2014年度春学期　統計学　第１３回　不確かな測定の不確かさを測る ― 不偏分散とt分布 (2014. 7. 10)

1. A.Asano,KansaiUniv. 2014 t

2. A.Asano,KansaiUniv. 3. A.Asano,KansaiUniv. 4. 2014 A.Asano,KansaiUniv. 5. 2014 A.Asano,KansaiUniv. 95% 6. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 7. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% X 8. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% X 9. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 10. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 11. 2014 A.Asano,KansaiUniv. 12. 2014 A.Asano,KansaiUniv. 13. 2014 A.Asano,KansaiUniv. X 14. 2014 A.Asano,KansaiUniv. X 15. 2014 A.Asano,KansaiUniv. X 16. 2014 A.Asano,KansaiUniv. X X 17. 2014 A.Asano,KansaiUniv. X X 18. 2014 A.Asano,KansaiUniv. X X X 19. 2014 A.Asano,KansaiUniv. X X X 20. 2014 A.Asano,KansaiUniv. X X X X 21. 2014 A.Asano,KansaiUniv. X X X X 22. 2014 A.Asano,KansaiUniv. X X X X 23. 2014 A.Asano,KansaiUniv. X X X X 24. 2014 A.Asano,KansaiUniv. X X X X 25. 2014 A.Asano,KansaiUniv. X X X X 26. 2014 A.Asano,KansaiUniv. X X X X 27. 2014 A.Asano,KansaiUniv. X X X X 28. 2014 A.Asano,KansaiUniv. X X X X 29. 2014 A.Asano,KansaiUniv. X X X X 30. 2014 A.Asano,KansaiUniv. X X X X 31. 2014 A.Asano,KansaiUniv. X X X X 32. 2014 A.Asano,KansaiUniv. X X X X 33. 2014 A.Asano,KansaiUniv. X X X X 34. 2014 A.Asano,KansaiUniv. X X X X 35. 2014 A.Asano,KansaiUniv. X X X X 36. 2014 A.Asano,KansaiUniv. X X X X 37. 2014 A.Asano,KansaiUniv. X X X X 38. 2014 A.Asano,KansaiUniv. X X X X 39. 2014 A.Asano,KansaiUniv. 95% X X X X 40. 2014 A.Asano,KansaiUniv. X X X X 95% 41. 2014 A.Asano,KansaiUniv. X X X X 95% 95% 42. 2014 A.Asano,KansaiUniv. X X X X 95% 95% 95% 43. A.Asano,KansaiUniv. 44. A.Asano,KansaiUniv. 45. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% X 46. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 47. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 48. 2014 A.Asano,KansaiUniv. 49. 2014 A.Asano,KansaiUniv. 50. 2014 A.Asano,KansaiUniv. 51. 2014 A.Asano,KansaiUniv. ()2 52. 2014 A.Asano,KansaiUniv. ()2 53. 2014 A.Asano,KansaiUniv. ()2 54. 2014 A.Asano,KansaiUniv. ()2 55. 2014 A.Asano,KansaiUniv. ()2 X1, ... Xn 56. 2014 A.Asano,KansaiUniv. ()2 X1, ... Xn 57. 2014 A.Asano,KansaiUniv. ()2 X1, ... Xn 58. 2014 A.Asano,KansaiUniv. ()2 X1, ... Xn X 59. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , X s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 n 1 1 60. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , X s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 n 1 1 61. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , X s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 n 1 1 62. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , X s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 n 1 1 ()2 63. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , X s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 n 1 1 ()2 64. 2014 A.Asano,KansaiUniv. n=2 X1, X2 65. 2014 A.Asano,KansaiUniv. n=2 X1, X2 66. 2014 A.Asano,KansaiUniv. n=2 X1, X2 X1 67. 2014 A.Asano,KansaiUniv. n=2 X1, X2 X1 X2 68. 2014 A.Asano,KansaiUniv. n=2 X1, X2 X1 X2 69. 2014 A.Asano,KansaiUniv. n=2 X1, X2 X1 X2 70. 2014 A.Asano,KansaiUniv. n=2 X1, X2 X1 X2 71. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 72. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 73. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 74. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 75. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X1 X2 76. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 77. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 78. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X 79. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X 80. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X 81. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X X1 82. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X X2X1 83. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X X2X1 84. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X X2X1 X 85. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X X2X1 X 86. 2014 A.Asano,KansaiUniv. n=2 X1, X2 XX1 X2 X1 X2 X X2X1 X 87. 2014 A.Asano,KansaiUniv. 88. 2014 A.Asano,KansaiUniv. 89. 2014 A.Asano,KansaiUniv. X 90. 2014 A.Asano,KansaiUniv. X 91. 2014 A.Asano,KansaiUniv. X 92. 2014 A.Asano,KansaiUniv. X X 93. 2014 A.Asano,KansaiUniv. X X 94. 2014 A.Asano,KansaiUniv. 95. 2014 A.Asano,KansaiUniv. 96. 2014 A.Asano,KansaiUniv. 97. 2014 A.Asano,KansaiUniv. 98. 2014 A.Asano,KansaiUniv. 99. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , Xn s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 1 100. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , Xn s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 1 101. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , Xn s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 1 - 1) 102. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , Xn s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 1 - 1) 103. 2014 A.Asano,KansaiUniv. n X1, X2, . . . , Xn s2 s2 = 1 n 1 (X1 X)2 + (X2 X)2 + + (Xn X)2 S2 = 1 n (X1 X)2 + (X2 X)2 + + (Xn X)2 n n 1 1 - 1) 104. 2014 A.Asano,KansaiUniv. 105. 2014 A.Asano,KansaiUniv. 106. 2014 A.Asano,KansaiUniv. 107. 2014 A.Asano,KansaiUniv. 108. 2014 A.Asano,KansaiUniv. 109. A.Asano,KansaiUniv. 110. A.Asano,KansaiUniv. 111. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% X 112. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 113. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 114. 2014 A.Asano,KansaiUniv. 115. 2014 A.Asano,KansaiUniv. 116. 2014 A.Asano,KansaiUniv. N(, 2) 117. 2014 A.Asano,KansaiUniv. N(, 2) 1/n 118. 2014 A.Asano,KansaiUniv. N(, 2) 1/n 119. 2014 A.Asano,KansaiUniv. N(, 2) 1/n 120. 2014 A.Asano,KansaiUniv. N(, 2) 1/n 121. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) 122. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X 123. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X 124. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X N(, 2) X1, ..., Xn Z = X 2/n Z 125. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X N(, 2) X1, ..., Xn Z = X 2/n Z 126. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X N(, 2) X1, ..., Xn Z = X 2/n Z 127. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X N(, 2) X1, ..., Xn Z = X 2/n Z 128. 2014 A.Asano,KansaiUniv. N(, 2) 1/n N(, 2/n) X N(, 2) X1, ..., Xn Z = X 2/n Z N(0, 1) 129. 2014 A.Asano,KansaiUniv. 130. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) 131. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) 132. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) 133. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) 134. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) Z = X 2/n (3) 2 2 t = X s2/n 135. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) Z = X 2/n (3) 2 2 t = X s2/n 136. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) Z = X 2/n (3) 2 2 t = X s2/n 137. 2014 A.Asano,KansaiUniv. Z = X 2/n Z Z 3(a) N(0, 1) Z = X 2/n (3) 2 2 t = X s2/n 138. 2014 A.Asano,KansaiUniv. t 2 t = X s2/n t Z n 1 t(n 1) t t 139. 2014 A.Asano,KansaiUniv. t 2 t = X s2/n t Z n 1 t(n 1) t t 140. 2014 A.Asano,KansaiUniv. t 2 t = X s2/n t Z n 1 t(n 1) t (n-1)t t 141. 2014 A.Asano,KansaiUniv. t t(n-1) 2 t = X s2/n t Z n 1 t(n 1) t (n-1)t t 142. 2014 A.Asano,KansaiUniv. t t(n-1) 2 t = X s2/n t Z n 1 t(n 1) t (n-1)t t t 143. 2014 A.Asano,KansaiUniv. t t(n-1) 2 t = X s2/n t Z n 1 t(n 1) t (n-1)t t t 144. 2014 A.Asano,KansaiUniv. t t(n-1) 2 t = X s2/n t Z n 1 t(n 1) t (n-1)t t t 145. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% X 146. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 147. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X 148. 2014 A.Asano,KansaiUniv. X1, ... Xn n 95% 2 X s2 149. 2014 A.Asano,KansaiUniv. t 0 t (n-1)t (3) 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 t(n-1) 150. 2014 A.Asano,KansaiUniv. t 0 t (n-1)t t(n-1) (3) 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 t(n-1) 151. 2014 A.Asano,KansaiUniv. t 0 t (n-1)t t(n-1) (3) 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 t(n-1) 152. 2014 A.Asano,KansaiUniv. t 0 t =95% (n-1)t t(n-1) (3) 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 2 n Z = X 2/n ( (0, 1) Z (3) 2 2 s2 t = X s2/n ( t Z t t(n-1) 153. 2014 A.Asano,KansaiUniv. t 0 t =95% (n-1)t t(n-1) =95% (3) 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 2 n Z = X 2/n ( (0, 1) Z (3) 2 2 s2 t = X s2/n ( t Z t t(n-1) 154. 2014 A.Asano,KansaiUniv. t 0 t =95% (n-1)t t(n-1) =95% (3) 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 2 n Z = X 2/n ( (0, 1) Z (3) 2 2 s2 t = X s2/n ( t Z t t(n-1) 155. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% 156. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% 157. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% 158. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% =2.5% 5% 159. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% =2.5% 5% 160. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% =2.5% 5% t0.025(n-1) 161. 2014 A.Asano,KansaiUniv. t 0 t 0 t =95% =2.5% 5% t0.025(n-1)2.5% 162. 2014 A.Asano,KansaiUniv. t 0 t t(n-1) 163. 2014 A.Asano,KansaiUniv. t 0 t t(n-1) 164. 2014 A.Asano,KansaiUniv. t 0 t =95% 2 n Z = X 2/n N(0, 1) (3) 2 2 s2 t = X s2/n t t Z n 1 t t(n 1) t t(n-1) 165. 2014 A.Asano,KansaiUniv. t 0 t =95% =95% 2 n Z = X 2/n N(0, 1) (3) 2 2 s2 t = X s2/n t t Z n 1 t t(n 1) t t(n-1) 166. 2014 A.Asano,KansaiUniv. t 0 t =95% =95% 2 n Z = X 2/n N(0, 1) (3) 2 2 s2 t = X s2/n t t Z n 1 t t(n 1) t t(n-1) 167. 2014 A.Asano,KansaiUniv. t 0 t =95% =95% 2 n Z = X 2/n N(0, 1) (3) 2 2 s2 t = X s2/n t t Z n 1 t t(n 1) t t0.025(n-1) t(n-1) 168. 2014 A.Asano,KansaiUniv. t 0 t =95% =95% 2 n Z = X 2/n N(0, 1) (3) 2 2 s2 t = X s2/n t t Z n 1 t t(n 1) t t0.025(n-1)-t0.025(n-1) t(n-1) 169. 2014 A.Asano,KansaiUniv. t -t0.025(n-1) t0.025(n-1) 95% (3) 2 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 170. 2014 A.Asano,KansaiUniv. t -t0.025(n-1) t0.025(n-1) 95% (3) 2 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 171. 2014 A.Asano,KansaiUniv. t -t0.025(n-1) t0.025(n-1) 95% (3) 2 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 t 0 t0.025(n 1)t0.025(n 1) t95% 2: t P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 172. 2014 A.Asano,KansaiUniv. t -t0.025(n-1) t0.025(n-1) 95% (3) 2 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 t 0 t0.025(n 1)t0.025(n 1) t95% 2: t P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 173. 2014 A.Asano,KansaiUniv. t -t0.025(n-1) t0.025(n-1) 95% (3) 2 2 s2 t = X s2/n (4) Z t n 1 t n 1) t 10 50 10 t 0 t0.025(n 1)t0.025(n 1) t95% 2: t P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) t 0 t0.025 (n 1)t0.025 (n 1) 0.025P(t t0.025(n 1)) = 0.025 t95% 2: t P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 174. 2014 A.Asano,KansaiUniv. t P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 175. 2014 A.Asano,KansaiUniv. t 95% P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 176. 2014 A.Asano,KansaiUniv. t 95% 95% P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 177. 2014 A.Asano,KansaiUniv. t 95% 95% P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 178. 2014 A.Asano,KansaiUniv. t 95% 95% =50 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 179. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 180. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 =10 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 181. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 =10 2.5%t0.025(n-1) P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 182. 2014 A.Asano,KansaiUniv. t 0 t =2.5% t0.025(n-1) 183. 2014 A.Asano,KansaiUniv. t 0 t =2.5% t0.025(n-1) 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.7062 31.8205 63.6567 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 t 100 t ( ) 184. 2014 A.Asano,KansaiUniv. t 0 t =2.5% t0.025(n-1) 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.7062 31.8205 63.6567 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 t 100 t ( ) 185. 2014 A.Asano,KansaiUniv. t 0 t =2.5% t0.025(n-1) 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.7062 31.8205 63.6567 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 t 100 t ( ) 186. 2014 A.Asano,KansaiUniv. t 0 t =2.5% n-1 = 9 t0.025(n-1) 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.7062 31.8205 63.6567 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 t 100 t ( ) 187. 2014 A.Asano,KansaiUniv. t 0 t =2.5% n-1 = 9 0.025 t0.025(n-1) 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.7062 31.8205 63.6567 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 t 100 t ( ) 188. 2014 A.Asano,KansaiUniv. t 0 t =2.5% n-1 = 9 0.025 t0.025(9)=2.262 t0.025(n-1) 0.40 0.30 0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.7062 31.8205 63.6567 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 t 100 t ( ) 189. 2014 A.Asano,KansaiUniv. t P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 190. 2014 A.Asano,KansaiUniv. t 95% P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 191. 2014 A.Asano,KansaiUniv. t 95% 95% P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 192. 2014 A.Asano,KansaiUniv. t 95% 95% P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 193. 2014 A.Asano,KansaiUniv. t 95% 95% =50 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 194. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 195. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 =10 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 196. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 =10 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 197. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 =10 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 t0.025(10-1)=2.262 198. 2014 A.Asano,KansaiUniv. t 95% 95% =50 =25 =10 P t0.025(n 1) X s2/n t0.025(n 1) = 0.95 2 P X t0.025(n 1) s2 n X + t0.025(n 1) s2 n = 0.95 95% (6) 100 t t() X = 50 s2 = 2 1) = 2.262 95%46.4 53.6 25 95%46.9 t0.025(10-1)=2.262 46.453.6 [46.4, 53.6] 199. 2014 A.Asano,KansaiUniv. =50 =10 200. 2014 A.Asano,KansaiUniv. =50 =10 =25 201. 2014 A.Asano,KansaiUniv. =50 =10 =25 95% [46.9, 53.1] 202. 2014 A.Asano,KansaiUniv. =50 =25 =10 =25 95% [46.9, 53.1] 203. 2014 A.Asano,KansaiUniv. =50 =25 =10 =25 95% [46.9, 53.1] [46.4, 53.6] 204. 2014 A.Asano,KansaiUniv. =50 =25 =10 =25 95% [46.9, 53.1] [46.4, 53.6] 205. 2014 A.Asano,KansaiUniv. =50 =25 =10 =25 95% [46.9, 53.1] [46.4, 53.6]

Education

2014年度春学期 統計学 第１３回 不確かな測定の不確かさを測る ― 不偏分散とt分布 (2014. 7. 10)

2014年度春学期　統計学　第１３回　不確かな測定の不確かさを測る ― 不偏分散とt分布 (2014. 7. 10)