Course Outcome 1
1. 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∑𝑥𝑖≤𝑥 𝑓(𝑥𝑖 )
2. 𝜇 = 𝐸(𝑋) = ∑𝑥 𝑥𝑓(𝑥)
3. 𝜎 2 = 𝑉(𝑋) = 𝐸[(𝑋 − 𝜇)2 ] = ∑𝑥(𝑥 − 𝜇)2 𝑓(𝑥)
4. 𝜎 2 = 𝑉(𝑋) = ∑𝑥 𝑥 2 𝑓(𝑥) − 𝜇2
5. 𝐸[ℎ(𝑋)] = ∑𝑥 ℎ(𝑥)𝑓(𝑥)
6. 𝑓(𝑥𝑖 ) = 1⁄𝑛 , 𝑥 = 𝑎, 𝑎 + 1, 𝑎 + 2, … , 𝑏, for 𝑎 ≤ 𝑏
7. 𝜇 = 𝐸(𝑋) = (𝑏 + 𝑎)⁄2
8. 𝜎 2 = [(𝑏 − 𝑎 + 1)2 − 1]⁄12
9. 𝑓(𝑥) = (𝑛𝑥)𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 , 𝑥 = 0, 1, … , 𝑛
10. 𝜇 = 𝐸(𝑋) = 𝑛𝑝
11. 𝜎 2 = 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝)
𝑥−1
12. 𝑓(𝑥) = (1 − 𝑝) 𝑝, 𝑥 = 1, 2, …
13. 𝜇 = 1⁄𝑝
14. 𝜎 2 = (1 − 𝑝)⁄𝑝2
15 𝑓(𝑥) = (𝑥−1
)(1 − 𝑝) 𝑥−𝑟 𝑝𝑟 , 𝑥 = 𝑟, 𝑟 + 1, 𝑟 + 2, …
𝑟−1
16. 𝜇 = 𝑟⁄𝑝
17. 𝜎 2 = 𝑟(1 − 𝑝)⁄𝑝2
18. 𝑓(𝑥) =
𝑁−𝐾
(𝐾
𝑥 )( 𝑛−𝑥 )
, x = max{0, n + K − N} to min{K, n}
(𝑁
𝑛)
22. 𝑓(𝑥) = [𝑒 −𝜆𝑇 (𝜆𝑇)𝑥 ]⁄𝑥! , 𝑥 = 0, 1, 2, …
23. 𝜇 = 𝐸(𝑋) = 𝜆𝑇
24. 𝜎 2 = 𝑉(𝑋) = 𝜆𝑇
𝑥
25. 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∫−∞ 𝑓(𝑢) 𝑑𝑢
∞
∞
29. 𝐸[ℎ(𝑋)] = ∫−∞ ℎ(𝑥)𝑓(𝑥) 𝑑𝑥
30. 𝑓(𝑥) = 1⁄(𝑏 − 𝑎) , 𝑎 ≤ 𝑥 ≤ 𝑏
31. 𝜎 2 = 𝑉(𝑋) = (𝑏 − 𝑎)2 ⁄12
32. 𝑓(𝑥) = 𝑁(𝜇, 𝜎 2 ) =
−(𝑥−𝜇)
1
2𝜎2
𝑒
√2𝜋𝜎
2
, −∞ < 𝑥 < ∞
33. Φ(𝑧) = 𝑃(𝑍 ≤ 𝑧) 34. 𝑍 = (𝑋 − 𝜇)⁄𝜎
35. 𝑓(𝑥) = 𝜆𝑒 −𝜆𝑥 for 0 ≤ 𝑥 < ∞
36. 𝜇 = 𝐸(𝑋) = 1⁄𝜆 37. 𝜎 2 = 𝑉(𝑋) = 1⁄𝜆2
38. 𝑃(𝑋 < 𝑡1 + 𝑡2 |𝑋 > 𝑡1 ) = 𝑃(𝑋 < 𝑡2 )
∞
39. Γ(𝑟) = ∫0 𝑥 𝑟−1 𝑒 −𝑥 𝑑𝑥, for 𝑟 > 0
40. 𝑓(𝑥) = 𝜆𝑟 𝑥 𝑟−1 𝑒 −𝜆𝑥 ⁄Γ(𝑟), for 𝑥 > 0, 𝜆 > 0, 𝑟 > 0
41. 𝜇 = 𝐸(𝑋) = 𝜆⁄𝑟 42. 𝜎 2 = 𝑉(𝑋) = 𝑟⁄𝜆2
𝛽 𝑥 𝛽−1
43. 𝑓(𝑥) = 𝛿 (𝛿 )
𝑥 𝛽
𝛽
44. 𝐹(𝑥) = 1 − 𝑒 −(𝑥⁄𝛿 ) 45. 𝜇 = 𝐸(𝑋) = 𝛿Γ (1 + 𝛽1 )
46. 𝜎 2 = 𝑉(𝑋) = 𝛿 2 Γ (1 + 𝛽2 ) − 𝛿 2 [Γ (1 + 𝛽1 )]
√
𝜃+𝜔2 ⁄2
(ln(𝑥)−𝜃)2
2𝜔2
]
49. 𝑉(𝑋) = 𝑒
2
0<𝑥<∞
2𝜃+𝜔2
(𝑒
𝜔2
− 1)
Γ(𝛼+𝛽)
50. 𝑓(𝑥) = Γ(𝛼)Γ(𝛽) 𝑥 𝛼−1 (1 − 𝑥)𝛽−1 , for x in [0, 1],
α > 0, β > 0
51. 𝜇 = 𝐸(𝑋) =
0.01
±2.33
±2.575
69. 𝜇: 𝑥̅ ± 𝑧𝛼⁄2 𝜎⁄√𝑛
70. 𝑛 = [(𝑧𝛼⁄2 𝜎)⁄𝐸 ]
71. 𝜇 ≤ 𝑢 = 𝑥̅ + 𝑧𝛼 𝜎⁄√𝑛 72. 𝑥̅ − 𝑧𝛼 𝜎⁄√𝑛 = 𝑙 ≤ 𝜇
73. 𝜇: 𝑥̅ ± 𝑧𝛼⁄2 𝑠⁄√𝑛 74. 𝜇: 𝑥̅ ± 𝑡𝛼⁄2,𝑛−1 𝑠⁄√𝑛
75. 𝑣 = 𝑛 − 1
2
76. [(𝑛 − 1)𝑠 2 ]⁄𝜒𝛼2⁄2,𝑛−1 ≤ 𝜎 2 ≤ [(𝑛 − 1)𝑠 2 ]⁄𝜒1−𝛼
⁄2,𝑛−1
2
77. [(𝑛 − 1)𝑠 2 ]⁄𝜒𝛼,𝑛−1
≤ 𝜎2
2
78. 𝜎 2 ≤ [(𝑛 − 1)𝑠 2 ]⁄𝜒1−𝛼,𝑛−1
79. 𝑝̂ = 𝑋⁄𝑛 80. 𝑝: 𝑝̂ ± 𝑧𝛼⁄2 √[𝑝̂ (1 − 𝑝̂ )]⁄𝑛
81. 𝑛 = [𝑧𝛼2⁄2 𝑝(1 − 𝑝)]⁄𝐸 2 82. 𝑛 = [𝑧𝛼2⁄2 (0.25)]⁄𝐸 2
83. 𝑝̂ − 𝑧𝛼 √[𝑝̂ (1 − 𝑝̂ )]⁄𝑛 ≤ 𝑝
84. 𝑝 ≤ 𝑝̂ + 𝑧𝛼 √[𝑝̂ (1 − 𝑝̂ )]⁄𝑛
2
𝛼
𝛼+𝛽
𝛼𝛽
52. 𝜎 2 = 𝑉(𝑋) = (𝛼+𝛽)2 (𝛼+𝛽+1)
53. 𝑓𝑋𝑌 (𝑥, 𝑦) = 𝑃(𝑋 = 𝑥, 𝑌 = 𝑦)
54. 𝑓𝑋 (𝑥) = ∑𝑦 𝑓𝑋𝑌 (𝑥, 𝑦)
55. 𝑓𝑌 (𝑦) = ∑𝑥 𝑓𝑋𝑌 (𝑥, 𝑦)
2
𝑧𝛼⁄2
̂ (1−𝑝
̂ ) 𝑧𝛼⁄2
𝑝
𝑝̂+
±𝑧𝛼⁄2 √
+ 2
85. 𝑝:
exp [− (𝛿 ) ], for 𝑥 > 0
exp [−
2𝜋
Critical values of z
α
0.10
0.05
0.025
±1.28 ±1.645 ±1.96
±1.645 ±1.96 ±2.24
2
∞
48. 𝐸(𝑋) = 𝑒
∞
Course Outcome 2
One-tailed
Two-tailed
28. 𝜎 2 = 𝑉(𝑋) = ∫−∞ 𝑥 2 𝑓(𝑥)𝑑𝑥 − 𝜇2
1
𝑏
59. 𝑃(𝑎 < 𝑋 < 𝑏) = ∫𝑎 ∫−∞ 𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 𝑑𝑥
60. 𝑓𝑌|𝑥 (𝑦) = 𝑓𝑋𝑌 (𝑥, 𝑦)⁄𝑓𝑋 (𝑥), 𝑓𝑋 (𝑥) > 0
61. 𝑓𝑋|𝑦 (𝑥) = 𝑓𝑋𝑌 (𝑥, 𝑦)⁄𝑓𝑌 (𝑦), 𝑓𝑌 (𝑦) > 0
62. 𝑓𝑋𝑌 (𝑥, 𝑦) = 𝑓𝑋 (𝑥)𝑓𝑌 (𝑦)
63. 𝐸[ℎ(𝑋, 𝑌)] = ∑ ∑ ℎ(𝑥, 𝑦)𝑓𝑋𝑌 (𝑥, 𝑦), X, Y discrete
64. 𝐸[ℎ(𝑋, 𝑌)] = ∫ ∫ ℎ(𝑥, 𝑦)𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦, X, Y
continuous
65. cov(𝑋, 𝑌) = 𝜎𝑋𝑌 = 𝐸[(𝑋 − 𝜇𝑥 )(𝑌 − 𝜇𝑦 )] = ∑𝑥 ∑𝑦(𝑥 −
𝜇𝑥 )(𝑦 − 𝜇𝑦 ) 𝑓𝑋𝑌 (𝑥, 𝑦)
66. cov(𝑋, 𝑌) = 𝜎𝑋𝑌 = 𝐸[(𝑋 − 𝜇𝑥 )(𝑌 − 𝜇𝑦 )] =
∞ ∞
∫−∞ ∫−∞(𝑥 − 𝜇𝑥 )(𝑦 − 𝜇𝑦 )𝑓𝑋𝑌 (𝑥, 𝑦)𝑑𝑥 𝑑𝑦
67. cov(𝑋, 𝑌) = 𝜎𝑋𝑌 = 𝐸(𝑋𝑌) − 𝜇𝑋 𝜇𝑌
68. 𝜌𝑋𝑌 = cov(𝑋, 𝑌)⁄√𝑉(𝑋)𝑉(𝑌) = 𝜎𝑋𝑌 ⁄(𝜎𝑋 𝜎𝑌 )
Type of Test
26. 𝜇 = 𝐸(𝑋) = ∫−∞ 𝑥𝑓(𝑥) 𝑑𝑥
∞
27. 𝜎 2 = 𝑉(𝑋) = ∫−∞(𝑥 − 𝜇)2 𝑓(𝑥)𝑑𝑥
47. 𝑓(𝑥) = 𝑥𝜔
𝑏
58. 𝑃(𝑎 < 𝑋 < 𝑏) = ∫𝑎 𝑓𝑋 (𝑥) 𝑑𝑥
𝑁−𝑛
20. 𝜎 2 = 𝑛𝑝(1 − 𝑝) ( 𝑁−1) 21. 𝑝 = 𝐾 ⁄𝑁
19. 𝜇 = 𝑛𝑝
56. 𝑓𝑋 (𝑥) = ∫𝑦 𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑦 57. 𝑓𝑌 (𝑦) = ∫𝑥 𝑓𝑋𝑌 (𝑥, 𝑦) 𝑑𝑥
2𝑛
𝑛
4𝑛
2 ⁄𝑛
1+𝑧𝛼
⁄2
𝑋̅−𝜇0
√𝑛
86. 𝑍0 = 𝜎⁄
87. 𝑇0 =
𝑋̅−𝜇0
𝑆⁄√𝑛
88. 𝜒02 = [(𝑛 − 1)𝑆 2 ]⁄𝜎02
89. 𝑍0 = (𝑋 − 𝑛𝑝0 )⁄√𝑛𝑝0 (1 − 𝑝0 )
100. 𝑍0 =
(𝑋̅1 −𝑋̅2 )−Δ0
√(𝜎12 ⁄𝑛1 )+(𝜎22 ⁄𝑛2 )
101. 𝜒02 = ∑𝑘𝑖=1
(𝑂𝑖 −𝐸𝑖 )2
102. 𝑣 = 𝑘 − 𝑝 − 1
𝐸𝑖
∑𝑐𝑗=1 𝑂𝑖𝑗
103. 𝑢̂𝑖 =
104. 𝑣̂𝑖 = 𝑛1 ∑𝑟𝑖=1 𝑂𝑖𝑗
1
105. 𝐸𝑖𝑗 = 𝑛𝑢̂𝑖 𝑣̂𝑗 = 𝑛 ∑𝑐𝑗=1 𝑂𝑖𝑗 ∑𝑟𝑖=1 𝑂𝑖𝑗
1
𝑛
2
106. 𝜒02 = ∑𝑟𝑖=1 ∑𝑐𝑗=1
(𝑂𝑖𝑗 −𝐸𝑖𝑗 )
𝐸𝑖𝑗
107. 𝑣 = (𝑟 − 1)(𝑐 − 1)
108. 𝜇1 − 𝜇2 : 𝑥̅1 − 𝑥̅2 ± 𝑧𝛼⁄2 √(𝜎12 ⁄𝑛1 ) + (𝜎22 ⁄𝑛2 )
2
138.
109. 𝑛 = [(𝑧𝛼⁄2 )⁄𝐸 ] (𝜎12 + 𝜎22 )
𝑏
110. 𝜇1 − 𝜇2 ≤ 𝑥̅1 − 𝑥̅2 + 𝑧𝛼 √(𝜎12 ⁄𝑛1 ) + (𝜎22 ⁄𝑛2 )
111. 𝑥̅1 − 𝑥̅2 − 𝑧𝛼 √(𝜎12 ⁄𝑛1 ) + (𝜎22 ⁄𝑛2 ) ≤ 𝜇1 − 𝜇2
112. 𝑇0 =
(𝑋̅1 −𝑋̅2 )−Δ0
116.
𝑇0∗
=
𝑎
√(𝑆12 ⁄𝑛1 )+(𝑆22 ⁄𝑛2 )
2
[(𝑠12 ⁄𝑛1 ) ⁄(𝑛1 −1)]+[(𝑠22 ⁄𝑛2 ) ⁄(𝑛2 −1)]
118. 𝜇1 − 𝜇2 : 𝑥̅1 − 𝑥̅2 ± 𝑡𝛼⁄2,𝑣 √(𝑠12 ⁄𝑛1 ) + (𝑠22 ⁄𝑛2 )
̅ −Δ0
∑ 𝑥−∑ 𝑦
𝐷
119. 𝑇0 =
120. 𝑑̅ =
𝑠𝐷 ⁄√𝑛
𝑛
𝑛(∑ 𝑥 2 −2 ∑ 𝑥𝑦+∑ 𝑦 2 )−(∑ 𝑥−∑ 𝑦)2
122. 𝜇𝐷 : 𝑑̅ ± 𝑡𝛼⁄2,𝑛−1 𝑠𝐷 ⁄√𝑛
123. 𝑓1−𝛼,𝑢,𝑣 = 1⁄𝑓𝛼,𝑣,𝑢
124. 𝐹0 = 𝑠12 ⁄𝑠22
125. (𝑠12 ⁄𝑠22 )𝑓1−𝛼⁄2,𝑛2 −1,𝑛1 −1 ≤ 𝜎12 ⁄𝜎22 ≤
(𝑠12 ⁄𝑠22 )𝑓𝛼⁄2,𝑛2 −1,𝑛1 −1
148. 𝛽̂0 =
126. 𝑍0 =
153. 𝑟 =
𝑃̂1 −𝑃̂2
1
1
√𝑃̂(1−𝑃̂)(𝑛 +𝑛 )
1
2
128. 𝑝̂2 = 𝑋2 ⁄𝑛2
𝑝̂1 (1−𝑝̂1 )
𝑝̂ (1−𝑝̂2 )
+ 2
𝑛1
𝑛2
129. 𝑝1 − 𝑝2 : 𝑝̂1 − 𝑝̂2 ± 𝑧𝛼⁄2 √
𝑋 +𝑋
130. 𝑃̂ = 1 2
𝑛1 +𝑛2
𝑎
𝑛
𝑎
𝑦⋅⋅2
2
𝑆𝑆𝑇 = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅⋅⋅ ) = ∑ ∑ 𝑦𝑖𝑗
−
𝑁
𝑖=1 𝑗=1
𝑖=1 𝑗=1
𝑎
𝑎
𝑆𝑆Treatments = 𝑛 ∑(𝑦̅𝑖⋅ − 𝑦̅⋅⋅
𝑖=1
133.
)2
𝑦𝑖⋅2 𝑦⋅⋅2
=∑ −
𝑛
𝑁
𝑖=1
𝑎
𝑛
2
𝑆𝑆𝐸 = ∑ ∑(𝑦𝑖𝑗 − 𝑦̅𝑖⋅ )
𝑖=1 𝑗=1
134. 𝑆𝑆𝑇 = 𝑆𝑆Treatments + 𝑆𝑆𝐸
135.
𝑆𝑆Treatments⁄(𝑎 − 1) 𝑀𝑆Treatments
𝐹0 =
=
𝑆𝑆𝐸 ⁄[𝑎(𝑛 − 1)]
𝑀𝑆𝐸
136.
𝑎
𝑏
𝑛
2
𝑆𝑆𝑇 = ∑ ∑ ∑ 𝑦𝑖𝑗𝑘
−
𝑖=1 𝑗=1 𝑘=1
137.
𝑎
2
𝑦𝑖⋅⋅
𝑦⋅⋅⋅2
𝑆𝑆𝐴 = ∑
−
𝑏𝑛 𝑎𝑏𝑛
𝑖=1
𝑦⋅⋅⋅2
𝑎𝑏𝑛
𝑛
2
𝑛 ∑𝑛
𝑖=1 𝑥𝑖 −(∑𝑖=1 𝑥𝑖 )
(𝑥
)
(𝑦
∑𝑛
̅)
−𝑥̅
−𝑦
𝑖
𝑖=1 𝑖
2
∑𝑛
𝑖=1(𝑥𝑖 −𝑥̅ )
𝑛
̂
∑𝑛
∑
𝑦
−𝛽
1 𝑖=1 𝑥𝑖
𝑖=1 𝑖
𝑛
𝑆𝑥𝑦
√𝑆𝑥𝑥 𝑆𝑦𝑦
147. 𝛽̂1 = 𝑆𝑥𝑦 ⁄𝑆𝑥𝑥
149. 𝛽̂0 = 𝑦̅ − 𝛽̂1 𝑥̅
∑ 𝑑𝑥𝑑𝑦
154. 𝑟 = (𝑠𝑑𝑥)(𝑠𝑑𝑦)(𝑛−1)
155. 𝑑𝑥 = 𝑥 − 𝑥̅
156. 𝑑𝑦 = 𝑦 − 𝑦̅
2
157. 𝑒𝑖 = 𝑦𝑖 − 𝑦̂𝑖 158. 𝑆𝑆𝐸 = ∑𝑛𝑖=1 𝑒𝑖2 = ∑𝑛𝑖=1(𝑦𝑖 − 𝑦̂𝑖 )
159. 𝑆𝑆𝐸 = 𝑛𝜎𝑦2 − 𝐵𝑛𝑟𝜎𝑥 𝜎𝑦 160. 𝑆𝑆𝐸 = 𝑆𝑆𝑇 − 𝛽̂1 𝑆𝑥𝑦
161. 𝜎 2 = 𝑆𝑆𝐸 ⁄(𝑛 − 2)
̂ −𝛽
𝛽
162. 𝑇0 = 1 1,0
163. 𝑠𝑒(𝛽̂1 ) = √𝜎̂ 2 ⁄𝑆𝑥𝑥
2
132.
𝑦⋅⋅⋅2
− 𝑆𝑆𝐴 − 𝑆𝑆𝐵
𝑎𝑏𝑛
150. 𝑆𝑥𝑥 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )2
151. 𝑆𝑦𝑦 = ∑𝑛𝑖=1(𝑦𝑖 − 𝑦̅)2 = 𝑆𝑆𝑇
152. 𝑆𝑥𝑦 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅)
164. 𝑇0 =
𝑛
𝑛
−
140. 𝑆𝑆𝐸 = 𝑆𝑆𝑇 − 𝑆𝑆𝐴𝐵 − 𝑆𝑆𝐴 − 𝑆𝑆𝐵
141. 𝐹0 = 𝑀𝑆𝐴 ⁄𝑀𝑆𝐸 142. 𝐹0 = 𝑀𝑆𝐵 ⁄𝑀𝑆𝐸
143. 𝐹0 = 𝑀𝑆𝐴𝐵 ⁄𝑀𝑆𝐸 144. 𝑦̂ = 𝛽̂0 + 𝛽̂1 𝑥
𝑛 ∑𝑛 𝑥 𝑦 −(∑𝑛 𝑥 )(∑𝑛 𝑦𝑖 )
145. 𝛽̂1 = 𝑖=1 𝑖 𝑖 𝑖=1 𝑖 𝑖=1
2
146. 𝛽̂1 =
𝑛(𝑛−1)
Course Outcome 3
131.
2
𝑦𝑖𝑗⋅
𝑖=1 𝑗=1
(𝑠12 ⁄𝑛1 +𝑠22 ⁄𝑛2 )
127. 𝑝̂1 = 𝑋1 ⁄𝑛1
𝑏
𝑆𝑆𝐴𝐵 = ∑ ∑
2
121. 𝑠𝐷2 =
𝑎𝑛
𝑦⋅⋅⋅2
𝑎𝑏𝑛
139.
(𝑋̅1 −𝑋̅2 )−Δ0
2
117. 𝑣 =
𝑗=1
−
113. 𝑣 = 𝑛1 + 𝑛2 − 2
𝑆𝑝 √(1⁄𝑛1 )+(1⁄𝑛2 )
2
𝑆𝑝 = [(𝑛1 − 1)𝑆12 + (𝑛2
− 1)𝑆22 ]⁄[𝑛1 + 𝑛2 − 2]
115. 𝜇1 − 𝜇2 : 𝑥̅1 − 𝑥̅2 ± 𝑡𝛼⁄2,𝑛1 +𝑛2 −2 𝑆𝑝 √(1⁄𝑛1 ) + (1⁄𝑛2 )
114.
𝑆𝑆𝐵 = ∑
2
𝑦⋅𝑗⋅
166. 𝑍0 =
̂ 2 ⁄𝑆𝑥𝑥
√𝜎
̂0 −𝛽0,0
𝛽
1 𝑥−2
√𝜎
̂ 2[ +
]
𝑛 𝑆𝑥𝑥
+
𝑅 −0.5𝑛
0.5√𝑛
1
𝑥 −2
165. 𝑠𝑒(𝛽̂0 ) = √𝜎̂ 2 [𝑛 + 𝑆 ]
𝑥𝑥
167. 𝑍0 =
𝑊 + −𝑛(𝑛+1)⁄4
√𝑛(𝑛+1)(2𝑛+1)⁄24