Appendix C

Answers to Selected Exercises

Chapter 2

Section 2-1

2-1. Let a, b denote a part above, below the specification, respectively

S = {aaa, aab, aba, abb, baa, bab, bba, bbb}

2-3. Let a denote an acceptable power supply

Let f, m, c denote a supply with a functional, minor, or cosmetic error, respectively.

S = {a, f, m, c}

2-5. Sequences of y or n of length 24 with 2²⁴ outcomes

2-7. S is the sample space of 100 possible two digit integers.

2-9. S = {0,1,2, ...,1E09} in ppb

2-11. S = {1.0,1.1,1.2,...,14.0}

2-13. S = {0,1,2,...,} in milliseconds

2-17. c = connect, b = busy,

S = {c, bc, bbc, bbbc, bbbbc,...}

2-21. (a) S = nonnegative integers from 0 to the largest integer that can be displayed by the scale S = {0,1,2,3,...}

(b) S

(c) {12,13,14,15}

(d) {0,1,2,...,11}

(e) S

(f) {0,1,2,...,7}

(g) ∅

(h) ∅

(i) {8,9,10,...}

2-23. Let d denoted a distorted bit and let o denote a bit that is not distorted.

(b) No, for example

(e) A₁ ∩ A₂ ∩ A₃ ∩ A₄ = {dddd}

(f) (A₁ ∩ A₂) ∪ (A₃ ∩ A₄) = {dddd, dodd, dddo, oddd, ddod, oodd, ddoo}

2-25. Let P denote being positive and let N denote being negative. The sample space is {PPP, PPN, PNP, NPP, PNN, NPN, NNP, NNN}.

(a) A = {PPP}

(b) B = {NNN}

(c) A ∩ B = ∅

(d) A ∪ B = {PPP, NNN}

2-27. (a) A′ ∩ B = 10, B′ = 10, A ∪ B = 92

2-29. (a) A′ = {x|x ≥ 72.5}

(b) B′ = {x|x ≤ 52.5}

(c) A ∩ B = {x|52.5 < x < 72.5}

(d) A ∪ B = {x|x > 0}

2-31. Let g denote a good board, m a board with minor defects, and j a board with major defects.

(a) S = {gg, gm, gj, mg, mm, mj, jg, jm, jj}

(b) S = {gg, gm, gj, mg, mm, mj, jg, jm}

2-35. 120

2-37. 144

2-39. 14,400

2-41. (a) 416,965,528

(b) 113,588,800

(c) 130,721,752

2-43. (a) 21   (b) 2520   (c) 720

2-45. (a) 1000   (b) 160   (c) 720

2-47. (a) 0.416   (b) 0.712   (c) 0.206

2-49. 41,947,059

2-51. 900

2-53. (a) 673   (b) 1672   (c) 6915

(d) 8399   (b) 1578

2-55. (a) 19   (b) 59

(c) 127   (d) 511

2-57. (a) 21   (b) 16

(c) 27   (d) 33

Section 2-2

2-59. (a) 0.4   (b) 0.8   (c) 0.6

(d) 1   (e) 0.2

2-61. (a) 1/10   (b) 5/10

2-63. (a) S = {1,2,3,4,5,6,7,8}

(b) 2/8   (c) 6/8

2-65. (a) 0.83   (b) 0.85

2-67. (1/10³)*(1/26³) = 5.7×10⁻⁸

2-69. (a) 4+4×3+4×3×3 = 52

(b) 36/52   (c) No

2-71. (a) 0.30   (b) 0.77   (c) 0.70

(d) 0.22   (e) 0.85   (f) 0.92

2-75. 0.9889

2-77. (a) 0.0792   (b) 0.1969

(c) 0.8142   (d) 0.9889

(e) 0.1858

2-79. (a) 0.00202   (b)

2-81. (a) 0.2448   (b) 4.58×10⁻⁷

(c) 0.7551   (d) 0.254

Section 2-3

2-83. (a) 0.9   (b) 0

(c) 0   (d) 0

(e) 0.1

2-85. (a) 0.70   (b) 0.95   (c) No

2-87. (a) 350/370   (b) 362/370

(c) 358/370   (d) 345/370

2-89. (a) 13/130   (b) 0.90, No

2-91. (a) 0.7255   (b) 0.8235

(c) 0.7255

2-93. (a) 0.2264   (b) 0.9680

(c) 0.9891

2-95. (a) 7/10   (b) 3/10

(c) 0.7   (d) 0.55

2-97. (a) 0.245   (b) 0.755

(c) 0.630

Section 2-4

2-99. (a) 86/100   (b) 79/100

(c) 70/79   (d) 70/86

2-101. (a) 0.903   (b) 0.591

2-103. (a) 12/100   (b) 12/28

(c) 34/122

2-105. (a) 0.5625   (b) 0.1918

(c) 0.3333

2-107. (a) 20/100   (b) 19/99

(c) 0.038   (d) 0.2

2-109. (a) 0.02   (b) 0.000458

(c) 0.9547

2-111. No

2-113. (a) 0.6087   (b) 0.3913

(c) 0.5   (d) 0.5

2-115. (a) 0.0987   (b) 0.0650

2-117. (a) 0.25   (b) 2/3

(c) 0.25

2-119. (a) 0.667   (b) 0.203

(c) 0.413   (d) 0.797

Section 2-5

2-121. (a) 0.2   (b) 0.3

2-123. 0.014

2-125. 0.028

2-127. (a) 0.2376   (b) 0.0078

2-129. (a) 0.2   (b) 0.2

2-133. (a) 0.0109   (b) 0.2264

(c) 0.9891   (d) 0.1945

2-135. (a) 0.0792   (b) 0.8142

(c) 0.9208   (d) 0.8031

2-137. 0.2

2-139. (a) 1 /126   (b) 4/9

(c) 1/6   (d) 1/6

2-141. (a) 0.245   (b) 0.755

(c) 0.336

Section 2-6

2-143. independent

2-145. (a) not independent. (b) yes

2-147. (a) not independent. (b) 0.733

2-149. (a) 0.59   (b) 0.328   (c) 0.41

2-151. (a) 0.00307   (b) 0.04096

2-153. (a) 0.01   (b) 0.49   (c) 0.09

2-155. (a) 0.00003   (b) 0.00024

(c) 0.00107

2-157. 0.9702

2-159. not independent.

2-161. independent.

2-163. 5.13×10⁻⁹

2-165. no change

Section 2-7

2-167. 0.89

2-169. (a) 0.97638   (b) 0.20755

2-171. (a) 0.615   (b) 0.618   (c) 0.052

2-173. (a) 0.9847   (b) 0.1184

2-175. 0.2540

2-177. 0.5

2-179. (a) 0.152   (b) 0.789

(c) 0.906

2-181. (a) 0.327

(b) 0.612

Supplemental Exercises

2-185. 0.014

2-187. (a) 0.82   (b) 0.90   (c) 0.18

(d) 0.80   (e) 0.92   (f) 0.98

2-191. (a) 0.2   (b) 0.202

(c) 0.638   (d) 0.2

2-193. (a) 0.03   (b) 0.97   (c) 0.40

(d) 0.05   (e) 0.012   (f) 0.018

(g) 0.0605

2-195. (a) 0.18143   (b) 0.005976

(c) 0.86494

2-197. 0.000008

2-199. (a) 50   (b) 37   (c) 93

2-201. S = {A, A′ D₁, A′ D₂, A′ D₃, A′ D₄, A′ D₅}

2-203. (a) 0.19   (b) 0.15   (c) 0.99

(d) 0.80   (e) 0.158

2-205. (a) No   (b) No

(c) 40/240   (d) 200/240

(e) 234/240   (f) 1

2-207. (a) 0.282   (b) 0.718

2-209. 0.996

2-211. (a) 0.0037   (b) 0.8108

2-213. (a) 0.0778   (b) 0.00108

(c) 0.947

2-215. (a) 0.9764   (b) 0.3159

2-217. (a) 0.207   (b) 0.625

2-219. (a) 0.453   (b) 0.262

(c) 0.881   (d) 0.547

(e) 0.783   (f) 0.687

2-221. 1.58×10⁻⁷

2-223. (a) 0.67336   (b) 2.646×10⁻⁸

(c) 0.99973

2-225. (a) 36⁷   (b) 70(26⁶)

(c) 100(26⁵)

2-227. (a) 0.994, 0.995

(b) 0.99, 0.985

(c) 0.998, 0.9975

2-229. (a) 0.421   (b) 0.699

(c) 0.338   (d) 0.948

Mind-Expanding Exercises

2-231. (a) n = 3   (b) n = 3

2-233. 0.306, 0.694

Chapter 3

Section 3-1

3-1. {0,1,2,...,1000}

3-3. {0,1,2,...,99999}

3-5. {1,2,...,491}

3-7. {0,1,2,...}

3-9. {0,1,2,...,15}

3-11. {0,1,2,...,10000}

3-13. {0,1,2,...,40000}

3-15. {1, 2, ..., 100}

Section 3-2

3-17. (a) 1   (b) 7/8   (c) 3/4   (d) 1/2

3-19. (a) 9/25   (b) 4/25   (c) 12/25   (d) 1

3-21. (a) 0.3   (b) 0.6

(c) 0.4   (d) 0.3

3-23. f(0) = 0.033, f(1) = 0.364,

f(2) = 0.603

3-25. P(X = 0) = 0.008,

P(X = 1) = 0.096,

P(X = 3) = 0.512

3-27. P (X = 50) = 0.5,

P(X = 25) = 0.3,

P(X = 10) = 0.2

3-29. P(X = 15) = 0.6,

P(X = 5) = 0.3,

P(X = −0.5) = 0.1

3-31. P(X = 0) = 0.00001,

P(X = 1) = 0.00167,

P(X = 2) = 0.07663,

P(X = 3) = 0.92169

Section 3-3

3-33. X = waiting time (hours)

3-35. X = Non-failed well depth P (X = 255) = (1515+1343)/7726 = 0.370

P(X = 218) = 26/7726 = 0.003

P(X = 317) = 3290/7726 = 0.426

P(X = 231) = 349/7726 = 0.045

P(X = 267) = (280+887)/7726 = 0.151

P(X = 217) = 36/7726 = 0.005

3-37. P(X = 0) = 0.72

P (X = 1) = 0.26

P (X = 2) = 0.02

3-39. (a) 7/8   (b) 1   (c) 3/4   (d) 3/8

where f_x(0) = 0.008, f_x (1) = 0.096.

f_x(2) = 0.384,

f_x(3) = 0.512

where P(X = 50 million) = 0.5,

P(X = 25 million) = 0.3,

P(X = 10 million) = 0.2

3-49. (a) 1   (b) 0.5   (c) 0.5   (d) 0.5

3-51. (a) 1   (b) 0.75   (c) 0.25   (d) 0.25

(e) 0   (f) 0

Section 3-4

3-57. μ = 2, σ² = 2

3-59. μ = 0, σ² = 1.5

3-61. μ = 2.8, σ² = 1.36

3-63. μ = 1.57, σ² = 0.311

3-65. 24

3-67. μ = 0.0004, σ² = 0.00039996

3-69. (a) μ = 18.694,

σ² = 735.9644,

σ = 27.1287

(b) μ = 37.172,

σ² = 2947.996,

σ = 54.2955

3-71. E(X) = 4.808, V(X) = 6.147

3-73. E(X) = 281.83, V(X) = 976.24

3-75. mean = 0.3

Section 3-5

3-77. μ = 2, σ² = 0.667

3-79. (a)2.40×10⁻¹⁸ (b) 0.9999

(c) 9.91×10⁻¹⁸   (d) 1.13810⁻⁴

3-81. (a) μ = 687.5, σ² = 56.25

(b) μ = 87.5, σ² = 56.25

3-83. E(X) = 4.5, E(Y) = 22.5,

σ_Y = 14.36

3-85. μ = 7, σ = 1.414

3-87. 0.0001

3-89. No

Section 3-6

3-93. (a) 0.9298   (b) 0(c) 0.0112

(d) 0.0016

3-95. (a) 2.40×10⁻¹⁸   (b) 0.9999

(c) 9.91×10⁻¹⁸   (d) 1.138×10⁻⁴

3-97. (a) 0   (b) 10

3-101. (a) 0.215   (b) 0.9999   (c) 4

3-103. (a) 0.410   (b) 0.218   (c) 0.37

3-105. (a) 1   (b) 0.999997

(c) E(X) = 12.244, V(X) = 2.179

3-107. (a) Binomial, p = 10⁴/36⁹,

n = 1E09   (b) 0

(c) E(X) = 4593.9, V(X) = 4593.9

3-109. (a) 0.9961   (b) 0.9886

3-111. (a) 0.142   (b) 0.322 (c) 0.963

3-113. (a) 0.009   (b) 0.382   (c) 0.972

(d) 3

3-115. (a) (1−0.01)¹⁰⁰⁰ ≅ 0   (b) 0.126

(c) 0.99

3-117. 14

Section 3-7

3-119. (a) 0.5   (b) 0.0625   (c) 0.0039

(d) 0.75   (e) 0.25

3-121. (a) 5   (b) 5

3-123. (a) 0.0064   (b) 0.9984 (c) 0.008

3-125. (a) 0.0167   (b) 0.9039 (c) 50

3-127. (a) 0.13   (b) 0.098   (c) 7.69 ≈ 8

3-129. (a) 3.91×10⁻¹⁹

(b) 200   (c) 2.56×10¹⁸

3-131. (a) 3×10⁸ (b) 3×10¹⁶

3-135. (a) 10   (b) 0.039

(c) 0.039   (d) 0.271

3-137. (a) 0.0604   (b) 0.1808

(c) 15

3-139. (a) 0.2273   (b) 0.185

(c) 43.99

Section 3-8

3-141. (a) 0.4191   (b) 0(c) 0.001236

(d) E(X) = 0.8, V(X) = 0.6206

3-145. (a) 0.12   (b) 0.852

3-147. (a) 0.087   (b) 0.9934   (c) 0.297

(d) 0.9998

3-149. (a) 0.7069   (b) 0.0607 (c) 0.2811

3-151. (a) 0.0041   (b) 0.3091   (c) 0.0165

3-153. (a) 0.25   (b) 0.72

(c) 0.16   (d) 5

3-155. (a) 0.599   (b) 0.176 (c) 18

Section 3-9

3-157. (a) 0.0183   (b) 0.2381

(c) 0.1954   (d) 0.0298

3-159. E(X) = V(X) = 2.996

3-161. (a) 0.264   (b) 48

3-163. (a) 0.4566   (b) 0.047

3-165. (a) 0.2   (b) 99.89%

3-167. (a) 0.6065   (b) 0.0067

(c) P(W = 0) = 0.0067,

P(W = 1) = 0.0437,

P(W ≤ 1) = 0.0504

3-169. (a) 0.026   (b) 0.287   (c) 0.868

3-171. (a) 0.435   (b) 0.487

(c) 13.816

3-173. (a) 0.010   (b) 0.005

(c) 2.56   (d) 3.15

Supplemental Exercises

3-175. E(X) = 1/4, V(X) = 0.0104

3-177. (a) n = 50, p = 0.1 (b) 0.112

(c) P(X ≥ 49) = 4.51×10⁻⁴⁸

3-179. (a) 0.000224   (b) 0.2256

(c) 0.4189

3-181. (a) 0.1024   (b) 0.1074

3-183. (a) 3000   (b) 1731.18

3-185. (a) P(X = 0) = 0.0498   (b) 0.5768

(c) P(X ≤ x) ≥ 0.9, x = 5

(d) σ² = λ = 6. Not appropriate.

3-187. (a) 0.1877   (b) 0.4148   (c) 15

3-189. (a) 0.0110   (b) 8/3

3-191. 40000

3-193. 0.1330

3-195. (a) 500   (b) 222.49

3-197. 0.1

3-199. f_X(0) = 0.16, f_X(1) = 0.19,

f_X(2) = 0.20, f_X(3) =0.31,

f_X(4) = 0.14

3-201. f_x(2) = 0.2, f_x(5.7) = 0.3,

f_x(6.5) = 0.3, f_x(8.5) = 0.2

3-203. (a) 0.0433   (b) 3.58

3-205. (a) f_X(0) = 0.2357,

f_X(1) = 0.3971,

f_X(2) = 0.2647,

f_X(3) = 0.0873,

f_X(4) = 0.01424,

f_X(5) = 0.00092

(b) f_X(0) = 0.0546,

f_X(1) = 0.1866,

f_X(2) = 0.2837,

f_X(3) = 0.2528,

f_X(4) = 0.1463,

f_X(5) = 0.0574,

f_X(6) = 0.0155,

f_X(7) = 0.0028,

f_X(8) = 0.0003,

f_X(9) = 0.0000,

f_X(10) = 0.0000

3-207. 37.8 seconds

3-209. (a) 2.8   (b) 0.061

(c) 13,900

Mind-Expanding Exercises

3-215. (a) 131   (b) 123

Chapter 4

Section 4-2

4-1. (a) 0.3679   (b) 0.2858   (c) 0

(d) 0.9817   (e) 0.0498

4-3. (a) 0.5   (b) 0.1464   (c) 0.7072

(d) 0.8536   (e) 1.12 radians

4-5. (a) 0.4375   (b) 0.7969

(c) 0.5625   (d) 0.7031   (e) 0.5

4-7. (a) 0.5   (b) 0.4375   (c) 0.125

(d) 0   (e) 1   (f) 0.9655

4-9. (a) 0.5   (b) 49.8

4-11. (a) 0.1

4-13. (a) 0.5625   (b) 1.1   (c) 0

4-15. (a) 0.221   (b)0.368   (c) 5.991

Section 4-3

4-17. (a) 0.56   (b) 0.7   (c) 0   (d) 0

4-19. F(x) = 0 for x ≤ 0; 1 −e^−x for x > 0

4-21. F(x) = 0 for x ≤ −,

4-23. (a) 0.56   (b) 0.7

(c) 0   (d) 0

4-25. 0.2

4-27. F(x) = 0 for x < 0; 0.25x² for 0 ≤ x < 2; 1 for 2 ≤ x

4-29. f(x) = 0.2 for 0 < x < 4;

f(x) = 0.04 for 4 ≤ x < 9

4-31. F(x) = 0 for x ≤ 0, x − 0.25x² for 0 < x ≤ 2, 1 for x > 2

4-33. F(x) = 0 for x ≤ 0, 1 − exp(−0.5x) for x > 0

P(40 < X ≤ 60) = 2.06E−9

Section 4-4

4-35. E(X) = 2, V(X) = 4/3

4-37. E(X) = 0, V(X) = 0.6

4-39. E(X) = 4, V(X) = 3.2

4-41. μ = 2 / 3, σ² = 2 / 9

4-43. μ = 2, σ² = 4

4-45. E(X) = 2

4-47. (a) E(X) = 109.39, V(X) = 33.19

(b) 54.70

4-49. (a) E(X) = 5.1, V(X) = 0.01

(b) 0.3679

Section 4-5

4-51. (a) E(X) = 0, V(X) = 0.577

(b) 0.90   (c) F(x) = 0 for x < −1;

0.5x+0.5 for −1 ≤ x < 1; 1 for 1 ≤ x

4-53. (a) F(x) = 0 for x < 0.95; 10x −9.5

for 0.95 ≤ x < 1.05; 1 for 1.05 ≤ x

(b) 0.3   (c) 0.96

(d) E(X) = 1.00, V(X) = 0.00083

4-55. (a) F(x) = 0 for x < 0.2050;

100x − 20.50 for

0.2050 ≤ x < 0.2150; 1 for

0.21.50 ≤ x

(b) 0.25   (c) 0.2140

(d) E(X) = 0.2100,

V(X) = 8.33×10⁻⁶

4-57. (a) F(X) = x/90 for 0 ≤ x ≤ 90

(b) E(X) = 45, V(X) = 675

(c) 1/3   (d) 0.333

4-59. (a) σ_x = 34.64 (b) 1/3 (c) 1/2

4-61. (a) Yes (b) μ = 5 (c) 0.3

Section 4-6

4-63. (a) 0.90658   (b) 0.99865

(c) 0.07353   (d) 0.98422

(e) 0.95116

4-65. (a) 0.90   (b) 0.5   (c) 1.28

(d) −1.28 (e) 1.33

4-67. (a) 0.93319 (b) 0.69146

(c) 0.9545   (d) 0.00132

(e) 0.15866

4-69. (a) 0.93319 (b) 0.89435

(c) 0.38292 (d) 0.80128

(e) 0.54674

4-71. (a) 0.99379 (b) 0.13591

(c) 5835

4-73. (a) 0.0228 (b) 0.019

(c) 152.028

(d) small (less than 5%)

4-75. (a) 0.0082 (b) 0.72109

(c) 0.564

4-77. (a) 12.309 (b) 12.1545

4-79. (a) 0.00621 (b) 0.308538

(c) 133.33

4-81. (a) 0.1587 (b) 1.3936

(c) 0.9545

4-83. (a) 0.00043 (b) 6016

(c) 1/8

4-85. (a) 0.02275 (b) 0.324

(c) 11.455

4-87. [23.5, 24.5], no effect from stdev

4-89. (a) 0.0248   (b) 0.1501

(c) 92.0213

4-91. σ = 0.912

4-93. (a) −1.645 (b)0.5   (c) 0.683

Section 4-7

4-95. (a) 0.0853 (b) 0.8293 (c) 0.0575

4-97. (a) 0.1446 (b) 0.4761 (c) 0.3823

4-99. (a) 0.2743 (b) 0.8413

4-101. 0.022

4-103. 0.5

4-105. (a) 0 (b) 0.156 (c) 10,233

(d) 8.3 days/year (e) 0.0052

4-107. (a) 0.012 (b) 0.9732 (c) 536.78

4-109. (a)0.07 (b) 0.997

4-111. (a) μ = 2500, σ = 50 (b) 0.9778

(c) 0.9778   (d) 0.473

Section 4-8

4-113. (a) 0.3679   (b) 0.1353

(c) 0.9502   (d) 0.95, x = 29.96

4-115. (a) 0.3679   (b) 0.2835

(c) 0.1170

4-117. (a) 0.1353 (b) 0.4866

(c) 0.2031 (d) 34.54

4-119. (a) 0.0498 (b) 0.8775

4-121. (a) 0.0025 (b) 0.6321

(c) 23.03 (d) same as part (c)

(e) 6.93

4-123. (a) 15.625 (b) 0.1629

(c) 3×10⁻⁶

4-125. (a) 0.2212 (b) 0.2865

(c) 0.2212 (d) 0.9179

(e) 0.2337

4-127. (a) 0.3528 (b) 0.04979

(c) 46.05 (d) 6.14×10⁻⁶

(e) e⁻¹² (f) same

4-129. (a) 0.3679 (b) 0.1353

(c) 0.0498

(d) does not depend on θ

4-131. (a) 0.435 (b) 0.135   (c) 0.369

(d) 0.865

4-133. (a) 0.26   (b) 0.51   (c) 0.71

4-135. (a) 0.544   (b) 0.65   (c) 2.63

Section 4-9

4-137. (a) 120   (b) 1.32934

(c) 11.6317

4-139. (a) Erlang λ = 5 calls/min, r = 10

(b) E(X) = 2, V(X) = 0.4

(c) 0.2 minute

(d) 0.1755   (e) 0.2643

4-141. (a) 50000   (b) 0.6767

4-143. v(a) 5×10⁵ (b) V(X) = 5×10¹⁰

σ = 223607 (c) 0.0803

4-145. (a) 0.1429   (b) 0.1847

4-149. (a) 1.54   (b) 0.632

μ = 11,494.25,

4-151. (a) σ² = 1,321,178.49

(b) ≈ 1

Section 4-10

4-153. E(X) = 12000, V(X) = 3.61×10¹⁰

4-155. 1000

4-157. (a) 803.68 hours

(b) 85319.64   (c) 0.1576

4-159. (a) 443.11   (b) 53650.5

(c) 0.2212

4-163. (a) 0.5698   (b) 0.1850

(c) 0.4724

4-165. (a) 0.0468   (b) 0.1388

4-167. (a) μ = 300, σ² = 90000.

(b) 0.449   (c) 416 s

4-169. δ = 40.93, β = 10.13

Section 4-11

4-171. (a) 0.0016   (b) 0.0029

(c) E(X) = 12.1825,

V(X) = 1202455.87

4-173. (a) 0.03593   (b) 1.65

(c) 2.718312.6965

4-175. (a) θ = 8.4056, ω² = 1.6094

(b) 0.2643   (c) 881.65

4-179. (a) E(X) = 4.855, V(X) = 4.090

(b) 0.9263 (c) 0.008

4-181. (a) 0.27   (b) 0.48

(c) μ = 4.48, σ² = 34.5126

4-183. (a) 0.01963 (b) 0.032

Section 4-12

4-185. (a) 0.0313   (b) 0.4559

(c) E(X) = 0.7143, V(X) = 0.0454

4-187. (a) Mode = 0.8333, E(X) = 0.6818, V(X) = 0.0402

(b) Mode = 0.6316, E(X) = 0.6154, V(X) = 0.0137

4-189. 0.0272

4-191. (a) α = 2.006, β = 4.04

(b) σ = 0.178

Supplemental Exercises

4-193. (a) 0.99379 (b) 0.621%

4-195. (a) 0.15866 (b) 90.0

(c) 0.9973   (d) (0.9973)¹⁰

(e) 9.973

4-197. (a) 0.0217 (b) 0.9566

(c) 229.5

4-199. 0.8488

4-201. (a) 620.4 (b) 105154.9

(c) 0.4559

4-203. (a) 0.0625 (b) 0.75 (c) 0.5

(d) F(x)= 0 for x < 2; x/4 − x+1 for 2 ≤ x < 4; 1 for 4 ≤ x

(e) E(X)= 10 / 3, V(X) = 0.2222

4-205. (a) 0.3935 (b) 0.9933

4-207. (a) θ = 3.43, ω² = 0.96

(b) 0.946301

4-209. (a) 0.6915 (b) 0.683

(c) 1.86

4-211. (a) 0.0062   (b) 0.012 (c) 5.33

4-215. (a) 0.5633 (b) 737.5

4-217. (a) 0.9906   (b) 0.8790

4-219. (a) μ = 0.167, σ² = 0.0198

(b) 0.03125 (c) 0.021

4-221. (a) 37.3892 (b) 9.42

4-223. (a) 0.01   (b) 0.979 (c) 2571

4-225. μ = −1.33, σ² = 430.2

Mind-Expanding Exercises

4-229. (a) kσ² (b) kσ²+k(μ − m)²

4-233. (a) μ = γ+1/ λ, σ² = 1/ λ²

(b) 1 − exp(−1)

Chapter 5

Section 5-1

5-1. (a) 3/8 (b) 5/8 (c) 3/8 (d) 1/8

(e) V(X) = 0.4961 V(Y) = 1.8594

(f) f(1) = 1/4, f(1.5) = 3/8, f(2.5) = 1/4, f(3) = 1/8

(g) f(2)= 1/3, f(3) = 2/3

(h) 1 (i) 2 1/3

(j) Not independent

5-3. (a) 3/8 (b) 3/8 (c) 7/8

(d) 5/8 (e) V(|X) = 0.4219

V(Y) = 1.6875

(f) f(−1) = 1/8,   f(−0.5)= 1/4,

f(0.5) = 1/2,   f(1) = 1/8

(g) 1   (h) 1   (i) 0.5

(j) Not independent

5-5. (b) f_X(0) = 0.970299,

f_X(1) = 0.029403,

f_X(2) = 0.000297,

f_X(3) = 0.000001

(c) 0.03   (d) f(0) = 0.920824,

f(1) = 0.077543,f(2) = 0.001632

(e) 0.080807

(g) Not independent

5-7. (b) f(0) = 2.40×10⁻⁹,

f(1) = 1.36×10⁻⁶,

f(2) = 2.899×10⁻⁴, f(3) = 0.0274,

f(4) = 0.972

(c) 3.972 (d) equals f(y)

(e) 3.988 (f) 0.0120

(g) Independent

5-9. (b) f_X(0) = 0.2511,

f_X(1) = 0.0405,

f_X(2) = 0.0063,

f_X(3) = 0.0009,

f_X(4) = 0.0001

(c) 0.0562

(d) f_Y|3(0) = 2/3, f_Y|3(1) = 1/3,

f_Y|3(2) = f_Y|3(3) = f_Y|3(4) = 0

(e) 0.0003 (f) 0.0741

(g) Not independent

5-11. (c) 0.308   (d) 5.7

5-13. (a) 0.4444 (b) 0.6944

(c) 0.5833   (d) 0.3733

(e) 2   (f) 0   (g) f_X (x) = 2x/9;

0 < x < 3 (h) f_Y|1.5(y) = 2y/9;

0 < y < 3 (i) 2 (j) 4/9

(k) f_X|2(x) = 2x/9; 0 < x < 3

5-15. (a) 1/81 (b) 5/27

(c) 0.790 (d) 16/81

(e) 12/5   (f) 8/5

(g) f(x) = 4x³ /81; 0 < x < 3

(h) f_Y|X=1(y) = 2y; 0 < y < 1

(i) 1   (j) 0

(k) f_X|Y=2(x) = 2x/5; 2 < x < 3

5-17. (a) 0.9879   (b) 0.0067

(c) 0.000308 (d) 0.9939

(e) 0.04 (f) 8/15 (g) f(x)= 5e^−5x; x > 0   (h) f_Y|X|=1 (y) = 3e^3−3y; 1 <   y (i) 4/3   (j) 0.9502, f_Y(2) = 15e⁻⁶/2; 0 < y

(k) f_X|Y=2(y) = 2e^−2x; 0 < x < 2

5-19. (a) 1/30   (b) 1/12 (c) 19/9

(d) 97/45 (g) 1   (h) 0.25

5-21. (a) P(X > 5, Y > 5) = 0.0439,

P(X > 10, Y > 10) = 0.0019

(b) 0.0655

5-23. (a) 0.25 (b) 0.0625 (c) 1

(d) 1   (e) 2/3 (f) 0.25

(g) 0.0625

(h) f_X|YZ(y, z) = 2x; 0 < x < 1

(i) 0.25

5-25. (a) 0.75   (b) 3/4   (c) 0.875

(d) 0.25   (g) 1 for x > 0

5-27. (a) 0.032   (b) 0.0267

5-29 (b) 1

(c) f_XY (x, y) = e^−y for 0 < x < y

(d) f_X|Y(x) = for 0 < x < y

(e) 1 − e⁻¹ (f)2   (g)1 −

(h) 1 − 3e⁻² (i) 3.9   (j) No

5-31 (a) 0.4352

(b) Yes, one solution is x = 1.625

and y = 0.703

Section 5-2

5-33. 0.8851

5-35. c = 1/36, ρ = −0.0435

5-37 σ_XY = −0.16, ρ_XY = −0.27

5-39. c = 8/81, ρ = 0.4924

5-41. σ_XY = ρ_XY = 0

5-43. σ_XY = ρ_XY = 0

5-45 σ_XY = 1, ρ_XY =

Section 5-3

5-49. (a) p₁ = 0.05, p₂ = 0.85, p₃ = 0.10

(d) E(X) = 1, V(X) = 0.95

(f) 0.07195

(g) 0.7358

(h) E(Y)= 17

(i) 0

(j) P(X = 2, Y = 17) = 0.0540,

P(X = 2|Y = 17) = 0.2224

(k) E(X|Y = 17) = 1

5-51. (b) 0.1944   (c) 0.0001

(e) E(X) = 2.4   (f) E(Y) = 1.2

(g) 0.7347   (h) 0

(i) P(X = 0|Y = 2) = 0.0204,

P(X = 1|Y = 2) = 0.2449,

P(X = 2|Y = 2) = 0.7347

(j) 1.7143

5-53. (a) 0.7887 (b) 0.7887

(c) 0.6220

5-55. 0.8270

5-57 (a) No   (b) 0.0668   (c) 6

(e) ρ = 0.894; = 10, μ_Y = 0

Section 5-4

5-63. (a) 18 (b) 77 (c) 0.5 (d) 0.873

5-65. (a) E(T)= 4, σ_T = 0.1414

5-67. (a) 0 (b) 1

5-69. E(X) = 1290, V(X) = 19600

5-71. (a) 0.002 (b) n = 6   (c) 0.9612

5-73. (a) 0.0027 (b) No (c) 0

5-75 0.36

5-77 (a) E(Z) = 13.2, V(Z) = 1.08

(b) 0.5   (c) 0.9412   (d) 15.62

Section 5-5

5-79. f_Y(y) = ¼; y = 3,5,7,9

5-81. (b) 18

Supplemental Exercises

5-101. (a) 3/8 (b) 3/4 (c) 3/4

(d) 3/8

(e) E(X)= 7/8, V(X)= 39/64,

E(Y) = 7/8, V(Y) = 39/64

(h) 2/3 (i) not independent

(j) 0.7949

5-103. (a) 0.0560 (b) Z~Bin(20, 0.1)

(c) 2   (d) 0.323

(e) Z|X ~ Bin(4, 0.25)

(f) 1 (g) not independent

5-105. (a) 1/108 (b) 0.5787

(c) 3/4 (d) 0.2199

(e) 9/4 (f) 4/3

5-107. 3/4

5-109. (a) 0.085 (b) Z~Bin(10,0.3) (c) 3

5-111. (a) 0.499 (b) 0.5

5-113. (a) 0.057 (b) 0.057

5-117. (a) E(T) = 1.5, V(T) = 0.078

(b) 0.0367

(c) E(P) = 4, V(P) = 0.568

5-121. (a) p₁ = 0.13, p₂ = 0.72, p₃ = 0.15,

x+y+z = 12

(b) not possible

(c) 0.736 (d) 0 (e) 0.970

(f) 0.285 (g) 0.345

5-123 f_P(p) = ,

where J =

5-125 202.949

Chapter 6

Section 6-1

6-1. No, usually not, Ex: {2,3}

6-3. No, usually not, Ex: {1,2,3,1000}.

6-5. Yes, Ex: {5,5,5,5,5,5,5}.

6-7. = 74.0044, s = 0.00473

6-9. = 7068.1, s = 226.5

6-11. = 43.975, s = 12.294

6-13. μ = 5.44

6-15. ₁ = 287.89, s₁ = 160.154

₂ = 325.01, s₂ = 121.20

6-17. = 7.184, s = 0.02066

6-19. (a) = 65.86, s = 12.16

(b) = 66.86, s = 10.74

6-21. = 4.7205, s = 0.6371

Section 6-2

6-25. = 5.328, min = 3.42

6-29. Symmetric with a single mode

6-31. = 1436.5, lower quartile:

Q₁ = 1097.8, and upper quartile:

Q₃ = 1735.0

6-33. = 89.250, lower quartile:

Q₁ = 86.100, and upper quartile:

Q₃ = 93.125

6-35. median: = 1436.5, mode: 1102, 1315, and 1750, mean: = 1403.7

6-37. = 366.57, s = 940.02, and = 41.455

6-39. 95th percentile = 5479

6-41. = 260.3, s = 13.41, = 260.85, and 90th percentile = 277.2

6-43. = 89.45, s = 2.8, = 90, and proportion = 22/40 = 55%

Section 6-4

6-65. (a) = 5.220, lower quartile:

Q1 = 4.538, upper quartile:

Q3 = 6.277

6-67. (a) unseeded, = 44.2, lower

quartile: Q1 = 23.7,

upper quartile: Q3 = 1202.6

(b) seeded, = 222, lower

quartile: Q1 = 79, upper quartile:

Q3 = 445

6-69. (a) = 2.415, s² = 0.285, and

s = 0.543

6-71. (a) = 952.44, s² = 9.55, and

s = 3.09

(b) = 953

6-73. (a) = 67.50, lower quartile:

Q₁ = 58.50, and upper quartile:

Q₃ = 75.00

(b) = 68.00, lower quartile:

Q₁ = 60.00, and upper quartile:

Q₃ = 75.00

Section 6-6

6-91. r = 0.876

Supplemental Exercises

6-107. (a) Sample 1 Range = 4

Sample 2 Range = 4

(b) Sample 1: s = 1.604

Sample 2: s = 1.852

6-113. (b) = 9.325, s = 4.486

6-125. (a) 34.232.05; 20,414.52

6-127. (b) 21.265; 6.422

Mind-Expanding Exercises

6-129. s²(old) = 50.61, s² (new) = 5061.1

6-133. = 431.89, = 34.028

6-135. = 69

6-137. (a) = 89.29 (b) = 89.19

(c) No

Chapter 7

Section 7-2

7-1. 8/103

7-3. 0.8186

7-5. 0.4306

7-7. 0.191

7-9. n = 12

7-11. 0.2312

7-13. (a) 0.5885 (b) 0.1759

7-15. 0.983

7.17. 0.67

7.19. (a) 4958   (b) 3420   (c) 684

(d) 3360   (e) 16/25

7.21. Sample averages more normally distributed

Section 7-3

7-23. (a) N = 25, Mean = 150.47,

S² = 105.06, SS = 2521.5

(b) 150.468

7-25. Bias = σ²

7-27. V (₁) = σ²/77 is smallest

7-33. Bias = σ²/n

7-35. (a) 423.33 (b) 9.08

(c) 1.85 (d) 424 (e) 0.2917

7-39. (d) 0.01 (e) 0.0413

7.41. Results can differ, obtained 2.85

Section 7-4

7-45.

7-49. unbiased

7-51. (a) = /(2n)

(b) same as part (a)

7-57. (b) λ₀ (m+x +1)/(m+λ₀+1)

7-59. (a) 5.046 (b) 5.05

Supplemental Exercises

7-65. 0.8664

7-67. 5.6569

7-69. n = 100

7-71. = /3

7-75. = 21.86, ₅₀₀₀ = 109,300, = 0.7

Chapter 8

Section 8-1

8-1. (a) 96.76% (b) 98.72%

(c) 93.56% (d) 97.72%

(e) 97.50%

8-3. (a) 1.29 (b) 1.65 (c) 2.33

8-5. (a) 1^st CI = 50, 2^nd CI = 50

(b) higher confidence implies a wider CI

8-7. (a) 4   (b) 7

8-9. (a) Longer (b) No (c) Yes

8-11. [87.85, 93.11]

8-13. (a) [74.0353, 74.0367]

(b) [74.035, ∞)

8-15. (a) [3232.11, 3267.89]

(b) [3226.4, 3273.6]

8-17. 267

8-19. 2²

8-21. (a) [13.383, 14.157]

(b) [13.521, ∞)   (c) 1   (d) 2

8-23. (a) (26.33, 29.67) (b) 66.4%

Section 8-2

8-25. (a) 2.179   (b) 2.064

(c) 3.012   (d) 4.073

8-27. (a) Mean = 25.1848,

Variance = 2.5760

(b) 24.037 ≤ μ ≤ 26.333

8-29. [58197.33, 62082.07]

8-31. [1.094, 1.106]

8-33. (−∞, 125.312]

8-35. [443.520, 528.080]

8-37. [301.06, 333.34]

8-39. (b) [2237.3, 2282.5]

(c) [2241.4, ∞)

8-41. (b) [4.051, 4.575]

(c) [4.099, ∞)

8-43. (b) [2.813, 2.991]

8-45 (a) (1.98, 2.18) (b) Yes

Section 8-3

8-47. (a) 13.85   (b) 21.67

(c) 10.12, 30.14

8-49. [0.0055, ∞)

8-51. [0.31, 0.46]

8-53. [0.626, 1.926]

8-55. σ ≤ 0.0122

8-57. (0.514, 0.811)

Section 8-4

8-59. (a) [0.02029, 0.06637]

(b) (−∞, 0.0627]

8-61. (a) [0.501, 0.571]

(b) [0.506, ∞)

8-63. (a) [0.225, 0.575]   (b) 2305

(c) 2401

8-65. 666

8-67. (a) (0.910, 0.945)   (b) 2440

(c) 9624

8-69. (0.025, 0.073)

8-71. (0.906, 0.951)

Section 8-6

8-73. [52131.1, 68148.3]

8-75. [1.068, 1.13]

8-77. [292.049, 679.551]

8-79. [263.7, 370.7]

8-81. [2193.5, 2326.5]

8-83. 90% PI = [2.71, 3.09]

90% CI = [2.85, 2.95]

99% CI = 2.81, 2.99]

8-85. [49555.54, 70723.86]

8-87. [1.06, 1.14]

8-89. TI = [237.18,734.42]

CI = 443.42,528.08]

8-91. TI = [247.60,386.60]

CI = [301.06,333.34]

8-93. TI = [2.49,3.31]

CI = [2.84,2.96]

Supplemental Exercises

8-97. (a) 0.0997 and 0.064

(b) 0.044 and 0.014

(c) 0.0051 and < 0.001

8-99. (a) Normality

(b) [16.99, ∞)

(c) [16.99, 33.25]

(d) (−∞, 343.74]

(e) [28.23, 343.74]

(f) 16.91 < μ < 29.09

15.85 < σ² < 192.97

(g) mean: [16.88, 33.12],

variance: [28.16, 342.94]

8-101. (a) [13.74, 16.92]

(b) [13.24, 17.42]

8-103. (a) Yes

(b) [197.84, 208.56]

(c) [185.41, 220.99]

(d) [171.21, 235.19]

8-105. [0.0956, ∞)

8-107. (a) Yes (b) [1.501, 1.557]

(c) [1370, 1.688]

(d) [1.339, 1.719]

8-109. (a) [0.0004505, 0.009549]

(b) 518 (c) 26002

8-111. (a) Normality

(c) [18.478, 26.982]

(e) [19.565, 123.289]

Mind-Expanding Exercises

8-113. (b) [28.62, 101.98]

8-115. (a) 46 (b) [10.19, 10.41],

p = 0.6004

8-117. 950 of CIs and 0.9963

Chapter 9

Section 9-1

9-1. (a) Yes (b) No (c) No

(d) No (e) No

9-3. (a) H₀ : μ = 20nm, H₁ : μ < 20nm

(b) No

9-5. (a) α = 0.02275.

(b) β = 0.15866

(c) β = 0.5

9-7. (a) 11.4175 ≤ _c ≤ 11.42

(b) 11.5875 ≤ _c ≤ 11.59

(c) 11.7087 ≤ _c ≤ 11.71

(d) 11.7937 ≤ _c ≤ 11.84

9-9. (a) P-value = 0.0135

(b) P-value ≤ 0.000034

(c) P-value = 0.158655

9-11. (a) 0.09296

(b) β = 0.04648

(c) β = 0.00005

9-13. (a) β ≅ 0.005543

(b) β ≅ 0.082264

(c) As n increases, β decreases

9-15. (a) α = 0.05705 (b) β = 0.5

(c) β = 0.05705.

9-17. (a) μ ≤ 191.40

(b) μ ≤ 185.37

(c) μ ≤ 186.6

(d) μ ≤ 183.2

9-19. (a) P-value = 0.2148

(b) P-value = 0.008894

(c) P-value = 0.785236

9-21. (a) α = 0.0164

(b) 1 − β = 0.21186

(c) α will increase and the power will increase with increased sample size.

9-23. (a) P-value = 0.238

(b) P-value = 0.0007

(c) P-value = 0.2585

9-25. (a) α = 0.29372

(b) β = 0.25721

9-27. (a) α ≈ 0   (b) α = 0.99506

9-29. (a) 0.02   (b) 0.03

Section 9-2

9-31. (a) H₀ : μ = 10, H₁:μ > 10

(b) H₀ : μ = 7, H₁ : μ ≠ 7

(c) H₀: μ = 5, H₁ : μ < 5

9-33. (a) a = z_α ≅ −2.33

(b) a = z_α ≅ −1.64

(c) a = z_α ≅ −1.29

9-35. (a) P-value ≅ 0.04

(b) P-value ≅ 0.066

(c) P-value ≅ 0.69

9-37. (a) P-value ≅ 0.98

(b) P-value ≅ 0.03

(c) P-value ≅ 0.65

9-39. (a) StDev = 0.7495, z₀ = −0.468,

P-value = 0.68, fail to reject H₀

(b) one-sided

(c) 19.42 ≤ μ ≤ 20.35

(d) 0.640

9-41. (a) 0.6827 (b) one-sided

(c) P-value = 0.002, reject H₀

(d) 98.8518 ≤ μ

(e) P-value = 0.15, fail to reject H₀

9-43. (a) z₀ = −0.95 > −1.96, fail to reject H₀

(b) β = 0.80939

(c) n ≅ 16

9-45. (a) z₀ = 1.26 < 1.65 fail to reject H₀

(b) P-value = 0.1038

(c) β ≅ 0.000325

(d) n ≅ 1

(e) 39.85 ≤ μ

9-47. (a) z₀ = 1.56 > −1.65, fail to reject H₀

(b) P-value ≅ 0.94

(c) Power = 0.97062

(d) n ≅ 5

(e) μ ≤ 104.53

9-49. (a) z₀ = 1.77 < 1.65, reject H₀

(b) P-value ≅ 0.04

(c) Power = 1

(d) n ≅ 2

(e) 4.003 ≤ μ

9-51. (a) one-sided

(b) z₀ = 2.95, reject H₀

(c) P-value = 0.002

Section 9-3

9-53. (a) critical value > 2.539

(b) critical value > 1.796

(c) critical value > 1.345

9-55. (a) 0.05 ≤ p ≤ 0.1

(b) 0.05 ≤ p ≤ 0.1

(c) 0.5 ≤ p ≤ 0.8

9-57. (a) 0.95 ≤ p ≤ 0.975

(b) 0.025 ≤ p ≤ 0.05

(c) 0.6 ≤ p ≤ 0.75

9-59. (a) 9

(b) .05 ≤ P-value ≤ = 0.1, fail to

reject H₀

(c) two-sided

(d) 11.89 ≤ μ ≤ 13.23

(e) t₀ = 1.905, reject H₀

(f) reject H₀

9-61. (a) t₀ = 0.6665, fail to reject H₀

(b) equal to P-value ≈ 0.5

(c) 246.84 ≤ μ ≤ 404.15, fail to reject H₀

9-63. (a) |t₀| = 3.48 > 2.064 reject H₀

P-value = 0.002

(b) Yes

(c) power ≅ 1

(d) n = 20

(e) 98.065 ≤ μ ≤ 98.463

9-65. (a) t₀ = −1.291, fail to reject H₀ at

α = 0.05, 0.2 < P-value < 0.5

(b) Yes

(c) power = 0.70

(d) n > 100

(e) 129.337 ≤ μ ≤ 130.157

9-67. (a) |t₀| = 1.55 < 2.861, fail to reject

H₀; 0.10 < P-value < 0.20

(b) Yes, see normal probability plot

(c) power = 0.30

(d) n = 40

(e) 1.9 ≤ μ ≤ 4.62

9-69. (a) t₀ = 0.15 < 1.753., fail to reject

H₀; P-value > 0.40.

(b) n = 4. Yes

9-71. t₀ = 3.46 > 1.833, reject H₀

0.0025 < P-value < 0.005

9-73. (a) t₀ = −14.69 < 1.6604, fail to

reject H₀; P-value > 0.995

(b) Yes

(c) power = 1

(d) n = 15

9-75. (a) H₀ : μ = 98.2, H₁ : μ ≠ 98.2

(b) z₀ = 0.98, P-value = 0.327, fail

to reject H₀

(c) (98.1151, 98.4549) contains

98.2, fail to reject H₀

Section 9-4

9-77. (a) critical values 6.84 and 38.58

(b) critical values 3.82 and 21.92

(c) critical values 6.57 and 23.68

9-79. (a) = 7.63

(b) = 4.57

(c) = 7.79

9-81. (a) 0.5 < P-value < 0.9

(b) 0.5 < P-value < 0.9

(c) 0.005 < P-value < 0.01

9-83. (a) = 0.23 < 26.30, fail to reject

H₀; P-value > 0.995

(b) 0.07 ≤ σ

9-85. (a) = 109.52 > 71.42, reject H₀;

P-value < 0.01

(b) 0.31 < σ < 0.46

9-87. (a) = 12.46 > 7.26, fail to reject

H₀; 0.1 < P-value < 0.4

(b) σ ≤ 5240

9-89. (a) = 11.52 < 19.02, fail to

reject H₀; 0.2 < P-value

(b) 0.45   (c) n = 30

Section 9-5

9-91. (a) one-sided

(b) appropriate

(c) = 0.3564, z₀ = −1.1867

P-value = 0.118, ≤ 0.6105

(d) 0.2354

9-93. (a) z₀ = −1.31 > −1.65, fail to reject

H₀; P-value = 0.095

(b) p ≤ 0.0303

9-95. (a) z₀ = 2.06 > 1.65, reject H₀;

P-value = 0.0196

(b) 0.7969 ≤ p

9-97. (a) z₀ = −0.94 > 2.33 fail to reject

H₀; P-value = 0.826

(b) 0.035 ≤ p

9-99. z₀ = 1.58 > −2.33, fail to reject H₀

9-101. (a) z₀ = 0.54 > 1.65, fail to reject

H₀; P-value = 0.295

(b) β = 0.639, n ≅ 118

9-103. H₀ : p = 0.90, H₁ : p > 0.9,

z₀ = 2.842, P-value =0.002,

reject H₀

9-105. (0.798 < μ), yes

Section 9-7

9-107. (a) = 6.955 < 15.09, fail to reject H₀

(b) P-value = 0.2237

9-109. (a) = 10.39 > 7.81 reject H₀

(b) P-value = 0.0155

9-111. (a) = 769.57 » 36.42,

reject H₀

(b) P-value ≅ 0

Section 9-8

9-113. = 146.3648, P-value ≈ 0

9-115. , fail to reject H₀;

P-value = 0.070

9-117. reject H₀;

P-value = 0.002

9-119. , fail to reject H₀;

P-value = 0.013

9-121. (a)

(b) P-value < 0.005

Section 9-9

9-123. (a) P-value = 2P(R⁺ ≤ 7|p = 0.5) = 0.132

(b) 0.180

9-125. (a) P-value= 2*P(R⁺ ≥ 3|p = 0.5) ≈ 1

(b) z₀ = 0, P-value ≈ 1

9-127. Ignore ties (a) w = 9 > = 5,

fail to reject H₀

(b) 0.110

9-129. Ignore ties

(a) = 27 < = 41, reject H₀

Section 9-10

9-131. (a) H₀ : μ = 3550, H₁ :μ ≤ 3550

and H₀ :μ = 3450, H₁ :μ ≥ 3450

(b) t₀ = 0, P-value =0.5 and

t₀ = 12.65 P-value ≈ 0, fail to

reject H₀ : μ = 3550,

not equivalent

9-133. (a) H₀ :μ = 9750, H₁ :μ ≥ 9750

(b) P-value > 0.5, fail to reject

H₀, not equivalent

Section 9-11

9-135. = 37.40, P-value = 0.002, reject H₀

9-137. = 30.57, P-value = 0.0007,

reject H₀

Supplemental Exercises

9-139. (a) 15

(b) SE Mean = 1.1525,

t₀ = −1.449, 0.1 < P-value < 0.2,

95.874 ≤ μ ≤ 100.786

(c) fail to reject H₀

(d) fail to reject H₀

9-141. (a) t₀ = 0.5694,0.25 < P-value < 0.4

(b) 10.726 ≤ μ ≤ 14.222

9-143. (a) n = 25, 0.9783;

−0.7495 ≤ μ₁ − μ₂, 0.9554; n = 400,

0.8599; n = 2500, 0.2119

(b) n = 25, 0.3783; n = 100, 0.2643;

n = 400, 0.1056; n = 2500, 0.0009;

significant when n = 2500.

9-145. (a) 0.4522 (b) 0.4404

(c) 0.3557 (d) 0.2981

9-147. (a) 0.6406 (b) 0.2877 (c) 0.0537

9-149. (a) n = 100, 0.3632; n = 150,

0.2119; n = 300, 0.0352

(b) n = 100, 0.6293; n = 150,

0.4522; n = 300, 0.1292

(c) 1   (d) 24, 5

9-151. (a) d = 2, β = 0; d = 3, β = 0 (b) 2

(c) d = 1, β ≈ 0.1; d = 1.5 β ≈ 0.04;

n = 4

9-153. = 0.000009, reject H₀

9-155. (a)

= 0.8897, fail to reject H₀

(b) 1.86 ≤ μ₁ −μ₂ ≤ 18.94

9-157. (a) normal distribution used

because sample size is large,

z₀ = 6.12, fail to reject H₀

(b) P-value ≈ 1

(c)

= 5.06, fail to reject H₀

9-159. (a) H₀: μ = 0.635 vs. H₁: μ > 0.635

(b) normal distribution used

because sample size is large,

z₀ = −5.31, fail to reject H₀

(c) P-value = 1

9-161. (a) t₀ = −6.10,

P-value<0.001, reject H₀

(b) d = 4.54, Power ≈ 1

(c) d = 2.27, n ≥ 5

(e) From a normal probability plot, assumption is reasonable

9-163. (a) t₀ = 0.37, fail to reject H₀

(b) From a normal probability plot, assumption is reasonable

(c) 0.25 < P-value < 0.4

9-165. (a) = 58.81, reject H₀

(b) P-value < 0.01

9-167. (a) = 0.509, fail to reject H₀

(b) = 0.509, P-value < 0.01, reject H₀

9-169. (a) H₀ : μ = 102, H₁ : μ ≤ 102 and

H₀ : μ = 98, H_l : μ ≥ 98

(b) Fail to reject H₀ : μ = 98, not equivalent

Chapter 10

Section 10-1

10-1. (a) −1.96 < z₀ = −0.9 < 1.96, do not

reject H₀; P-value = 0.368

(b) −9.79 ≤ μ₁ − μ₂ ≤ 3.59

(c) Power = 0.14

(d) n₁ = n₂ = 180

10-3. (a) z₀ = 0.937 < 2.325, do not

reject H₀; P-value = 0.174

(b) μ₁ − μ₂ ≥ −4.74

(c) Power = 0.04

(d) Use n₁ = n₂ = 339

10-5 (a) z₀ = −5.84 < 1.645, do not reject

H₀; P-value = 1

(b) μ₁ − μ₂ ≥ 6.8

(c) Power = 0.9988

(d) The sample size is adequate

10-7 (a) z₀ = −7.25 < −1.645 reject H₀;

P-value ≅ 0

(b) −3.684 ≤ μ₁ − μ₂ ≤ −2.116

(c) n₁ = n₂ = 11

10-9. (a) −5.83 ≤ μ₁ −μ₂ ≤ −0.57;

P-value = 0.0173

(b) Yes

(c) Power = 0.9616; n ≅ 10

(d) Normal

10-11 (−1.42, 1.02)

10-13 z₀ = −4.716, reject H₀

Section 10-2

10-15. (a) df = 26.45 ≈ 26,

μ₁ − μ₂ ≤ −1.688,

0.0025 < P-value < 0.005, one-sided

(b) reject H₀ (c) Yes

(d) 0.005 < P-value < 0.01, reject H₀

10-17. (a) t₀ = −1.94 < −1.70

0.025 < P-value < 0.05

(b) μ₁ − μ₂ ≤ −0.196

(c) Power = 0.95

(d) n = n₁ = n₂ = 21

10-19. (a) −2.042 < t₀ = 0.230 < 2.042,

do not reject H₀ P-value > 0.80

(b) −0.394 ≤ μ₁ − μ₂ ≤ 0.494

10-21. (a) t₀ = −3.11 < −2.485, reject H₀

(b) −5.688 ≤ μ₁ − μ₂ ≤ −0.3122

10-23. (a) Assumptions verified

(b) t₀ = −2.83 < − 2.101 reject H₀;

0.010 < P-value < 0.020

(c) −0.7495 ≤ μ₁ − μ₂ ≤ −0.1105

10-25. (a) t₀ = −5.498 < −2.021 reject H₀;

P-value < 0.0010

(b) n₁ = n₂ = 38

10-27. (a) t₀ = 3.03 > 2.056 reject H₀;

0.005 < P-value < 0.010

(b) t₀ = 3.03 > 1.706, reject H₀

10-29. (a) t₀ = 7.0 > 2.048, reject

H₀; P-value ≅ 0

(b) 14.93 ≤ μ₁ − μ₂ ≤ 27.28

(c) n > 8

10-31. (a) t₀ = 2.82 > 2.326 reject H₀;

P-value ≅ 0.025

(b) μ₁ − μ₂ ≥ 0.178

10-33. (a) Normal

(b) t₀ = 2.558 > 2.101 reject H₀;

P-value ≈ 0.02

(c) 1.86 ≤ μ₁ − μ₂ ≤ 18.94

(d) Power = 0.05

(e) n = 51

10-37. Heat 5 vs. 7: t₀ = 4.28, reject H₀;

Heat 6 vs. 7: t₀ = −1.37, P-value = 0.194, fail to reject H₀

10-39 (a) t₀ = −2.349, reject H₀

(b) μ₁ − μ₂ ≤ −0.021

(c) no, reject H₀

Section 10-3

10-41. (a) w₂ = 75 > = 51, fail to reject H₀

(b) P-value = 2[1 − P(Z < 0.58)] = 0.562

10-43. (a) w₁ = 77 ≤ = 78, reject H₀

(b) P-value = 2[1 − P(Z < 2.19)] = 0.034

10-45. (a) Min (258, 207) > = 185,

fail to reject H₀

(b) P-value = 0.0155

10.47 E = 5, fail to reject H₀

Section 10-4

10-49. (a) 0.1699 ≤ μ_d ≤ 0.3776

(b) t-test is appropriate.

10-51. −727.46 ≤ μ_d ≤ 2464.21

10-53. (a) t₀ = 5.465 > 1.761 reject H₀;

P-value ≅ 0

(b) 18.20 ≤ μ_d

10-55. (a) t₀ = 8.387 > 1.833 reject H₀

(b) t₀ = 3.45 > 1.833 reject H₀

(c) Yes

10-57. (a) Normal

(b) −0.379 ≤ μ_d ≤ 0.349

(c) 6 ≤ n

10-59. (a) P-value = P(R⁺ ≥ r⁺ = 141, p = 0.5) = 0.0005

reject H₀

10.61. (a) No

(b) t₀ = 3.36, P-value = 0.004,

reject H₀

Section 10-5

10-63. (a) f_0.25,5,10 = 1.59

(b) f_0.10,24,9 = 2.28

(c) f_0.05,8,15 = 2.64

(d) f_0.75,5,10 = 0.529

(e) f_0.90,24,9 = 0.525

(f) f_0.95,8,15 = 0.311

10-65. f₀ = 0.805 > 0.166, fail to reject

H₀; σ₁/σ₂ ≤ 2.20

10-67. (a) f₀ = 1.21 > 0.333, fail to

reject H₀; 0.403 ≤ ≤ 3.63

(b) Power = 0.65

(c) n ≈ 31

10-69. (a) f₀ = 0.923 > 0.365, do not reject H₀

(b) (0.323, 2.527)

10-71. (a) (0.607, 1.463)

(b) (0.557, 1.559)

(c) 0.669 <

10-73. 0.4058 < f₀ = 1.78 < 2.46, do not reject H₀

10-75. 0.248 < f₀ = 0.640 < 4.04, do not reject H₀;

0.159 ≤ ≤ 2.579

10-77. 0.333 < f₀ = 1.35 < 3, do not

reject H₀;

0.45 ≤ ≤ 4.05

10-79. (a) 0.248 < f₀ = 3.337 < 4.03, do not reject H₀

(b) No

10-81. f₀ = 1.78, fail to reject H₀

Section 10-6

10-83. (a) one-sided

(b) z₀ = 1.4012, P-value = 0.0806,

−0.0085 ≤ p₁ − p₂

(c) reject H₀ at α = 0.10, fail to

reject H₀ at α = 0.05

10-85. (a) z₀ = 4.45 > 1.96 reject H₀;

P-value ≈ 0

(b) 0.039 ≤ p₁ − p₂ ≤ 0.1

10-87. (a) z₀ = 5.36 > 2.58 reject H₀;

P-value ≈ 0

10−89. H₀: p_WTC = p_other H₁ : p_WTC > p_other,

z₀ = 2.712, P-value = 0.003,

reject H₀

10-91. (0.0383, 0.0997)

Supplemental Exercises

10-93. (a) Confidence interval −2.622 < μ₁; − μ₂ < 0.902

(b) two-sided

(c) P-value = 0.329

10-95. (a) normality, equality of variance, and independence of the observations.

(b) 1.40 ≤ μ₁ − μ₂ ≤ 8.36

(c) Yes

(d) (0.113, 3.673)

(e) No

10-97. (a) t₀ = 2.554 > 1.895, reject H₀

(b) t₀ = 2.554 < 2.998, do not reject H₀

(c) t₀ = −1.986 < 1.895, do not reject H₀

(d) t₀ = − 1.986 < 2.998, do not reject H₀

10-99. (a) z₀ = 6.55 > 1.96, reject H₀

(b) z₀ = 6.55 > 2.58, reject H₀

(c) z₀ is so large

10-101. (a) −0.0335 ≤ p₁ − p₂ ≤ 0.0329

(b) −0.0282 ≤ p₁ − p₂ ≤ 0.0276

(c) 95% CI: −0.0238 ≤ p_l − p₂

≤ 0.0232 90% CI:

−0.0201 ≤ p₁ − p₂ ≤ 0.0195

10-103. (a) Yes

(b) Yes if similar populations

10-105. (a) 0.0987 ≤ μ₁ − μ₂ ≤ 0.2813

(b) 0.0812 ≤ μ₁ − μ₂ ≤ 0.299

(c) μ₁ − μ₂ ≤ 0.2813

(d) z₀ = 3.42 > 1.96, reject H₀;

P-value = 0.00062

(e) n₁ = n₂ = 9

10-107. (a) z₀ = −5.36 < −2.58, reject H₀

(b) conclusions are the same

(c) n = 60

10-109. (a) No

(b) data appear normal with equal variances

(c) It is more apparent the data follow normal distributions.

(d) One-sided confidence

interval > 22.93

(e) one-sided test, reject H₀,

f₀ = 72.78 > 3.28

10-111. (a) Normality appears valid.

(b) 0.50 < P-value < 0.80, do not reject H₀

(c) n = 30

10-113. (a) It may not be assumed that

(b) t₀ = −2.74 < −2.131, reject H₀

(c) Power = 0.95

(d) n = 26

10-115. f₀ = 0.317, fail to reject H₀

10-117. Traditional interval: (−0.0335, 0.0329); Other interval:

(−0.0337, 0.0327)

Mind-Expanding Exercises

10-123. (c) 0.519 ≤ θ ≤ 3.887

Chapter 11

Section 11-2

11-1 (a) ₁ = 1.846, ₀ =−27.643

(b) 27.726   (c) 6.50

11-3. (a) ₀ = 48.013, ₁ = −2.330

(b) 37.99   (c) 39.39 (d) 6.71

11-5. (a) ₀ = 14.195, ₁ = 10.092,

σ² = 27.24

(b) 89.95   (c) −10.092

(d) 0.99   (e) −7.98,3.13

11-7. (a) ₀ = −6.3355, ₁ = 9.20836,

σ² = 3.7746

(b) 500.124   (c) 9.20836

(d) −1.618

11-9. (a) ₀ = −16.5093, ₁ = 0.0693554,

σ² = 7.3212

11-11. (b) ₀ = 234.071, ₁ = −3.50856,

σ² = 398.25

(c) 128.814

(d) 156.833 and 15.1175

11-13. (b) ₀ = 2625.39, ₁ = 36.962,

σ² = 9811.2

(c) 1886.15

11-15. (a) ₀ = 0.658, ₁ = 0.178,

σ² = 0.083

(b) 3.328 (c) 0.534

(d) 1.726 and 0.174

11-17. (b) ₀ = 2.02, ₁ = 0.0287,

σ² = 0.0253

11-19. (a) = 39.2 − 0.0025x

(b) ₁ = −0.0025

11-21. (b) = 2132.41, = −36.9618

Section 11-4

11-23. (a) = 5.546   (b) se₁ = 0.105

(c) 17.58   (d) P-value ≈ 0

11-25. (a) Divided by 2.20462 pounds,

₁ = 0.0046, ₀ = 80.571

(b) = 12.288

(c) se₁ = 0.061

(d) No change, t = 0.076

(e) Conclusions do not change,

P-value ≈ 1

11-27. (a) t₀ = 12.4853, P-value < 0.001;

t₁ = 20.387, P-value < 0.001;

MS_E = 2.194; f₀ = 415.91,

P-value ≈ 0

(b) reject H₀: β = 0

(c) 2.194

11-29. (a) f₀ = 74.63, P-value = 0.000002,

reject H₀

(b) estimate of σ² = 1.8436

(c) se (₀) = 0.9043

11-31. (a) f₀ = 61.41, P-value ≅ 0,

reject H₀

(b) se(₁) = 1.288 se(₀) = 9.059

(c) t₀ = 0.072, fail to reject H₀

11-33. (a) f₀ = 74334.4, P-value ≈ 0,

reject H₀

(b) se(₀) = 1.66765,

se(₁) = 0.0337744

(c) t₀ = −23.37, P-value = 0.000,

reject H₀

(d) t₀ = −3.8, P-value < 0.005,

reject H₀

11-35. (a) f₀ = 44.0279,

P-value = 0.00004, reject H₀

(b) se(₀) = 9.84346,

se(₁) = 0.0104524

(c) t₀ = −1.67718,

P-value = 0.12166, fail to reject H₀

11-37. (a) f₀ = 155.2, P-value < 0.00001,

reject H₀

(b) se(₀) = 2.96681,

se(₁) = 45.3468

(c) t₀ = −2.3466, P-value = 0.0306,

fail to reject H₀

(d) t₀ = 57.8957,

P-value < 0.00001, reject H₀

(e) t₀ = 2.7651, P-value = 0.0064,

reject H₀

11-39. (a) P-Value = 0.0000, reject H₀

(b) ² = 0.083 se(₀) = 0.1657,

se(₁) = 0.014

(c) Reject H₀

11-41. (a) P-value = 0.310, No

(b) ² = 30.69

(c) se(₀) = 9.141

11-43. 0.55

Sections 11-5 and 11-6

11-45. (a) (1.639, 2.052)

(b) (17.805, 19.191)

(c) (7.553, 29.443)

11-47. (a) (−2.9713, −1.7423)

(b) (46.7145, 49.3115)

(c) (41.3293, 43.0477)

(d) (39.1275, 45.2513)

11-49. (a) [7.464, 12.720]

(b) [−4.30, 32.69]

(c) [91.698, 98.164]

(d) [83.801, 106.061]

11-51. (a) Confidence interval

9.101 < β₀ < 9.932

(b) (−11.6219, −1.04911)

(c) (498.72024, 501.52776)

(d) (495.57344, 504.67456)

11-53. (a) (0.03689, 0.10183)

(b) (−47.0877, 14.0691)

(c) (44.0897, 49.1185)

(d) (37.8298, 55.3784)

11-55. (a) (201.552, 266.590)

(b) (−4.67015, −2.34696)

(c) (111.8339, 145.7941)

11-57. (a) (−43.1964, −30.7272)

(b) (2530.09, 2720.68)

(c) (1823.7833, 1948.5247)

(d) (1668.9013, 2103.4067)

11-59. (a) (0.1325, 0.2235)

(b) (0.119, 1.196)

(c) (1.87, 2.29)

Section 11-7

11-61. R² decreases

11-63. (a) R² = 67.2%

11-65. (a) R² = 99.986%

11-67. (a) R² = 87.94%

11-69. (a) R² = 85.22%

11-71. (a) R² = 89.6081%

(c) R² = 95.73%

(d) ² old = 9811.21,

² new = 4022.93

11-75. (a) f₀ = 207, reject H₀

Section 11-8

11-77. (a) t₀ = 4.81, P-value < 0.0005,

reject H₀

(b) z₀ = 1.747, P-value = 0.04,

reject H₀

(c) ρ ≥ 2.26, reject H₀

11-79. (a) t₀ = 5.475, P-value = 0.000,

reject H₀

(b) (0.3358, 0.8007)

(c) Yes

11-81. (a) = −0.0280411+0.990987x

(b) f₀ = 79.838, reject H₀

(c) 0.903

(d) t₀ = 8.9345, reject H₀

(e) z₀ = 3.879, reject H₀

(f) (0.7677, 0.9615)

11-83. (a) = 5.50+6.73x

(b) 0.948

(c) t₀ = 8.425, reject H₀

(d) (0.7898, 0.9879)

11-85. (a) r = 0.82

(b) t₀ = 7.85, reject H₀,

P-value < 0.005

(c) [0.660, 0.909]

(d) z₀ = 1.56, fail to reject H₀,

P-value = 0.119

Section 11-9

11-87. (a) Yes    (b) No   (c) Yes

(d) Yes

11-89. (b) = −0.8819+0.00385x

(c) f₀ = 122.03, reject H₀

Section 11-10

11-91. (a) ₀ = 5.340, ₁ = −0.00155

(b) Test that all slopes zero:

P-value ≈ 0

11-93. (a) ₀ = −7.047, ₁ = −0.00074,

₂ = −0.9879

(b) Test that all slopes zero:

P-value = 0.036

(d) 0.771

Supplemental Exercises

11-97. (b) = −0.966824+1.54376x

(c) f₀ = 252263.9, P-value ≈ 0,

reject H₀

(d) [1.5367, 1.5509]

(e) t₀ = −199.34, reject H₀

11-99. y* = 1.2232+0.5075x where

y* = 1/y

11-101. = 0.7916x

11-103. (b) = 0.6714+− 2964x

(c) R² = 21.5%

11-105. (b) = −0.699+1.66x

(c) f₀ = 22.75, reject H₀,

P-value = 0.001

(d) (3.399, 5.114)

11-107. (c) all data: (7741.74, 10956.26),

outlier removed: (8345.22, 11272.79)

Chapter 12

Section 12-1

12.1. (a) = −3.10088 −0.60154x₁ +1.77309x₂

(c) = 13.87

12.3. (a) = 1.051+0.09514x₁

+0.000004106x₂

−0.1673x₃

(b) = 1.08 (c) e = 0.42

12-5. (b) = 171.055+3.714x₁ −1.126x₂

(c) 189.49

12-7. (b) 2

12.9 (a) Satisfaction = 144 - 1.11Age -

0.585 Severity+ 1.30 Anxiety

(b) 49.5

(c) 5.90, 0.13, 0.13, 1.06

(d) No

12-11. (a) = 49.90 − 0.01045x₁

−0.0012x₂ − 0.00324x₃

+0.292x₄ −3.855x₅+0.1897x₆

(b) ² = 4.965 se(₀) = 19.67,

se(₁) = 0.02338,

se(₂) = 0.01631,

se(₃) = 0.0009459,

se(₄) = 1.765, se(₅) = 1.329,

se(₆) = 0.273

(c) 29.867

12-13. (a) y = 47.8 − 9.60x₁

+0.415x₂+18.3x₃

(b) ² = 12

(c) se(₀) = 49.94, se(₁) = 3.723,

se(₂) = 0.2261, and

se(₃) = 1.323

(d) 91.38

12-15. (a) = −440.39+19.147x₁

+68.080x₂

(b) ² = 55563 se(₀) = 94.20,

se(₁) = 3.460, and se(₂) = 5.241

(c) 186.675

12-17. (a) = −0.1105+0.4072x₁

+2.108x₂

(b) ² = 0.00008 se(₀) = 0.2501,

se(₁) = 0.1682, and

se(₂) = 5.834

(c) 0.97074

12-19. (a) = 238.56+0.3339x₁

−2.7167x₂

(b) ² = 1321

(c) se(₀) = 45.23,

se(₁) = 0.6763, and

se(₂) = 0.6887

(d) 61.5195

12-21. (a) = 2.99+1.20x₃

+4.60x₇ − 3.81x₁₀

(b) ² = 4.14

(c) se(₀) = 5.877, se(₃) = 0.974,

se(₇) = 0.385, se(₁₀) = 0.486

(d) 81.96

12-23. (a) = 383.80 − 3.6381x₁ −0.1119x₂

(b) ² = 153.0, se(₀) = 36.22,

se(₁) = 0.5665, se(₂) = 0.04338

(c) 180.95   (d) = 484.0 − 7.656x₁

−0.222x₂ − 0.0041x₁₂

(e) ² = 147.0, se(₀) = 101.3,

se(₁) = 3.846, se(₂) = 0.113,

se(₁₂) = 0.0039

(f) 173.1

Section 12-2

12-25. (a) P-value = 0.6438 > 0.05

(b) P-value = 1.09E-07 < 0.05

12-27. (a) H₀ : β₁ = β₂ = β₃ = 0

H₁ : β_j ≠ 0 for at least one j

P-value = 0.04157 < 0.05, reject H₀

(b) 16.22%

12-29. (a) t₀ = 53.087, P-value ≈ 0;

t₁ = 15.02, P-value ≈ 0; t₂=−23.43,

P-value≈0; MS_E =25.582

df Residual = 12, SS

Residual = 307, R-Sq, = 0.9867

(b) f₀ = 445.2899, reject H₀

(c) t-test for each regressor is significant

12-31. (a) f₀ = 263.26, P-value ≈ 0,

reject H₀

(b) t₀(₁) = 19.32, P-value ≈ 0,

reject H₀

t₀(₂) = −13.16, P-value ≈ 0,

reject H₀

12-33. (a) P-value = 9.18E-5, 5.42E-8, and 0.3378

(b) t₀ = 0.98, fail to reject H₀

12-35. (a) f₀ = 19.53, reject H₀

12-37. (a) f₀ = 828.31, reject H₀

(b) t₀ = −2.58, reject H₀ t₀ = 1.84,

fail to reject H₀ t₀ = 13.82, reject H₀

12-39. (a) f₀ = 99.67, reject H₀,

P-value ≈ 0

(b) t₀ = 5.539, reject H₀ t₀ = 12.99,

reject H₀

12-41. (a) f₀ = 9.28, reject H₀,

P-value = 0.015

(b) t₀ = 0.49, fail to reject H₀

t₀ = −3.94, reject H₀

12-43. (a) f₀ = 191.09

(b) t₀ = 12.30, reject H₀ t₀ = 11.94,

reject H₀ t₀ = −7.84, reject H₀

(c) f₀ = 142.66, reject H₀

12-45. (a) f₀ = 97.59, reject H₀,

P-value = 0.002

(b) t₀ = −6.42, reject H₀

t₀ = −2.57, fail to reject H₀

(c) f₀ = 6.629, fail to reject H₀

(d) f₀ = 7.714, fail to reject H₀

(e) f₀ = 1.11, fail to reject H₀

(f) 147.0

12-47. (a) f₀ = 65.55, P-value ≈ 0,

reject H₀

(b) Age: t₁ = −8.40, Severity:

t₂ = −4.43, Anxiety: t₃ = 1.23, not all necessary

12-49. (a) (−0.818 ≤ β₁ ≤ −0.385)

(b) (13.1 ≤ μ_Y|x0 < 14.65)

(c) (5.05 < Y₀ < 22.69)

(d) Prediction interval

12-51. (a) (−0.690 ≤ β₃ ≤ 0.355) (b) Yes

Sections 12-3 and 12-4

12-53. (a) (49.927 < β₀ < 292.183)

(0.033 < β₁ < 7.393)

(−2.765 < β₂ < 0.513)

(b) (158.82, 220.13)

(c) (126.06, 252.88)

12-55. (a) (−20.477 < β₁ < 1.269)

(−0.245 < β₂ < 1.076)

(14.428 < β₃ < 22.159)

(b) (77.582, 105.162)

(c) (82.133, 100.611)

12-57. (a) (−6.9467 < β₁ < −0.3295)

(−0.3651 < β₂ < 0.1417)

(b) (−45.8276 < β₁ < 30.5156)

(−1.3426 < β₂ < 0.8984)

(−0.03433 < β₃ < 0.04251)

12-59. (a) (12.1363 < β₁ < 26.1577)

(b) (−233.4, 63.2)

(c) (−742.09,571.89)

12-61. (a) (0.0943 < β₁ < 0.7201)

(−8.743 < β₂ < 12.959)

(b) (0.861, 0.896)

(c) (0.855, 0.903)

12-63. (a) (−2.173 < β₁ < 2.841)

(−5.270 < β₂ < −0.164)

(b) (−36.7, 125.8)

(c) (−112.8, 202.0)

(d) CI: (107.4, 267.2)PI: (30.7, 344.0)

12-65. (a) −10.18 < β₀ < 16.62

(1.00 < β₃ < 1.45)

(3.85 < β₇ < 5.00)

(−5.11 < β₁₀ < − 3.07)

(b) 0.3877

(c) (81.3, 82.9)

12-67. (a) −8.658 < β₀ < 108.458;

−0.08 < β₂ < 0.059;

−0.05 < β₃ < 0.047;

−0.06 < β₇ < 0;

−4.962 < β₈ < 5.546;

−7.811 < β₉ < 0.101;

−1.102 < β₁₀ < −5.023;

(b) 25.557 < μ _Y|x0 < 33.863

(c) 50.355 < β₀ < 71.147;

−0.036 < β₂ < −0.006;

−0.006 < β₇ < −0.001;

−6.485 < β₉ < −0.429;

(d) Intervals in part (c) are narrower so that part (c) may be a preferable model.

Section 12-5

12-69. (a) 0.893

12-71. (a) 0.95

12-73. (a) 0.843

12-75. (a) 0.997

12-77. (a) 0.756

12-79. (a) 0.985   (b) 0.99

12-81. (b) 0.9937

12-83. (a) 0.12   (b) Yes

Section 12-6

12-85. (a) = −1.633+1.232x −1.495x²

(b) f₀ = 1,858,613, reject H₀

(c) t₀ = −601.64, reject H₀

12-87. (a) = −1.769+0.421x₁

+ 0.222x₂ − 0.128x₃

−0.02x₁₂+0.009x₁₃

+0.003x₂₃ − 0.019

−0.007+0.001

(b) f₀ = 19.628, reject H₀

(d) f₀ = 1.612, fail to reject H₀

12-89. (b) = 56.677 − 0.1457x₁

−0.00525x₂ −0.138x₃ −4.179x₄

12-91. (a) Min C_p : x₁, x₂ C_p = 3.0,

MS_E = 55563.92

= −440.39+19.147x₁,

+68.080x₂ Min MS_E is same as

Min C_p

(b) Same as part (a)

(c) Same as part (a)

(d) Same as part (a)

(e) All models are the same

12-93. (a) Min C_p : x₁ C_p = 1.1,

MS_E = 0.0000705 = −0.20052

+0.467864x₁ Min MS_E is same as

Min C_p

(b) Same as part (a)

(c) Same as part (a)

(d) Same as part (a)

(e) All models are the same

12-95. (a) Min C_p : x₂ C_p = 1.2,

MS_E = 1178.55

= 253.06 − 2.5453x₂ Min MS_E

is same as Min C_p

(b) Same as part (a)

(c) Same as part (a)

(d) Same as part (a)

(e) All models are the same

12-97. (a) Min C_p : x₁, x₂

C_p = 2.9, MS_E = 10.49

= −50.4+0.671x₁+1.30x₂Min

MS_E is same as Min C_p

(b) Same as part (a)

(c) Same as part (a)

(d) Same as part (a)

(e) All models are the same

12-99. (a) Min. C_p = 5.1, Att, PctComp, Yds, YdsperAtt, TD, PctTD, PctInt

(b) PctComp, YdsperAtt, PctTD, PctInt

(c) PctComp, YdsperAtt, PctTD, PctInt

(d) Att, PctComp, Yds, YdsperAtt, TD, PctInt

12-101. (a) Min C_p = 1.3:

= 61.001 − 0.02076x_cid

−0.00354x_etw −3.457x_axle

Min MS_E = 4.0228

= 49.5 − 0.017547x_cid

−0.0034252x_etw+l.29x_cmp

− 3.184x_axle −0.0096x_c02

(b) = 63.31 − 0.0178x_cid

−0.00315x_etw − 3.3x_axle

−0.0084x_c02

(c) Same as Min MSE model in part (a)

(d) = 45.18 − 0.00321x_etw

−4.4x_axle+0.385x_n/v

(e) Min C_p model is preferred

(f) Min C_p = 4.0,

Min MS_E = 2.267

= 10 − 0.0038023x_etw

+ 3.936x_cmp+15.216 x_co

−0.011118x_c02 − 7.401x_trans

+ 3.613lx_drv1+2.342x_drv2

Stepwise: = 39.12 −0.0044x_etw

+ 0.271x_n/v − 4.5x_trns

+ 3.2x_drvl+1.7x_drv2 Forward

selection: = 41.12 −0.00377x_etw

+ 0.336x_n/v − 2.1x_axle

−3.4x_trans+2.1x_drv1+2x_drv2

Backward elimination is same as

Min C_p and Min MS_E

12-103. (a) = −0.304+0.083x₁

−0.031x₃+0.004

C_p = 4.04 MS_E = 0.004

(b) = −0.256+0.078x₁

+0.022x₂ − 0.042x₃+0.0008

C_p = 4.66 MS_E = 0.004

(c) Prefer the model in part (a)

Supplemental Exercises

12-107. (a) 2

(b) 0.0666, 0.0455

(c) 6.685

12-109. (a) f₀ = 1321.39, reject H₀,

P-value < 0.00001

(b) t₀ = −1.45, fail to reject H₀

t₀ = 19.95, reject H₀ t₀ = 2.53,

fail to reject H₀

12-111. (a) y* = −0.908+5.48

+ 1.3 −3.92 −1.14

(b) f₀ = 109.02, P-value = 0.000;

t = 11.27, P-value = 0.000;

t = 14.59, P-value = 0.000;

t = −6.98, P-value = 0.000;

t = −8.11, P-value = 0.000

12-113. (a) = 21.068 − 1.404

+ 0.0055x₄+0.000418x₅

MS_E = 0.013156   C_p = 4.0

(b) Same as part (a)

(c) x₄, x₅ with C_p = 4.1 and

MS_E = 0.0134

(d) The part (c) model is preferable

(e) Yes

12-115. (a) = 300.0+0.85x₁

10.4x₂+ = 405.8

(b) f₀ = 55.37, reject H₀

(c) 0.9022

(d) MS_E = 10.65

(e) f₀ = 0.291, fail to reject H₀

12-117. (a) f₀ = 18.28, reject H₀

(b) f₀ = 2, do not reject H₀

(c) MS_E (reduced) = 0.005

MS_E (Full) = 0.004

Mind-Expanding Exercises

12-121. R² ≥ 0.449

Chapter 13

Section 13-2

13-1. (a) 4

(b) 5

(c) f₀ = 1.58, 0.1 < P-value < 0.25

(d) fail to reject H₀

13-3. (a) 5

(b) df_E = 24, MS_E = 1.1408,

SS_Treatments = 38.96, MS_Treatments

= 7.792, F = 6.83, P-value = 0.0004

(c) Significantly different

(d) 1.1408

13-5. (a) f₀ = 14.76, reject H₀

13-7. (a) f₀ = 12.73, reject H₀

(b) P-value ≅ 0

13-9. (a) f₀ = 16.35, reject H₀

(c) 95%:(140.71, 149.29)99%:

(7.36, 24.14)

13-11. (a) f₀ = 1.86, fail to reject H₀

(b) P-value = 0.214

13-13. (a) f₀ = 8.30, reject H₀

(b) P-value = 0.002

(d) (69.17, 81.83)

(e) (8.42, 26.33)

13-15. (a) f₀ = 0.72, fail to reject H₀

(b) P-value = 0.486, SS_E = 0.146

13-17. (a) f₀ = 2.62, fail to reject H₀

(b) P-value = 0.083

13-19. (b) f₀ = 35.448, P-value = 2.07E-6

(e)(89.273, 90.794)

13-31. n = 3

13-33. 3.5

Section 13-3

13-35. (a) f₀ = 5.77, reject H₀

(b) 0.01412 (c) 0.0148

13-37. (a) f₀ = 0.75, fail to reject H₀

(b) 0   (c) 24

13-39. (a) 0.0289 (b) 48.8%

(c) 48.8%

13-41. (a) H₀: = 0 H₁ ≠ 0

(c) Set equal sample sizes,

σ² = 0.164, = 0

13-43 (a) f₀ = 76.09, reject H₀ (b)(132.97, 147.83)

Section 13-4

13-47. (a) f₀ = 8.92, reject H₀

13-49. (a) f₀ = 3.00, fail to reject H₀

13-51. (a) f₀ = 1.61, fail to reject H₀

13-53. (a) 3   (b) 4

(c) f₀ = 23.15, P-value < 0.01

(d) reject H₀

Supplemental Exercises

13-55. (a) f₀ = 76.09, reject H₀

(c) (132.97, 147.83)

13-57. (a) f₀ = 7.84, reject H₀

(b) P-value = 0.007

13-59. (a) f₀ = 6.23, reject H₀

(c) Algorithm 5

13-61. (a) Power = 0.2 (b) n = 50

13-63. (a) f₀ = 4.656, P-value = 0.006

13-65. (a) Random effects

(c) = 62.227, = 16.21667

Chapter 14

Section 14-3

14-1. (a) H₀ : τ₁ = τ₂ = 0

H₁ : at least one τ_i ≠ 0

(b) H₀ : β₁ = β₂ = β₃ = 0

H₁: at least one βj ≠ 0

(c) H₀ : τβ₁₁ = τβ_l2 = ···

τβ₂₃ = 0 H₁ : at least one

τβ_ij ≠ 0

(b) f(G) = 273.79, f(P) = 8.84,

f(GP) = 1.26 reject H₀ for only main effects

14-3. (a) f(M) = 7.91, f(T) = 28.97,

f(MT) = 3.56 reject H₀ for both main effects and the interaction

14-5. (a) Y_ijk = μ+τ_i+β_j+(τβ)_ij

(b) f(I) = 40.07, f(T) = 0.32,

f(IT) = 1.70 reject H₀ for only insulation

14-7. (a) f(D) = 25.23, f(A) = 543.52,

f(DA) = 3.54 reject H₀ for both main effects and the interaction

14-9. (−3.40, 7.64)

14-11. (a) H₀ : τ₁ = τ₂ = τ₃ = 0,

H₁ : at least one τ_i ≠0

H₀ : β₁ = β₂ = β₃ = 0,

H₁ : at least one β_i ≠ 0

(b) Mg+: f₀ = 0.55,

P-value = 0.617; K+,f₀ = 0.65,

P-value = 0.570

Section 14-4

14-13. (a) f(H) = 7.64, f(C) = 31.66,

f(F) = 19.92, f(HC) = 2.92,

f(HF) = 2.97, f(CF) = 0.96

H, C, F, HF are significant at α = 0.05. The P-value for HC is 0.075.

Section 14-5

14-15. Significant effects:A = 17.00,

C = 10.875, D = 8.375,AD = 9.125

14-17. (a) Cleaning Method = −5.593

Test Position = −1.280

Clean * test = −1.220

(b) Cleaning Method is the only significant factor

14-19. (a) Significant effects:

A = 11.8125, B = 33.9375,

C = 9.6875, AB = 7.9375

14-21. None of the effects are significant

14-23. (b) A, B and AB

(c) = 400+40.124x₁

−32.75x₂+26.625x₁x₂

14-25. f₀ = 5.11, do not reject H₀

14-27. (a) For model with A, B, C, AB, s = 2.92; s (center pts) = 2.86

(b) F₀ = 192.71, curvature is significant

14-29. (a) Large effects: C = −39.79,

D = −198.47, E = −64.86

(b) y = 546.90 − 39.79x_c

−198.47x_D − 64.86x_E

(d) none

14-31. (a) 3-way and higher interactions

(b) SE coef = 0.9, t = 1.25,

p = 0.267, df_A = 1, SS_A = 20.25,

MS_A = 20.25, F =1.57, df_Error = 5,

MS_Error = 12.9

14-33. (a) Large effects: B = −0.000750

(b) with all effects in error except

B the P-value = 0.007

14-35. (a) Effects: Number of visits = 38.12, Length of stay = 22.50,

Case manager = 35.50,

(b) df_Error = 28, MS_Error = 584.8, SE

Coef = 4.275

Section 14-6

14-37. Significant effects: A = 15.625,

C = 10.625, D = 8.875,

AD = 8.875, CD = −3.125,

ACD = 1.875

14-43. (a) Effect JED is confounded with blocks

(b) Marginal significant effects:

J = 19.0, D = 14.75, JD = −18.0,

ED = −25.5,

14-45. (a) Blocking important, SS_Blocks large relative to SS_Error

(b) ABC, ABD, ACD, and BCD

(c) Coefficient for AD = 15.14,

t-statistic for AD = 1.525,

dfBlocks = 1, MS for

interactions = 1165.33

Section 14-7

14-49. (a) 2^5-1

(b) E = ABCD

(c) A = 10.8750, B = 33.6250,

C = 10.6250, D = −0.6250,

E = 0.3750, AB+CDE = 7.1250,

14-51. (a) E = −ABCD

(b) Resolution V

(c) E = −0.4700, BE = −0.4050,

DE = −0.3150

14-53. (c) A = 1.435, B = −1.465,

D = 4.545, AB = 1.15, AD = −1.23

14-55. (b) Design Generator: D = BE,

E = AC Defining Relation:

I = ACE = BDE = ABCDE

A = CE = BCDE = ABDE

B = DE = ACDE = ABCE

AliasesC = AE = ABDE = BCDE

D = BE = ABCE = ACDE

E = AC = BD = ABCD

(c) A = −1.525, B = −5.175,

C = 2.275, D = −0.675, E = 2.275

14-57. (a) Alias Structure:

I+ABD+ACE+BCF

+DEF+ABEF+ACDF

+BCDE A+BD+CE

B+AD+CF C+AE+BF

D+AB+EF E+AC+DF

F+BC+DE AF+BE+CD

(b) Alias Structure: I+ABCG

+ ABDH + ABEF

+ ACDF + ACEH

+ ADEG + AFGH

+ BCDE + BCFH

+ BDFG + BEGH

+ CDGH + CEFG+ DEFH

A

B

C

D

E

F

G

H

AB+CG+DH+EF

AC+BG+DF+EH

AD+BH+CF+EG

AE+BF+CH+DG

AF+BE+CD+GH

AG+BC+DE+FH

AH+BD+CE+FG

14-59. (a) Generators are E = BCD,

F = ACD, G = ABC, and

H = ABD, I = BCDE =

ACDF = ABEF = ABCG =

ADEG = BDFG = CEFG =

ABHD = ACEH = BCFH =

DEFH = CDGH =

BEGH = AFGH

(b) Glassware = 1.4497,

Reagent = −0.8624,

Prep = 0.6034, Tracer = 0.6519,

Dissolution = −0.8052,

Hood = 1.3864,

Chemistry = 0.0591,

Ashing = −0.0129

14-61. (a) E=ABCD

(b) 5

Section 14-8

14-63. (b) = 82.024 − 1.115x₁

−2.408x₂+0.861

−1.59 −1.801x₁x₂

14-65. Path of steepest ascent passes through the point (0, 0) and has a slope −0.8/1.5 = −0.533

14-67. (a) Central composite design, not rotatable

(b) = 150.04 − 58.47x₁

+ 3.35x₂ − 6.53+10.58

+ 0.50 x₁x₂

The linear terms are significant (P − value = 0.001), while both the square terms and interaction terms are insignificant

14-69. (a) along the vector (1.22, −2.13, 1.62, −0.61)

(b) (1.22, −2.13, 1.62, −0.61)

14-71. (a) Yes

(b) x₁ = 1, x₂ = 41 with y₁ = 7

14-73. (a) Central composite design with α = 1

(b)

Supplemental Exercises

14-75. (a) t = 2.54, t = 5.02 The main effect of pH and the interaction of pH and Catalyst Concentration (CC) are significant

14-77. (a) f(L) = 63.24, f(S) = 39.75,

f(LS) = 5.29 L, S, LS are significant

14-79. (a) A = −2.74, B = −6.66,

C = 3.49, AB = −8.71, AC=7.04,

BC = 11.46, ABC = −6.49, not significant

14-81. (a) V = −15.75, P=10.75,

G = −25.00, PG = −19.25,

(b) = 102.75 − 7.87x₁

+ 5.37x₃−12.50x₄ −9.63x₃₄

14-83. V = 3.25, F = −10.25, P = −10.25,

G = −23.75, PG = −11.75

14-85. Design Generators: D = AB E = AC

Alias Structure

I+ABD+ACE+BCDE

A+BD+CE+ABCDE

B+AD+CDE+ABCE

C+AE+BDE+ABCD

D+AB+BCE+ACDE

E+AC+BCD+ABDE

BC+DE+ABE+ACD

BE+CD+ABC+ADE

14-87. (a) 2² factorial with two replicates

(b) Significant effects: x₁ = 0.795,

x² = −1.160

(c) = 0.1994+0.07688x₁

14-89. (a) H₀ : τ₁ = τ₂ = τ₃ = 0

H₀ : β₁ = β₂ = β₃ = 0

H₀ : (τβ)₁₁ = ... = (τβ)₃₃ = 0

(b) f(N) = 311.71, f(P) = 119.17,

f(PN) = 92.94

(d) = 1.22

14-91. (a) Generators are E = ABC,

F = ABD, and G = ACD;

I = ABCE = ABDF = CDEF

= ACDG = BDEG = BCFG

= AEFG

(b) A = −74.9, B = 76.1,

C = −366.4, D = 236.9,

E = −213.4, F = 119.9, G = 101.9

14-93. (a) Block 1: (1), bc, ac, ab; Block 2: a,b,c, abc

(b) A = −0.500, B = 13.000,

C = 6.500, AB = −2.000

AC = −14.500, BC = −5.000

14-95. (a) A: 4 levels, B: 3 levels

(b) 1

(c) AB interaction not significant

(d) dfError = 6, SS_B = 34670882,

MS_E = 29736583, f₀ =1.36

14-97 (a) For seed type

TP Moschofilero

Chapter 15

Section 15-3

15-1. (a) chart: UCL = 242.78,

CL = 223, LCL = 203.22, R

chart: UCL = 72.51, CL = 34.286,

LCL = 0

(b) = 223 = 14.74

15-3. (a) chart: UCL = 245.11,

CL = 223, LCL = 200.89S chart:

UCL = 30.77, CL = 13.58, LCL = 0

(b) = 223, = 14.74

15-5. (a) chart: UCL = 4.930,

CL = 4.668, LCL = 4.406,

R chart: UCL = 0.961, CL = 0.454,

LCL = 0

(b) chart: UCL = 4.910,

CL = 4.668, LCL = 4.425, S chart:

UCL = 0.355, CL = 0.17, LCL = 0

15-7. (a) chart: UCL = 37.5789,

CL = 34.32, LCL = 31.0611 R

chart: UCL = 11.9461, CL = 5.65,

LCL = 0

(b) chart: UCL = 37.4038,

CL = 34.0947, LCL = 30.7857

R chart: UCL = 12.1297,

CL = 5.73684,

LCL = 0 = 2.4664

15-9. (a) chart: UCL = 17.4,

CL = 15.09, LCL = 12.79 R chart:

UCL = 5.792, CL = 2.25, LCL = 0

(b) chart: UCL = 17.96,

CL = 15.78, LCL = 13.62 R chart:

UCL = 5.453, CL = 2.118, LCL = 0

(c) chart: UCL = 17.42,

CL = 15.09, LCL = 12.77 S chart:

UCL = 3.051, CL = 1.188, LCL = 0

Revised chart: UCL = 17.95,

CL = 15.78, LCL = 13.62 S chart:

UCL = 2.848, CL = 1.109, LCL = 0

15-11. (a) chart: UCL = 0.0635,

CL = 0.0629, LCL = 0.0624

R chart: UCL = 0.0020,

CL = 0.0009, LCL = 0

(b) chart: UCL = 0.0634,

CL = 0.0630, LCL = 0.0626 R

chart: UCL = 0.0014, CL = 0.0007, LCL = 0

(c) chart: UCL = 0.0634,

CL = 0.0630, LCL = 0.0626

S chart: UCL = 0.00058,

CL = 0.00027, LCL = 0

Revised:

chart: UCL = 0.0633,

CL = 0.0630, LCL = 0.0626

S chart: UCL = 0.00058,

CL = 0.00028, LCL = 0

15-13. (a) 2.73   (b) no

15-17. s = 2.956, /d₂ = 1.251

Section 15-4

15-19. (a) Individual chart:

UCL = 60.889, CL = 53.05,

LCL = 45.211 MR chart:

UCL = 9.634, CL = 2.94737,

LCL = 0 The process appears to be in control.

(b) = 53.05, = 2.613

15-21. (a) Individual chart:

UCL = 10.5358, CL = 10.0272,

LCL = 9.5186 MR chart:

UCL = 0.625123, CL = 0.19125,

LCL = 0 The process appears to be in control.

(b) = 10.0272, = 0.1696

15-23. (a) Initial study: Individual chart: UCL = 130.5,

CL = 100.78, LCL = 71.06

MR chart: UCL = 36.51,

CL = 11.18, LCL = 0 Revised:

Individual chart UCL = 127.08,

CL = 99.48, LCL = 71.88 MR

chart: UCL = 33.91, CL = 10.38,

LCL = 0

(b) = 99.4792, = 9.20059

15-25. (a) X chart: UCL = 116.43,

CL = 82.0, LCL = 47.57, R chart:

UCL = 42.30, CL = 12.95,

LCL = 0; in control

(b) 82.0, 11.48

15-27. (a) X chart:

UCL = 0.299, CL = 0.119,

LCL = −0.061,

MR chart:

UCL = 0.221, CL = 0.068,

LCL = 0

(b) Sample 17 out of control, revised X-Chart:

UCL = 0.266, CL = 0.110,

LCL = −0.046;

MR chart:

UCL = 0.192, CL = 0.059, LCL = 0

Section 15-5

15-29. (a) 1.3333 (b) 26

15-31. (a) PC = PCR_K = 1.5 (b) 0

15-33. Proportion nonconforming is 0.00779

PCR = 0.905

PCR_K = 0.837

15-35. 0.0009

PCR = 1.13

PCR_K = 1.06

15-37. (a) PCR = 1.35 PCR_K = 1.217

(b) 0.00013

15-41. (a) = 2.46

(b) PCR = 1.02, PCR_K = 0.94

15-43. (a) Fraction defective = 0.002,

PCR = 1.03, PCR_K = 1.03

(b) Fraction defective = 0.057,

PCR = 1.03, PCR_K = 0.526

15-45. (a) 1.4

(b) PCR = 0.95, PCR_K = 0.64

15-47 (a) PCR = 1.33, PCR_K = 0.83

(b) 0.64%

Section 15-6

15-49. (a) not in control UCL = 0.0835,

CL = 0.0645, LCL = 0.0455

(b) Revised P-chart:

UCL = 0.1252, CL = 0.0561,

LCL = 0

15-51. (a) P chart: UCL = 0.1986,

CL = 0.1506, LCL = 0.1026

(b) Revised P chart:

UCL = 0.2062, CL = 0.1573,

LCL = 0.1085

15-53. (a) The limits need to be revised.

UCL = 3.811, CL = 1.942,

LCL = 0.072, sample 5 and 24

exceed limits

(b) U chart: UCL = 3.463,

CL = 1.709, LCL = 0

15-55. (a) UCL = 0.3528, CL = 0.2598,

LCL = 0.1667, not in control

(b) UCL = 0.6694, CL = 0.5195,

LCL = 0.3696, not in control,

points 5, 9, 12, 20 exceed the limits

15-57 (a) C chart,

UCL = 23.08, CL = 12.48,

LCL = 1.88

(b) In control

(c) No samples removed.

(d) X chart:

UCL = 23.67, CL = 12.48,

LCL = 1.29, both X and C charts

indicate the process is in control

Section 15-7

15-59. (a) 4   (b) 0.0228. (c) 43.8596

15-61. (a) 0.2177   (b) ARL = 4.6

15-63. (a) 0.1515   (b) ARL = 6.6

15-65. (a) 0.1020   (b) ARL = 9.8

15-67. (a) 0.2877   (b) ARL = 3.48

15-69. (a) = 6.928 (b) 0.0401

(c) 24.94

15-71 (a) UCL = 0.26, LCL = −0.02

(b) 22.10

UCL = 0.22, LCL = 0.02,

(c)

ARL = 8.04

Section 15-8

15-73. (a) h = 4, k = 0.5, UCL = 3.875,

LCL = −3.875 Yes, this process is in-control.

(b) Observation 20 is out of control, CUSUM = 6.08

15-75. (a) Sigma estimate from s = 0.1736, estimate from moving range = 0.1695, compare to exercise 15-67

(b) h = 4, k = 0.5, UCL = 0.0678,

LCL = −0.0678, out of control at the specified target level

15-77. (a) ARL = 38.0 (b) ARL = 10.4

15-79. (a) 0.169548

(b) The process appears to be in control. UCL = 10.17, CL = 10,

LCL = 9.83

(c) Out of control at observation 13, UCL = 10.29, CL = 10,

LCL = 9.71

15-81. (a) prefer λ = 0.1 and L = 2.81

(b) prefer λ = 0.5 and L = 3.07

(c) 9

15-83. (a) UCL = 12, LCL = −12, not in control

(b) For h = 4, 2.01, < ARL < 2.57

15-85. (a) UCL = 71.94, LCL = 68.07,

sample 13 is out of control

(b) UCL = 75.32, LCL = 64.68,

sample 19 is out of control

(c) ARL ≈ 1 for part (a);

ARL = 1.53 for part (b)

Section 15-9

15-87. Minimax criteria: purchase; Most probable criteria: no purchase;

Expected cost criteria: no purchase

Supplemental Exercises

15-91. (a) chart: UCL = 64.019,

CL = 64, LCL = 63.982 R chart:

UCL = 0.046, CL = 0.018,

LCL = 0

(b) = 64, = 0.011

(c) PCR = 0.63

(d) PCR_k = 0.63

(e) σ² = (0.0033)² = 0.000011

(f) ARL = 12.9

15-93. (a) The process appears to be in control.

(b) P chart: UCL = 0.1717,

CL = 0.1063, LCL = 0.04093

15-95. (a) Individual chart:

UCL = 60.327362, CL = 60.32641,

LCL = 60.325458 MR chart:

UCL = 0.001169, CL = 0.000358,

LCL = 0

(b) Individual chart:

UCL = 0.001362, CL = 00.00041,

LCL = 0.000542 MR chart:

UCL = 0.001169, CL = 0.000358,

LCL = 0

(c) = 60.3264 = 0.0003173

PCR = 1.0505 PCR_k = 0.9455

15-97. (a) Trial control limits

S chart: UCL = 170.25,

CL = 86.42, LCL = 2.59 chart:

UCL = 670.00, CL = 558.77,

LCL = 447.53

Revised S chart: UCL = 158.93,

CL = 80.68, LCL = 2.42 chart:

UCL = 655.79, CL = 551.95,

LCL = 448.10

(b) PCR_K = 0.8725

(c) = 36.9917

(d) ARL = 18.6

15-107. (a) ARL = 43.9

(b) ARL = 6.30

(c) ARL = 2.00

15-109. (a) chart: UCL = 140.168,

CL = 139.49, LCL = 138.812

R chart: UCL = 2.48437,

CL = 1.175,

LCL = 0

(b) Revised: chart:

UCL = 140.417, CL = 139.709,

LCL = 139.001

R chart: UCL = 2.596,

CL = 1.227, LCL = 0 = 0.5276

(c) PCR = 1.26 PCR_k = 1.08

(d) σ² = 0.081 (e) ARL = 5.55

15-111. P (X < USL) = 0.00135

15-113. (a) P( > 12.24 when λ = 16) = 0.96995

(b) P(U > 10.68 when λ = 16) = 1

15-115. (a) ARL = 17.5

(b) ARL = 3.63

15-119. (a) = 3.0411,

probability = 0.0197

(b) 50.8

15-121. (a) X chart:

UCL = 1.563, CL = 1.507,

LCL = 1.452,

samples 20-24 are out of control; MR chart:

UCL = 0.068, CL = 0.020,

LCL = 0,

samples 20 and 25 are out of control

(b) Revised X Chart:

UCL = 1.546, CL = 1.498,

LCL = 1.451;

Revised MR chart:

UCL = 0.058, CL = 0.018, LCL = 0

(c) = 1.49V, = 0.0157V.

(d) Results do not support

σ = 0.02V

15-123. (a) 4

(b) USL = 26.78, LSL = 10.82

15-125. 1.38

15-127. (a) X chart:

UCL = 52.31, CL = 45.74,

LCL = 39.16;

MR chart:

UCL = 8.08, CL = 2.47, LCL = 0

(b) No

15-129. $270

Mind-Expanding Exercises

15-131. 0.125, 0.004

15-133. (b) ARL = 1/p where

p = 1 − Φ(k − )

(c) ARL = 22.0 for k = 2

(d) ARL = 4.47

15-135. n =

15-137. (a) chart CL = 8 UCL = 16.49

LCL = 0

(b) Yes