C ϭ {SSF, SFS, SSS} ϭ C

5. a. S ϭ {(1, 1, 1), (1, 1, 2), (1, 1, 3), (1, 2, 1), (1, 2, 2), (1, 2, 3),

(1, 3, 1), (1, 3, 2), (1, 3, 3), (2, 1, 1), (2, 1, 2), (2, 1, 3),

(2, 2, 1), (2, 2, 2), (2, 2, 3), (2, 3, 1), (2, 3, 2), (2, 3, 3), (3, 1, 1),

(3, 1, 2), (3, 1, 3), (3, 2, 1), (3, 2, 2), (3, 2, 3), (3, 3, 1),

(3, 3, 2), (3, 3, 3)}

b. {(1, 1, 1), (2, 2, 2), (3, 3, 3)}

c. {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)}

d. {(1, 1, 1), (1, 1, 3), (1, 3, 1), (1, 3, 3), (3, 1, 1), (3, 1, 3),

(3, 3, 1), (3, 3, 3)}

7. a. There are 35 outcomes in S.

b. {AABABAB,

AABAABB, AAABBAB, AAABABB, AAAABBB}

11. a. .07



b. .30



c. .57



13. a. .36

d. .47



b. .64

e. .17



c. .53

f. .75



15. a. .572



b. .879



17. a. There are statistical software packages other than SPSS

and SAS.

b. .70

c. .80

d. .20



19. a. .8841



b. .0435



21. a. .10

e. .31



b. .18, .19

f. .69



23. a. .067



b. .400



25. a. .98

27. a. .1



c. .41

c. .933



b. .02

b. .7



d. .59



c. .03



d. .533

d. .24



c. .6



29. a. 676; 1296

b. 17,576; 46,656

1,679,616

d. .942

31. a. 243



b. 3645 days (roughly 10 yr)



33. a. 362,880

35. a. .0048

37. a. 60

39. a. .0839

41. a. .929



c. 456,976;



b. 131,681,894,400

b. .0054



b. 10



c. .9946



c. 2100

d. .2885



c. .0456



b. .24975

b. .0714



c. .99997520



43. .000394, .00394, .00001539

45. a. .447, .500, .200

47. a. .50

d. .375



b. .50

e. .769



b. .400, .447



c. .211



c. .625



49. .217, .178

51. .436, .581

53. .083

55. .236

59. a. .21



b. .455



61. a. .578, .278, .144

63. b. .54



c. .68



c. .264, .274

b. 0, .457, .543

d. .74



e. .7941



65. P(Mean⏐S) ϭ .3922, P(Median⏐S) ϭ .2941, so Mean and

Median are most and least likely, respectively.

67. .000329; very uneasy.



Answers to Selected Odd-Numbered Exercises



69. a. .126

e. .5325



b. .05

c. .1125

f. .2113



71. a. .300



b. .820



d. .2725



93. .45, .32

95. a. .0083



c. .146



b. .2



99. a. .956

b. .00421



b. .994



101. .926



79. .0059



103. a. .018



81. a. .95



105. a. .883, .117



83. a. .10, .20



b. 0



b. .601

b. 23



c. .156



107. 1 Ϫ (1 Ϫ p1)(1 Ϫ p2) и и и и и (1 Ϫ pn)



c. (1 Ϫ p)3

85. a. p(2 Ϫ p)

b. 1 Ϫ (1 Ϫ p)n

3

d. .9 ϩ (1 Ϫ p) (.1)

e. .1(1 Ϫ p)3/[.9 ϩ .1(1 Ϫ p)3] ϭ .0137 for p ϭ .5



109. a. .0417



b. .375



111. P(hire #1) ϭ 6/24 for s ϭ 0, ϭ 11/24 for s ϭ 1,

ϭ 10/24 for s ϭ 2, and ϭ 6/24 for s ϭ 3, so s ϭ 1 is best.



87. .8588, .9897



113. 1/4 ϭ P(A1 ʝ A2 ʝ A3)

P(A1) и P(A2) и P(A3) ϭ 1/8



89. [2␲(1 Ϫ ␲)]/(1 Ϫ ␲ 2)

91. a. .333, .444



c. .2



97. .905



75. .401, .722

77. a. .06235



695



b. .150



c. .291



Chapter 3

1. x ϭ 0 for FFF; x ϭ 1 for SFF, FSF, and FFS; x ϭ 2 for SSF,

SFS, and FSS; and x ϭ 3 for SSS

3. Z ϭ average of the two numbers, with possible values 2/2,

3/2, . . . , 12/2; W ϭ absolute value of the difference, with

possible values 0, 1, 2, 3, 4, 5

5. No. In Example 3.4, let Y ϭ 1 if at most three batteries are

examined and let Y ϭ 0 otherwise. Then Y has only two

values.

7. a. {0, 1, . . . , 12}; discrete

c. {1, 2, 3, . . . }; discrete

e. {0, c, 2c, . . . , 10,000c}, where c is the royalty per book;

discrete g. {xϺ m Յ x Յ M} where m (M) is the minimum (maximum) possible tension; continuous

9. a. {2, 4, 6, 8, . . . }, that is, {2(1), 2(2), 2(3), 2(4), . . . }, an

infinite sequence; discrete

b. {2, 3, 4, 5, 6, . . . }, that is, {1 ϩ 1, 1 ϩ 2, 1 ϩ 3, 1 ϩ

4, . . . }, an infinite sequence; discrete

11. a. p(4) ϭ .45, p(6) ϭ .40, p(8) ϭ .15, p(x) ϭ 0 for x

6, or 8

c. .55, .15

13. a. .70

d. .71



b. .45

e. .65



4,



c. .55

f. .45



15. a. (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4),

(3, 5), (4, 5)

b. p(0) ϭ .3, p(1) ϭ .6, p(2) ϭ .1

c. F(x) ϭ 0 for x Ͻ 0, ϭ .3 for 0 Յ x Ͻ 1, ϭ .9 for 1 Յ

x Ͻ 2, and ϭ 1 for 2 Յ x

17. a. .81

b. .162

UUUAA; .00405



c. It is A; AUUUA, UAUUA, UUAUA,



b. F(x) ϭ 0 for x Ͻ 1, ϭ .477 for 1 Յ x Ͻ 2, ϭ .602 for

2 Յ x Ͻ 3, . . . , ϭ .954 for 8 Յ x Ͻ 9, ϭ 1 for x Ն 9

c. .602, .301

23. a. .20



b. .33



25. a. p(y) ϭ (1 Ϫ p)y и p



c. .78



d. .53



for y ϭ 0, 1, 2, 3, . . .



27. a. 1234, 1243, 1324, . . . , 4321

b. p(0) ϭ 9/24, p(1) ϭ 8/24, p(2) ϭ 6/24, p(3) ϭ 0,

p(4) ϭ 1/24

29. a. 2.06



b. .9364



c. .9677



d. .9364



31. .74, .8602, .85

33. a. p



b. p(1 Ϫ p)



c. p



35. E[h3(X)] ϭ 2.4667, E[h4(X)] ϭ 2.667, so 4 copies is better.

37. E(X ) ϭ (n ϩ 1)/2, E(X 2) ϭ (n ϩ 1)(2n ϩ 1)/6, V(X ) ϭ

(n2 Ϫ 1)/12

39. 2.3, .81, 88.5, 20.25

43. E(X Ϫ c) ϭ E(X) Ϫ c, E(X Ϫ ␮) ϭ 0

47. a. .515

f. .000



b. .218

g. .595



49. a. .354



b. .115



c. .011



d. .480



e. .965



c. .918



51. a. 6.25



b. 2.17



c. .030



53. a. .403



b. .787



c. .774



55. .1478

57. .407, independence



19. p(0) ϭ .09, p(1) ϭ .40, p(2) ϭ .32, p(3) ϭ .19



59. a. .017



21. a. p(x) ϭ .301, .176, .125, .097, .079, .067, .058, .051, .046

for x ϭ 1, 2, . . . , 9



61. When p ϭ .9, the probability is .99 for A and .9963 for B. If

p ϭ .5, these probabilities are .75 and .6875, respectively.



b. .811, .425



c. .006, .902, .586



696



Answers to Selected Odd-Numbered Exercises



63. The tabulation for p Ͼ .5 is unnecessary.



89. a. 4



65. a. 20, 16



91. a. .221



b. 70, 21



67. P(⏐X Ϫ ␮⏐Ն 2␴) ϭ .042 when p ϭ .5 and ϭ .065 when

p ϭ .75, compared to the upper bound of .25. Using k ϭ 3 in

place of k ϭ 2, these probabilities are .002 and .004, respectively, whereas the upper bound is .11.

69. a. .114

b. .879

c. .121

distribution with n ϭ 15, p ϭ .10



d. Use the binomial



75. a. nb(x; 2, .5)



b. .033

b. .188



c. .688



d. 2, 4



77. nb(x; 6, .5), 6

79. a. .932

e. .251



b. .065



81. a. .011



c. .068



b. .441



83. Poisson(5)



85. a. .122, .809, .283

c. .530, .011

87. a. .099



b. .135



d. .492



c. .554, .459



a. .492



b. 6,800,000



c. p(x; 20.106)



95. b. 3.114, .405, .636

97. a. b(x; 15, .75)

b. .686

c. .313

d. 11.25, 2.81



e. .310



99. .991

101. a. p(x; 2.5)



b. .067



c. .109



103. 1.813, 3.05



71. a. h(x; 15, 10, 20) for x ϭ 5, . . . , 10

b. .0325

c. .697

73. a. h(x; 10, 10, 20)



c. At least Ϫln(.1)/2 Ϸ 1.1513 years



b. .215



d. .945



105. p(2) ϭ p2, p(3) ϭ (1 Ϫ p)p2, p(4) ϭ (1 Ϫ p)p2, p(x) ϭ

[1 Ϫ p(2) Ϫ . . . Ϫ p(x Ϫ 3)](1 Ϫ p)p2 for x ϭ 5, 6, 7, . . . ;

.99950841

107. a. .0029

109. a. .135



b. .0767, .9702

b. .00144



c.



Α x∞ϭ0[p(x; 2)]5



111. 3.590

113. a. No



b. .0273



115. b. .6p(x; ␭) ϩ .4p(x; ␮)

c. (␭ ϩ ␮)/2

d. (␭ Ϫ ␮)2/4 ϩ (␭ ϩ ␮)/2



b. .133

b. 12, 3.464



117.



Α i10ϭ1(piϩjϩ1 ϩ piϪjϪ1)pi, where pk ϭ 0 if k Ͻ 0 or k Ͼ 10.



121. a. 2.50



c. 2



b. 3.1



Chapter 4

1. a. .25



b. .50



3. b. .5



c. .6875



25. b. 1.8(90th percentile for X) ϩ 32

c. a(X percentile) ϩ b



c. .4375

d. .6328

d. .578



27. 0, 1.814



7. a. f(x) ϭ .1 for 25 Յ x Յ 35 and 0 otherwise

b. .20

c. .40

d. .20



29. a. 2.14

d. .97



9. a. .562



31. a. 2.54



5. a. .375



b. .125



c. .297



b. .438, .438



c. .071



11. a. .25

b. .1875

c. .9375

e. f(x) ϭ x/2 for 0 Ͻ x Ͻ 2

f. 1.33

g. .222, .471

h. 2



d. 1.4142



13. a. 3

b. 0 for x Յ 1, 1 Ϫ x᎐3 for x Ͼ 1

c. .125, .088

d. 1.5, .866

e. .924

x9



x 10



c. 1.17

c. Ϫ.42



b. 1.34



33. a. .9918



b. .0082



35. a. .3336

c. .5795



b. Approximately 0

d. 6.524

e. .8028



37. a. 0, .5793, .5793

39. a. 36.7



15. a. F(x) ϭ 0 for x Յ 0, ϭ 90[ᎏ9ᎏ Ϫ ᎏ1ᎏ0 ] for 0 Ͻ x Ͻ 1, ϭ 1 for

xՆ1

b. .0107

c. .0107, .0107

d. .9036

e. .818, .111

f. .3137



41. .002



17. a. A ϩ (B Ϫ A)p

b. E(X) ϭ (A ϩ B)/2, ␴⌾ ϭ

(B Ϫ A)/͙12

ෆ

c. [Bn ϩ 1 Ϫ An ϩ 1]/[(n ϩ 1)(B Ϫ A)]



45. 7.3%



19. a. .597

b. .369

c. f(x) ϭ .3466 Ϫ .25 ln(x) for 0 Ͻ x Ͻ 4



b. .81

e. 2.41



c. .8664



c. Ͻ 87.6 or Ͼ 120.4



b. .3174, no



b. 22.225



c. 3.179



43. 10, .2



47. 21.155

49. a. .1190, .6969



b. .0021



c. .7054



21. 314.79



d. Ͼ5020 or Ͻ1844 (using z.0005 ϭ 3.295)



23. 248, 3.60



e. Normal, ␮ ϭ 7.576, ␴ ϭ 1.064, .7054



Answers to Selected Odd-Numbered Exercises



697



51. .3174 for k ϭ 1, .0456 for k ϭ 2, .0026 for k ϭ 3, as compared to the bounds of 1, .25, and .111, respectively.



97. There is substantial curvature in the plot. ␭ is a scale

parameter (as is ␴ for the normal family).



53. a. Exact: .212, .577, .573; Approximate: .211, .567, .596

b. Exact: .885, .575, .017; Approximate: .885, .579, .012

c. Exact: .002, .029, .617; Approximate: .003, .033, .599



99. a. F(y) ϭ ᎏ48ᎏ (y 2 Ϫ y 3/18) for 0 Յ y Յ 12

b. .259, .5, .241

c. 6, 43.2, 7.2

d. .518

e. 3.75



55. a. .9409



101. a. f(x) ϭ x2 for 0 Յ x Ͻ 1 and ϭ ᎏ4ᎏ Ϫ ᎏ4ᎏx for 1 Յ x Յ ᎏ3ᎏ

b. .917

c. 1.213

7



b. .9943



57. b. Normal, ␮ ϭ 239, ␴ 2 ϭ 12.96

59. a. 1



b. 1



1



c. .982



61. a. .449, .699, .148



d. .129



b. .05, .018



63. a. short d plan #1 better, whereas long d plan #2 better

b. 1\␭ ϭ 10 d E[h1 (X)] ϭ 100, E[h2(X)] ϭ 112.53

1\␭ ϭ 15 d E[h1 (X)] ϭ 150, E[h2(X)] ϭ 138.51

65. a. .238

f. .713



b. .238



c. .313



67. a. .424



~ Ͻ 24

b. .567, ␮



d. .653

c. 60



e. .653



d. 66



103. a. .9162



b. .9549



c. 1.3374



b. .0663



c. (72.97, 119.03)



107. b. F(x) ϭ 0 for x Ͻ Ϫ1, ϭ (4x Ϫ x3/3)/9 ϩ

x Յ 2, and ϭ 1 for x Ͼ 2

~Ͼ0

c. No. F(0) Ͻ .5 d ␮

5

d. Y ϳ Bin(10, ᎏ2ᎏ7 )



73. a. .826, .826, .0636



b. .664



c. 172.727



77. a. 123.97, 117.373



b. .5517



c. .1587



113. b.

c.

d.

e.



b. .9573

f. 125.90



c. .0414



83. ␣ ϭ ␤

85. b. [⌫(␣ ϩ ␤) и ⌫(m ϩ ␤)]/[⌫(␣ ϩ ␤ ϩ m) и ⌫(␤)], ␤/(␣ ϩ ␤)

87. Yes, since the pattern in the plot is quite linear.

89. Yes

91. Yes

93. Plot ln(x) vs. z percentile. The pattern is straight, so a lognormal population distribution is plausible.



11

ᎏᎏ

27



for Ϫ1 Յ



109. a. .368, .828, .460

b. 352.53

c. 1/␤ и exp[Ϫexp(Ϫ(x Ϫ ␣)/␤)] и exp(Ϫ(x Ϫ ␣)/␤)

~ ϭ 182.99

d. ␣

e. ␮ ϭ 201.95, mode ϭ 150, ␮

111. a. ␮

b. No

c. 0

d. (␣ Ϫ 1)␤

e. ␯ Ϫ 2



81. a. 149.157, 223.595

d. 148.41

e. 9.57



7



105. a. .3859



69. a. ʝ Ai

b. Exponential with ␭ ϭ .05

c. Exponential with parameter n␭



79. a. 68.0, 122.1

b. .3204

c. .7257, skewness



3



p(1 Ϫ exp(Ϫ␭1x)) ϩ (1 Ϫ p)(1 Ϫ exp(Ϫ␭2x)) for x Ն 0

p/␭1 ϩ (1 Ϫ p)/␭2

V(X) ϭ 2p/␭12 ϩ 2(1 Ϫ p)/␭22 Ϫ ␮2

1, CV Ͼ 1

f. CV Ͻ 1



115. a. Lognormal



b. 1



c. 2.72, .0185



119. a. Exponential with ␭ ϭ 1

c. Gamma with parameters ␣ and c␤

121. a. (1/365)3



b. (1/365)2



c. .000002145



123. b. Let u1, u2, u3, . . . be a sequence of observations from a

Unif[0, 1] distribution (a sequence of random numbers).

Then with xi ϭ (Ϫ.1)ln(1 Ϫ ui), the xi’s are observations

from an exponential distribution with ␭ ϭ 10.

125. g(E(X)) Յ E(g(X))

127. a. 710, 84.423, .684



b. .376



95. The pattern in the plot is quite linear; it is very plausible that

strength is normally distributed.



Chapter 5

1. a. .20

b. .42

c. At least one hose is in use at each

pump; .70.

d. pX (x) ϭ .16, .34, .50 for x ϭ 0, 1, 2,

respectively; pY (y) ϭ .24, .38, .38 for y ϭ 0, 1, 2, respectively; .50

e. No; p(0, 0) pX (0) и pY (0)



b. eϪ␭Ϫ␮ и [1 ϩ ␭ ϩ ␮]

11. a. eϪ␭Ϫ␮ и ␭x и ␮y/x!y!

Ϫ(␭ϩ␮)

m

и (␭ ϩ ␮) /m!; Poisson (␭ ϩ ␮)

c. e



3. a. .15



15. a. F(y) ϭ 1 Ϫ eϪ␭y ϩ (1 Ϫ eϪ␭y)2 Ϫ (1 Ϫ eϪ␭y)3 for y Ն 0

b. 2/3␭



b. .40



5. a. .054



b. .00018



7. a. .030

d. .380



b. .120

e. Yes



c. .22



d. .17, .46



c. .300



9. a. 3/380,000

b. .3024

c. .3593

d. 10Kx2 ϩ .05 for 20 Յ x Յ 30

e. No



13. a. eϪxϪy for x Ն 0, y Ն 0

d. .330



b. .400



c. .594



17. a. .25

b. .318

c. .637

ෆ2ෆϪ

ෆෆx2ෆ/␲R2 for ϪR Յ x Յ R; no

d. fX(x) ϭ 2͙R

19. a. K(x2 ϩ y2)/(10Kx2 ϩ .05); K(x2 ϩ y2)/(10Ky2 ϩ .05)

b. .556, .549

c. 25.37, 2.87



698



Answers to Selected Odd-Numbered Exercises



21. a. f(x1, x2, x3)/f X1 ,X2 (x1, x2)



b. f(x1, x2, x3)/f X1(x1)



23. .15



59. a. .9986, .9986

b. .9015, .3970

c. .8357

d. .9525, .0003



25. L2



61. a. 3.5, 2.27, 1.51



27. .25 hr



63. a. .695



b. 4.0675 Ͼ 2.6775



65. a. .9232



2



29. Ϫᎏ3ᎏ

31. a. Ϫ.1082



37. a. xෆ

p(xෆ )

b. s2

p(s2)



|



0



112.5



| .38



312.5 800



.20



|



Probability



.30

.1



.2



.3



.4



.5



| .000



.000



.000



.001



.005



.027



|



.6



| .088



Probability



|



|



.12



, E(S 2) ϭ 212.25 ϭ ␴ 2



0



Proportion



p(xෆ )

b. .85



, E(X

ෆ) ϭ ␮ ϭ 44.5



.04 .20 .25 .12 .30 .09



39. Proportion



41. a. xෆ



69. a. 2400



25 32.5 40 45 52.5 65



|



.7



.8



.9



1.0



.201



.302



.269



.107



73. a. Approximately normal with mean ϭ 105, SD ϭ 1.2649;

Approximately normal with mean ϭ 100, SD ϭ 1.0142

b. Approximately normal with mean ϭ 5, SD ϭ 1.6213

c. .0068

d. .0010, yes

75. a. .2, .5, .3 for x ϭ 12, 15, 20; .10, .35, .55 for y ϭ 12, 15, 20

b. .25

c. No

d. 33.35

e. 3.85



.01



2



3



79. Ϸ 1



.30



.40



.22



.08



2.5



.16



.24



.25



.20



|



b. .9788



1



2



p(r)



71. a. 158, 430.25



c. 2400, 41.77



0



1.5



|



b. 1205; independence



77. a. 3/81,250

b. fX (x) ϭ k(250x Ϫ 10x2) for 0 Յ x Յ 20

1

and ϭ k(450x Ϫ 30x2 ϩ ᎏ2ᎏx3) for 20 Ͻ x Յ 30; fY (y) results

from substituting y for x in fX (x). They are not independent.

c. .355

d. 25.969

e. 204.6154, Ϫ.894

f. 7.66



1



c. r



b. .9660



67. .1588



b. Ϫ.0131



|



b. 15.4, 75.94, 8.71



3

.10



3.5



4



.04



81. a. 400 min



47. a. .6826



b. .1056



83. 97



49. a. .6026



b. .2981



85. .9973



b. 70



89. b, c. Chi-squared with ␯ ϭ n.



51. .7720

53. a. .0062



b. 0



91. a. ␴ 2W /(␴ 2W ϩ ␴ 2E )



55. a. .9838



b. .8926



93. 26, 1.64

95. a. .6



57. .9616



b. .9999



ෆ

b. U ϭ ␳X ϩ ͙1

Ϫ ␳2 Y



Chapter 6

~

1. a. 8.14, X

b. .77, X

ෆ

d. .148

e. .204, S/X

ෆ



15. a. ␪ˆ ϭ ΑX 2i /2n



c. 1.66, S



17. b. .444



3. a. 1.348, X

b. 1.348, X

ෆ

ෆ

c. 1.781, X

ϩ

1.28S

ෆ

d. .6736

e. .0905

5. N xෆ ϭ 1,703,000;

1,601,438.281

7. a. 120.6



΄



΅



19. a. pˆ ϭ 2␭ˆ Ϫ .30 ϭ .20

b. pˆ ϭ (100␭ˆ Ϫ 9)/70

21. b. ␣ˆ ϭ 5, ␤ˆ ϭ 28.0/⌫(1.2)



T Ϫ N dෆ ϭ 1,591,300;



b. 1,206,000



9. a. 2.11

b. .119

p1q1

p q 1/2

11. b. ᎏ ᎏ ϩ ᎏ2 ᎏ2

n1

n2



c. .80



T и (xෆ / ෆy) ϭ



d. 120.0



c. Use pˆ i ϭ xi/ni and qˆ i ϭ 1 Ϫ pˆ i



23. ␭ˆ 1 ϭ xෆ, ␭ˆ 2 ϭ yෆ, estimate of (␭1 Ϫ ␭2) is ෆx Ϫ ෆy.

25. a. 384.4, 18.86

b. 415.42

29. a. ␪ˆ ϭ min(Xi), ␭ˆ ϭ n/Α[Xi Ϫ min(Xi)]

b. .64, .202

33. With xi ϭ time between birth i Ϫ 1 and birth i, ␭ˆ ϭ

6/Α6i ϭ1 ixi ϭ .0436.



in place of pi and qi in part (b) for i ϭ 1, 2.



35. 29.5



d. Ϫ.245



37. 1.0132



e. .041



b. 74.505



Answers to Selected Odd-Numbered Exercises



699



Chapter 7

1. a. 99.5%



b. 85%



3. a. Narrower



c. 2.96



b. No



5. a. (4.52, 5.18)

c. .55

d. 94



c. No



d. 1.15

d. No



35. a. 95% CI: (23.1, 26.9)

b. 95% PI: (17.2, 32.8), roughly 4 times as wide

37. a. (.888, .964)

c. (.634, 1.218)



b. (4.12, 5.00)



7. By a factor of 4; the width is decreased by a factor of 5.

9. a. (xෆ Ϫ 1.645␴/͙n

ෆ, ∞); (4.57, ∞)

ෆ, ∞)

c. (Ϫ∞, ෆx ϩ z␣ и ␴/͙n

ෆ);

b. (xෆ Ϫ z␣ и ␴/͙n

(Ϫ∞, 59.7)

11. 950, .8714



b. (.752, 1.100)



39. a. Yes

b. (6.45, 98.01)

c. (18.63, 85.83)

41. All 70%; (c), because it is shortest

43. a. 18.307



b. 3.940



c. .95



d. .10



45. (3.6, 8.1); no



13. a. (608.58, 699.74)



b. 189



47. a. 95% CI: (6.702, 9.456)



15. a. 80%



c. 75%



49. a. There appears to be a slight positive skew in the middle

half of the sample, but the lower whisker is much longer

than the upper whisker. The extent of variability is rather

substantial, although there are no outliers.

b. Yes. The pattern of points in a normal probability plot is

reasonably linear.

c. (33.53, 43.79)



b. 98%



17. 134.53

19. (.513, .615)

21. .218

23. a. (.438, .814)

25. a. 381

29. a. 2.228

e. 2.485



b. 659



b. 339

b. 2.086

f. 2.571



51. a. (.624, .732)

c. 2.845



d. 2.680



31. a. 1.812

b. 1.753

c. 2.602

d. 3.747

e. 2.1716 (from MINITAB)

f. Roughly 2.43

33. a. Reasonable amount of symmetry, no outliers

b. Yes (based on a normal probability plot)

c. (430.5, 446.1), yes, no



b. 1080



b. (.166, .410)



c. No



53. (Ϫ.84, Ϫ.16)

55. 246

57. (2tr /␹21Ϫ␣/2,2r, 2tr /␹2␣/2,2r) ϭ (65.3, 232.5)

59. a. (max(xi)/(1 Ϫ ␣/2)1/n, max(xi)/(␣/2)1/n)

c. (b); (4.2, 7.65)

b. (max(xi), max(xi)/␣1/n)

61. (73.6, 78.8) versus (75.1, 79.6)



Chapter 8

1. a. Yes

d. Yes



b. No

e. No



c. No

f. Yes



5. H0: ␴ ϭ .05 versus Ha: ␴ Ͻ .05. I: conclude variability in

thickness is satisfactory when it isn’t. II: conclude variability in thickness isn’t satisfactory when in fact it is.

7. I: concluding that the plant isn’t in compliance when it is; II:

concluding that the plant is in compliance when it isn’t.

9. a. R1

b. I: judging that one of the two companies is

favored over the other when that is not the case; II: judging

that neither company is favored over the other when in fact

one of the two really is preferred.

c. .044

d. ␤(.3) ϭ ␤(.7) ϭ .488, ␤(.4) ϭ ␤(.6) ϭ .845

e. Reject H0 in favor of Ha.

b. .01

11. a. H0: ␮ ϭ 10 versus Ha: ␮ 10

c. .5319, .0078

d. 2.58

e. 10.1032 is replaced by 10.124, and 9.8968 is replaced by

9.876.

f. xෆ ϭ 10.020, so H0 should not be rejected.

g. z Ն 2.58 or Յ Ϫ2.58



13. b. .0004, 0, less than .01

15. a. .0301



b. .003



c. .004



17. a. z ϭ 2.56 Ն 2.33, so reject H0.

d. .0052

19. a. z ϭ Ϫ2.27, so don’t reject H0.

21. a.

b.

c.

d.



b. .8413

b. .2266



c. 143

c. 22



t.025,12 ϭ 2.179 Ͼ 1.6, so don’t reject H0: ␮ ϭ .5.

Ϫ1.6 Ͼ Ϫ2.179, so don’t reject H0.

Don’t reject H0.

Reject H0 in favor of Ha: ␮ .5.



23. t ϭ 2.24 Ն 1.708, so H0 should be rejected. The data does

suggest a contradiction of prior belief.

25. a. z ϭ Ϫ3.33 Յ Ϫ2.58, so reject H0.

b. .1056

c. 217



x ϭ .640, s ϭ .3025, fs ϭ .480. A boxplot

27. a. ෆx ϭ .750, ~

shows substantial positive skew; there are no outliers.

b. No. A normal probability plot shows substantial curvature. No, since n is large.



700



Answers to Selected Odd-Numbered Exercises



c. z ϭ Ϫ5.79; reject H0 at any reasonable significance

level; yes.

d. .821

29. a. .498 Ͻ 1.895, so do not reject H0.



b. .72



31. Ϫ1.24 Ͼ Ϫ1.397, so prior belief does not appear to be

contradicted.



57. t Ϸ 1.9, so P-value Ϸ .041. Since P-value Յ ␣, H0: ␮ ϭ 25

should be rejected in favor of Ha: ␮ Ͼ 25.

59. t Ϸ 1.9, so P-value Ϸ .116. H0 should therefore not be

rejected.

b. P-value Ϸ 0. Yes.



61. a. .8980, .1049, .0014



35. Yes, because Ϫ2.47 Յ Ϫ1.96.



63. z ϭ Ϫ3.12 Յ Ϫ1.96, so H0 should be rejected.



37. z ϭ 3.67 Ն 2.58, so reject H0: p ϭ .40. No.



65. a. H0: ␮ ϭ .85 versus Ha: ␮ .85

b. H0 cannot be rejected for either ␣.



c. No



39. a. H0: p ϭ .02 vs Ha : p Ͻ .02, z ϭϪ1.01 Ͼ Ϫ1.645, don’t

reject H0, carry out inventory.

b. .1949

c. Ϸ 0



67. a. Yes, because t ϭ 12.9 Ն 2.228.

b. Normal population distribution



41. a. z ϭ 3.07 Ն 2.58, reject H0 and the company’s premise.

b. .0332



69. a. No; no

b. No, because z ϭ .44 and P-value ϭ .33 Ͼ .10.



43. No, no, yes. R ϭ {5, 6, . . . , 24, 25}, ␣ ϭ .098, ␤ ϭ .090



71. a. Approximately .6; approximately .2 (from Appendix

Table A.17)

b. n ϭ 28



45. a. Reject H0.

c. Don’t reject H0.

e. Don’t reject H0.

47. a. .0778

d. .0066



b. Reject H0.

d. Reject H0. (a close call)



b. .1841

e. .5438



49. a. 0.40

e. Ͻ.005



73. a. z ϭ 1.64 Ͻ 1.96, so H0 cannot be rejected; Type II

b. .10. Yes.

75. Yes. z ϭ Ϫ3.32 Յ Ϫ3.08, so H0 should be rejected.



c. .0250



77. No, since z ϭ 1.33 Ͻ 2.05.



b. .018

c. .130

f. Ϸ .000



d. .653



79. P-value Ϸ 0, so reject H0; it appears that ␮ Ͼ 15.



51. P-value Ͼ ␣, so don’t reject H0; no apparent difference.



81. a. .01 Ͻ P-value Ͻ .025, so do not reject H0; no extradiction



53. P-value Ͻ .0004 Ͻ .01, so H0: ␮ ϭ 5 should be rejected in

favor of Ha: ␮ 5.



83. a. For H2: ␮ Ͻ ␮0, reject H0 if z Α xi /␮0 Յ ␹21Ϫ␣, 2n

b. Test statistic value ϭ 19.65 Ͼ 8.260, so do not reject H0.



55. No; P-value Ϸ .2



85. a. Yes, ␣ ϭ .002



Chapter 9

1. a. Ϫ.4 hr; it doesn’t



b. .0724, .2691



c. No



3. z ϭ 1.76 Ͻ 2.33, so don’t reject H0.

5. a. z ϭ Ϫ2.90, so reject H0.

c. .8212

d. 66



b. .0019



7. Yes, since z ϭ 1.83 Ն 1.645.

9. a. 6.2; yes

b. z ϭ 1.14, P-value Ϸ .25, no

c. No

d. A 95% CI is (10.0, 21.8).



29. t ϭ Ϫ2.10, df ϭ 25, P-value ϭ .023. At significance level

.05, we would conclude that cola results in a higher average

strength, but not at significance level .01.

31. a. Virtually identical centers, substantially more variability in

medium range observations than in higher range observations

b. (Ϫ7.9, 9.6), based on 23 df; no

33. t ϭ 1.33, P-value ϭ .094, don’t reject H0, no



11. A 95% CI is (.99, 2.41).



35. t ϭ Ϫ2.2, df ϭ 16, P-value ϭ .021 Ͼ .01 ϭ ␣, so don’t

reject H0.



13. 50



37. a. (Ϫ.561, Ϫ.287)



15. b. It increases.

17. a. 17



b. 21



c. 18



d. 26



19. t ϭ Ϫ1.20 Ͼ Ϫt.01,9 ϭ Ϫ2.821, so do not reject H0.

21. Yes; Ϫ2.64 Յ Ϫ2.602, so reject H0.

23. b. No

c. t ϭ Ϫ.38 Ͼ Ϫt␣/2,10 for any reasonable ␣, so

don’t reject H0 (P-value Ϸ .7).

25. (.3, 6.1), yes, yes

27. (6.5, 31.3) based on 9 df; yes, yes



b. Between Ϫ1.224 and .376



39. a. Yes

b. t ϭ 2.7, P-value ϭ .018 Ͻ .05 ϭ ␣, so H0 should be

rejected.

41. t ϭ 1.9, P-value ϭ .047. H0 cannot be rejected at significance level .01, but is barely rejected at ␣ ϭ .05.

43. a. No



b. Ϫ49.1



c. 49.1



45. a. 95% CI: (Ϫ2.52, 1.05); plausible that they are identical

b. Linear pattern in npp implies normality of difference distribution is plausible.



Answers to Selected Odd-Numbered Exercises



47. H0 is rejected because Ϫ4.18 Յ Ϫ2.33



69. (Ϫ299.3, 1517.9)



49. P-value ϭ .4247, so H0 cannot be rejected.



71. (1024.0, 1336.0), yes



51. a. z ϭ .80 Ͻ 1.96, so don’t reject H0.

b. n ϭ 1211

53. a. The CI for ln(␪) is ln(␪ˆ) Ϯ z␣/2[(m Ϫ x)/(mx) ϩ

(n Ϫ y)/(ny)]1/2. Taking the antilogs of the lower and upper

limits gives a CI for ␪ itself.

b. (1.43, 2.31); aspirin appears to be beneficial.



73. Yes. t ϭ Ϫ2.25, df ϭ 57, P-value Ϸ .028



55. (Ϫ.35, .07)

57. a. 3.69

e. 4.30



b. 4.82

f. .212



c. .207

g. .95



d. .271

h. .94



701



75. a. No. t ϭ Ϫ2.84, df ϭ 18, P-value Ϸ .012

b. No. t ϭ Ϫ.56, P-value Ϸ .29

77. Not at significance level .05. t ϭ Ϫ1.76 Ͼ Ϫt.05,4 ϭ Ϫ2.015

79. No, nor should the two-sample t test be used, because a normal probability plot suggests that the good-visibility distribution is not normal.

81. Unpooled: df ϭ 15, t ϭ Ϫ1.8, P-value Ϸ .092

Pooled: df ϭ 24, t ϭ Ϫ1.9, P-value Ϸ .070



59. f ϭ .384; since .167 Ͻ .384 Ͻ 3.63, don’t reject H0.

61. f ϭ 2.85 Ն 2.08, so reject H0; there does appear to be more

variability in low-dose weight gain.

63. (s22 F1Ϫ␣/2 /s12 , s22 F␣/2 /s12 ); (.023, 1.99)

65. No. t ϭ 3.2, df ϭ 15, P-value ϭ .006, so reject H0: ␮1 Ϫ ␮2 ϭ

0 using either ␣ ϭ .05 or .01.

67. z Ͼ 0 d P-value Ͼ .5, so H0: p1 Ϫ p2 ϭ 0 cannot be

rejected.



83. a. m ϭ 141, n ϭ 47



b. m ϭ 240, n ϭ 160



85. z ϭ .83, P-value Ϸ .20, no.

87. .9015, .8264, .0294, .0000; true average IQs; no

89. Yes; z ϭ 4.2, P-value Ϸ 0

91. a. Yes. t ϭ Ϫ6.4, df ϭ 57, and P-value Ϸ 0

b. t ϭ 1.1, P-value ϭ .14, so don’t reject H0.

93. (Ϫ1.29, Ϫ.59)



Chapter 10

1. a. f ϭ 1.85 Ͻ 3.06 ϭ F.05,4,15, so don’t reject H0.

b. P-value Ͼ .10



19. Any value of SSE between 422.16 and 431.88 will work.



3. f ϭ 1.30 Ͻ 2.57 ϭ F.10,2,21, so P-value Ͼ .10. H0 cannot be

rejected at any reasonable significance level.

5. f ϭ 1.73 Ͻ 5.49 ϭ F.01,2,27, so the three grades don’t appear

to differ.

7. f ϭ 1.70 Ͻ 2.46 ϭ F.10,3,16, so P-value Ͼ .10. H0 cannot be

rejected at any reasonable significance level.

9. f ϭ 3.96 and F.05,3,20 ϭ 3.10 Ͻ 3.96 Ͻ 4.94 ϭ F.01,3,20, so

.01 Ͻ P-value Ͻ .05. Thus H0 can be rejected at significance

level .05; there appear to be differences among the grains.

11. w ϭ 36.09



3

1

4

2

5

437.5

462.0

469.3 512.8 532.1

Brands 2 and 5 don’t appear to differ, nor does there appear

to be any difference between brands 1, 3, and 4, but each

brand in the first group appears to differ significantly from

all brands in the second group.



13.



21. a. f ϭ 22.6 and F.01,5,78 Ϸ 3.3, so reject H0.

b. (Ϫ99.16, Ϫ35.64), (29.34, 94.16)

23.

1

2

3

4



1

Ϫ

Ϫ

Ϫ

Ϫ



2

3

2.88 Ϯ 5.81 7.43 Ϯ 5.81

Ϫ

4.55 Ϯ 6.13

Ϫ

Ϫ

Ϫ

Ϫ

4

3

2



4

12.78 Ϯ 5.48

9.90 Ϯ 5.81

5.35 Ϯ 5.81

Ϫ

1



25. a. Normal, equal variances

b. SSTr ϭ 8.33, SSE ϭ 77.79, f ϭ 1.7, H0 should not be

rejected (P-value Ͼ .10)

27. a. f ϭ 3.75 Ն 3.10 ϭ F.05,3,20, so brands appear to differ.

b. Normality is quite plausible (a normal probability plot of

the residuals xij Ϫ ෆxiи shows a linear pattern).

c. 4 3 2 1 Only brands 1 and 4 appear to differ

significantly.



3



1



4



2



5



427.5



462.0



469.3



502.8



532.1



ෆnෆ)

33. arcsin(͙x/



4

33.84



35. a. 3.68 Ͻ 4.94, so H0 is not rejected.

b. .029 Ͼ .01, so again H0 is not rejected.



15. w ϭ 5.94



2

24.69



1

26.08



3

29.95



The only significant differences are between 4 and both

1 and 2.

17. (Ϫ.029, .379)



31. Approximately .62



37. f ϭ 8.44 Ͼ 6.49 ϭ F.001, so P-value Ͻ .001 and H0 should

be rejected.

5 3 1 4 2 This underscoring pattern is a bit awkward

to interpret.