1. Administrative Matters Midterm II Results Take max of two midterm scores:

2. Administrative Matters Midterm II Results Take max of two midterm scores Approx grades: 92-100A 82-92B 70-82C 60-70 D 0 – 60F

3. Last Time Confidence Intervals –For proportions (Binomial) Choice of sample size –For Normal Mean –For proportions (Binomial) Interpretation of Confidence Intervals

4. Reading In Textbook Approximate Reading for Today’s Material: Pages 493-501, 422-435, 447-467 Approximate Reading for Next Class: Pages 422-435, 372-390

5. Sample Size for Proportions i.e. find so that: Now solve to get: (good candidate for list of formulas)

6. Sample Size for Proportions i.e. find so that: Now solve to get: Problem: don’t know

7. Sample Size for Proportions Solution 1: Best Guess Use from: –Earlier Study –Previous Experience –Prior Idea

8. Sample Size for Proportions Solution 2: Conservative Recall So “safe” to use:

9. Interpretation of Conf. Intervals Mathematically: pic 1 pic 2 3 rd interpretation

10. Interpretation of Conf. Intervals Frequentist View: If repeat the experiment many times

11. Interpretation of Conf. Intervals Frequentist View: If repeat the experiment many times, About 95% of the time, CI’s will contain μ

12. Interpretation of Conf. Intervals Frequentist View: If repeat the experiment many times, About 95% of the time, CI’s will contain μ (and 5% of the time it won’t)

13. Interpretation of Conf. Intervals Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals

14. Interpretation of Conf. Intervals

23. Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals Shows proper interpretation

24. Interpretation of Conf. Intervals Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals Shows proper interpretation: –If repeat drawing the sample

25. Interpretation of Conf. Intervals Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals Shows proper interpretation: –If repeat drawing the sample –Interval will cover truth 95% of time

26. Interpretation of Conf. Intervals Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals Lower Confidence Level (95%  80%)

27. Interpretation of Conf. Intervals Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals Lower Confidence Level (95%  80%): –Shorter confidence intervals

28. Interpretation of Conf. Intervals Nice Illustration: Publisher’s Website Statistical Applets Confidence Intervals Lower Confidence Level (95%  80%): –Shorter confidence intervals –Leads to lower hit rate

29. Interpretation of Conf. Intervals Recall Class HW: Estimate % of Male Students at UNC

30. Interpretation of Conf. Intervals Recall Class HW: Estimate % of Male Students at UNC Revisit Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls

31. Interpretation of Conf. Intervals Estimate % of Male Students at UNC

32. Interpretation of Conf. Intervals Recall Class HW: Estimate % of Male Students at UNC Recall: Q1: Sample of 25 from Class

33. Interpretation of Conf. Intervals Recall Class HW: Estimate % of Male Students at UNC Recall: Q1: Sample of 25 from Class Q2: Sample of 25 from any doorway

34. Interpretation of Conf. Intervals Recall Class HW: Estimate % of Male Students at UNC Recall: Q1: Sample of 25 from Class Q2: Sample of 25 from any doorway Q3: Sample of 25 think of names

35. Interpretation of Conf. Intervals Recall Class HW: Estimate % of Male Students at UNC Recall: Q1: Sample of 25 from Class Q2: Sample of 25 from any doorway Q3: Sample of 25 think of names Q4: Random sample (from phone book)

36. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q1: Sample from Class

37. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q1: Sample from Class: - Compare to theoretical

38. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q1: Sample from Class: - Compare to theoretical - Some bias

39. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q1: Sample from Class: - Compare to theoretical - Some bias - less variation

40. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q2: From Doorways

41. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q2: From Doorways: - No bias

42. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q2: From Doorways: - No bias - More variation

43. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q3: Think up names

44. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q3: Think up names: - Upwards bias

45. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q3: Think up names: - Upwards bias - More variation

46. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q4: Random Sample

47. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q4: Random Sample: - Looks better?

48. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q4: Random Sample: - Looks better? - Reasonable variation?

49. Interpretation of Conf. Intervals Histogram analysis: Class Example 7 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg7.xls Q4: Random Sample: - Looks better? - Reasonable variation? - Really need CIs etc.

50. Interpretation of Conf. Intervals Now consider C.I. View: Class Example 13 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls

51. Interpretation of Conf. Intervals Now consider C.I. View: Class Example 13 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls Explore idea: CI should cover 90% of time

52. Interpretation of Conf. Intervals Class Example 13

53. Interpretation of Conf. Intervals Class Example 13

54. Interpretation of Conf. Intervals Class Example 13

55. Interpretation of Conf. Intervals Class Example 13

56. Interpretation of Conf. Intervals Class Example 13

57. Interpretation of Conf. Intervals Class Example 13 Q1: Summarize Coverage

58. Interpretation of Conf. Intervals Class Example 13 Q1: Summarize Coverage 94% > 90% (since sd too small)

59. Interpretation of Conf. Intervals Class Example 13 Q2: Summarize Coverage 77% < 90% (since too variable)

60. Interpretation of Conf. Intervals Class Example 13 Q3: Summarize Coverage 77% < 90% (since too biased)

61. Interpretation of Conf. Intervals Class Example 13 Q4: Summarize Coverage 87% ≈ 90% (seems OK?)

62. Interpretation of Conf. Intervals Class Example 13 Simulate from Binomial 87% ≈ 90% (shows within expected range)

63. Interpretation of Conf. Intervals Class Example 13: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg13.xls Q1: SD too small  Too many cover Q2: SD too big  Too few cover Q3: Big Bias  Too few cover Q4: Good sampling  About right Q5: Simulated Bi  Shows “natural var’n”

64. Interpretation of Conf. Intervals HW: 6.20 ($1260, $1540), 6.21 6.28 (but use Excel & make histogram)

65. Research Corner Another SiZer analysis: British Incomes Data

66. Research Corner Another SiZer analysis: British Incomes Data o Annual Survey (1985)

67. Research Corner Another SiZer analysis: British Incomes Data o Annual Survey (1985) o Done in Great Britain

68. Research Corner Another SiZer analysis: British Incomes Data o Annual Survey (1985) o Done in Great Britain o Variable of Interest: Family Income

69. Research Corner Another SiZer analysis: British Incomes Data o Annual Survey (1985) o Done in Great Britain o Variable of Interest: Family Income o Distribution?

70. Research Corner British Incomes Data SiZer Results:  1 bump at coarse scale (expected)

71. Research Corner British Incomes Data SiZer Results:  1 bump at coarse scale  2 bumps at medium scale

72. Research Corner British Incomes Data SiZer Results:  1 bump at coarse scale  2 bumps at medium scale (Quite a radical statement)

73. Research Corner British Incomes Data SiZer Results:  1 bump at coarse scale  2 bumps at medium scale  Finer scale bumps not statistically significant

74. Research Corner British Incomes Data  2 bumps at medium scale  Usual models for Incomes (one bump only)

75. Research Corner British Incomes Data  2 bumps at medium scale  Usual models for Incomes (one bump only)  2 bumps were verified

76. Research Corner British Incomes Data  2 bumps at medium scale  Usual models for Incomes (one bump only)  2 bumps were verified (in PhD dissertation)

77. Research Corner British Incomes Data  2 bumps at medium scale  Usual models for Incomes (one bump only)  2 bumps were verified (in PhD dissertation)  But when worth looking?

78. Next time Add multiple year plots as well In: IncomesAllKDE.mpg

79. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests

80. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution

81. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution Usually σ is unknown

82. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution Usually σ is unknown, so replace with an estimate, s.

83. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution Usually σ is unknown, so replace with an estimate, s. For n large, should be “OK”

84. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution Usually σ is unknown, so replace with an estimate, s. For n large, should be “OK”, but what about: n small?

85. Deeper look at Inference Recall: “inference” = CIs and Hypo Tests Main Issue: In sampling distribution Usually σ is unknown, so replace with an estimate, s. For n large, should be “OK”, but what about: n small? How large is n “large”?

86. Unknown SD Goal: Account for “extra variability in the s ≈ σ approximation”

87. Unknown SD Goal: Account for “extra variability in the s ≈ σ approximation” Mathematics: Assume individual

88. Unknown SD Goal: Account for “extra variability in the s ≈ σ approximation” Mathematics: Assume individual I.e. Data have mound shaped histogram

89. Unknown SD Goal: Account for “extra variability in the s ≈ σ approximation” Mathematics: Assume individual I.e. Data have mound shaped histogram Recall averages generally normal

90. Unknown SD Goal: Account for “extra variability in the s ≈ σ approximation” Mathematics: Assume individual I.e. Data have mound shaped histogram Recall averages generally normal But now must focus on individuals

91. Unknown SD Then

92. Unknown SD Then So can write:

93. Unknown SD Then So can write: (recall: standardization (Z-score) idea)

94. Unknown SD Then So can write: (recall: standardization (Z-score) idea, used in an important way here)

95. Unknown SD Then So can write: Replace

96. Unknown SD Then So can write: Replace by

97. Unknown SD Then So can write: Replace by, then

98. Unknown SD Then So can write: Replace by, then has a distribution named

99. Unknown SD Then So can write: Replace by, then has a distribution named: “t-distribution with n-1 degrees of freedom”

100. t - Distribution Notes: 1.n is a parameter

101. t - Distribution Notes: 1.n is a parameter (like )

102. t - Distribution Notes: 1.n is a parameter (like ) (Recall: these index families of probability distributions)

103. t - Distribution Notes: 1.n is a parameter (like ) that controls “added variability that comes from the s ≈ σ approximation”

104. t - Distribution Notes: 1.n is a parameter (like ) that controls “added variability that comes from the s ≈ σ approximation” View: Study Densities, over degrees of freedom… http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/EgTDist.mpg

105. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7

106. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7

107. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 t is more spread

108. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 t is more spread: - Lower Peak

109. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 t is more spread: - Lower Peak - Fatter Tails

110. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 t is more spread smaller 5%-tile

111. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 t is more spread smaller 5%-tile larger 99%-tile

112. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 t is more spread Makes sense, since s ≈ σ  more variation

113. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 3 All effects are magnified Since s ≈ σ approx gets worse

114. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 1 Extreme Case Have terrible s ≈ σ approx

115. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 7 Now try larger d.f.

116. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 14 All approximations are better

117. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 25 Even better

118. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 25 Even better - Densities almost on top

119. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 25 Even better - Densities almost on top - Quantiles very close

120. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 100 Hard to see any difference

121. t - Distribution Compare N(0,1) distribution, to t-distribution, d.f. = 100 Hard to see any difference Since excellent s ≈ σ approx

122. t - Distribution Notes: 2.Careful: set “degrees of freedom” = = n – 1

123. t - Distribution Notes: 2.Careful: set “degrees of freedom” = = n – 1 (not n)

124. t - Distribution Notes: 2.Careful: set “degrees of freedom” = = n – 1 (not n) Easy to forget later

125. t - Distribution Notes: 2.Careful: set “degrees of freedom” = = n – 1 (not n) Easy to forget later Good to add to sheet of notes for exam

126. t - Distribution Notes: 3.Must work with standardized version of

127. t - Distribution Notes: 3.Must work with standardized version of i.e.

128. t - Distribution Notes: 3.Must work with standardized version of i.e. (will affect how we compute probs….)

129. t - Distribution Notes: 3.Must work with standardized version of i.e. No longer can plug mean and SD into EXCEL formulas

130. t - Distribution Notes: 3.Must work with standardized version of i.e. No longer can plug mean and SD into EXCEL formulas In text standardization was already done

131. t - Distribution Notes: 3.Must work with standardized version of i.e. No longer can plug mean and SD into EXCEL formulas In text standardization was already done, since used in Normal table calc’ns

132. t - Distribution Notes: 4.Calculate t probs (e.g. areas & cutoffs),

133. t - Distribution Notes: 4.Calculate t probs (e.g. areas & cutoffs), using TDIST

134. t - Distribution Notes: 4.Calculate t probs (e.g. areas & cutoffs), using TDIST & TINV

135. t - Distribution Notes: 4.Calculate t probs (e.g. areas & cutoffs), using TDIST & TINV Caution: these are set up differently from NORMDIST & NORMINV

136. t - Distribution Notes: 4.Calculate t probs (e.g. areas & cutoffs), using TDIST & TINV Caution: these are set up differently from NORMDIST & NORMINV See Class Example 14 http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg14.xls

137. t - Distribution Class Example 14:

138. t - Distribution Class Example 14: Calculate Upper Prob

139. t - Distribution Class Example 14: Calculate Upper Prob

140. t - Distribution Class Example 14: Calculate Upper Prob Using TDIST

141. t - Distribution Class Example 14: Calculate Upper Prob Using TDIST (Check TDIST menu)

142. t - Distribution Class Example 14: Calculate Upper Prob Using TDIST - cutoff

143. t - Distribution Class Example 14: Calculate Upper Prob Using TDIST - cutoff - d. f.

144. t - Distribution Class Example 14: Calculate Upper Prob Using TDIST - cutoff - d. f. - upper prob. only

145. t - Distribution Class Example 14: Careful: opposite from NORMDIST Using TDIST - cutoff - d. f. - upper prob. only

146. t - Distribution Class Example 14: Careful: opposite from NORMDIST use upper Using TDIST - cutoff - d. f. - upper prob. only

147. t - Distribution Class Example 14: Careful: opposite from NORMDIST use upper, NOT lower probs Using TDIST - cutoff - d. f. - upper prob. only

148. t - Distribution Class Example 14: To compute lower prob

149. t - Distribution Class Example 14: To compute lower prob Use “1 – trick”, i.e. Not Rule of probability

150. t - Distribution Class Example 14: How about upper prob of negative?

151. t - Distribution Class Example 14: How about upper prob of negative? Give it a try

152. t - Distribution Class Example 14: How about upper prob of negative? Give it a try Get an error message in response

153. t - Distribution Class Example 14: How about upper prob of negative? Give it a try Get an error message in response (Click this for sometimes useful info)

154. t - Distribution Class Example 14: Reason: TDIST tuned for 2-tailed

155. t - Distribution Class Example 14: Reason: TDIST tuned for 2-tailed (where need cutoff > 0)

156. t - Distribution Class Example 14: Reason: TDIST tuned for 2-tailed (where need cutoff > 0) (correct version for CIs and H. tests)

157. t - Distribution Class Example 14: Approach:

158. t - Distribution Class Example 14: Approach: Use “1 – trick”

159. t - Distribution Class Example 14: Approach: Use “1 – trick” (to write as prob. can compute)

160. t - Distribution Class Example 14: For Two-Tailed Prob

161. t - Distribution Class Example 14: For Two-Tailed Prob TDIST is very convenient

162. t - Distribution Class Example 14: For Two-Tailed Prob TDIST is very convenient (much better than NORMDIST)

163. t - Distribution Class Example 14: For Interior Prob

164. t - Distribution Class Example 14: For Interior Prob Use “1 – trick”

165. t - Distribution Class Example 14: For Interior Prob Use “1 – trick” TDIST again very convenient

166. t - Distribution Class Example 14: For Interior Prob Use “1 – trick” TDIST again very convenient (again better than NORMDIST)

167. t - Distribution Class Example 14: Now try increasing d.f.

168. t - Distribution Class Example 14: Now try increasing d.f. Big difference for small n

169. t - Distribution Class Example 14: Now try increasing d.f. Big difference for small n But converges for larger n

170. t - Distribution Class Example 14: Now try increasing d.f. Big difference for small n But converges for larger n To Normal(0,1)

171. t - Distribution Class Example 14: Now try increasing d.f. Big difference for small n But converges for larger n To Normal(0,1) (as expected)

172. t - Distribution HW: C23 For T ~ t, with degrees of freedom: (a) 3 (b) 12 (c) 150 (d) N(0,1) Find: i.P{T> 1.7} (0.094, 0.057, 0.046, 0.045) ii.P{T < 2.14} (0.939, 0.973, 0.983, 0.984) iii.P{T < -0.74} (0.256, 0.237, 0.230, 0.230) iv.P{T > -1.83} (0.918, 0.954, 0.965, 0.966)

173. t - Distribution HW: C23 v.P{|T| > 1.18} (0.323, 0.261, 0.240, 0.238) vi.P{|T| < 2.39} (0.903, 0.966, 0.982, 0.983) vii.P{|T| < -2.74} (0, 0, 0, 0)

174. And now for something completely different “Thinking Outside the Box” Also Called: “Lateral Thinking”

175. And now for something completely different Find the word or simple phrase suggested: death..... life

176. And now for something completely different Find the word or simple phrase suggested: death..... life life after death

177. And now for something completely different Find the word or simple phrase suggested: ecnalg

178. And now for something completely different Find the word or simple phrase suggested: ecnalg backward glance

179. And now for something completely different Find the word or simple phrase suggested: He's X himself

180. And now for something completely different Find the word or simple phrase suggested: He's X himself He's by himself

181. And now for something completely different Find the word or simple phrase suggested: THINK

182. And now for something completely different Find the word or simple phrase suggested: THINK think big ! !

183. And now for something completely different Find the word or simple phrase suggested: ababaaabbbbaaaabbbb ababaabbaaabbbb..

184. And now for something completely different Find the word or simple phrase suggested: ababaaabbbbaaaabbbb ababaabbaaabbbb.. long time no 'C'

185. t - Distribution Class Example 14: Next explore TINV (Inverse function)

186. t - Distribution Class Example 14: Next explore TINV (Inverse function) (Given cutoff, find area)

187. t - Distribution Class Example 14: Next explore TINV

188. t - Distribution Class Example 14: Next explore TINV Given prob. (area)

189. t - Distribution Class Example 14: Next explore TINV Given prob. (area) & d.f.

190. t - Distribution Class Example 14: Next explore TINV Given prob. (area) & d.f., find cutoff

191. t - Distribution Class Example 14: Next explore TINV Given prob. (area) & d.f., find cutoff (next think carefully about interpretation)

192. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above:

193. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Now invert this,

194. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Now invert this, i.e. given prob.

195. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Now invert this, i.e. given prob., find cutoff

196. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: For same d.f.

197. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: For same d.f., use resulting prob. as input

198. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: For same d.f., use resulting prob. as input But new answer is different

199. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Maybe due to rounding?

200. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Maybe due to rounding? Try exact value

201. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Maybe due to rounding? Try exact value

202. t - Distribution Class Example 14: Next explore TINV Recall TDIST e.g. from above: Maybe due to rounding? Try exact value Still get wrong answer

203. t - Distribution Class Example 14: Next explore TINV Reason for inconsistency:

204. t - Distribution Class Example 14: Next explore TINV Reason for inconsistency: Works via 2-tailed

205. t - Distribution Class Example 14: Next explore TINV Reason for inconsistency: Works via 2-tailed, not 1-tailed, probability

206. t - Distribution Class Example 14: Explore TINV Works via 2-tailed, not 1-tailed, probability

207. t - Distribution Class Example 14: Explore TINV Works via 2-tailed, not 1-tailed, probability Check by inverting 2-tailed answer above:

208. t - Distribution Class Example 14: Explore TINV Works via 2-tailed, not 1-tailed, probability Check by inverting 2-tailed answer above:

209. t - Distribution Class Example 14: Explore TINV Works via 2-tailed, not 1-tailed, probability Check by inverting 2-tailed answer above: Get:

210. t - Distribution Class Example 14: Explore TINV Works via 2-tailed, not 1-tailed, probability Check by inverting 2-tailed answer above: Get: plug in above output

211. t - Distribution Class Example 14: Explore TINV Works via 2-tailed, not 1-tailed, probability Check by inverting 2-tailed answer above: Get: plug in above output, to return to input

212. EXCEL Functions Summary: Normal:

213. EXCEL Functions Summary: Normal: plug in: get out:

214. EXCEL Functions Summary: Normal: plug in: get out: NORMDIST: cutoff

215. EXCEL Functions Summary: Normal: plug in: get out: NORMDIST: cutoff area

216. EXCEL Functions Summary: Normal: plug in: get out: NORMDIST: cutoff area NORMINV: area

217. EXCEL Functions Summary: Normal: plug in: get out: NORMDIST: cutoff area NORMINV: area cutoff

218. EXCEL Functions Summary: Normal: plug in: get out: NORMDIST: cutoff area NORMINV: area cutoff (but TDIST is set up really differently)

219. EXCEL Functions t distribution: 1 tail:

220. EXCEL Functions t distribution: 1 tail: plug in: get out:

221. EXCEL Functions t distribution: 1 tail: plug in: get out: TDIST: cutoff

222. EXCEL Functions t distribution: 1 tail: plug in: get out: TDIST: cutoff area

223. EXCEL Functions t distribution: 1 tail: plug in: get out: TDIST: cutoff area EXCEL notes: - no explicit inverse

224. EXCEL Functions t distribution: 1 tail: plug in: get out: TDIST: cutoff area EXCEL notes: - no explicit inverse - backwards from Normal…

225. EXCEL Functions t distribution: 2 tail:

226. EXCEL Functions t distribution: 2 tail: plug in: get out:

227. EXCEL Functions t distribution: 2 tail: plug in: get out: TDIST: cutoff

228. EXCEL Functions t distribution: Area 2 tail: plug in: get out: TDIST: cutoff area

229. EXCEL Functions t distribution: Area 2 tail: plug in: get out: TDIST: cutoff area TINV: area

230. EXCEL Functions t distribution: Area 2 tail: plug in: get out: TDIST: cutoff area TINV: area cutoff

231. EXCEL Functions t distribution: Area 2 tail: plug in: get out: TDIST: cutoff area TINV: area cutoff (EXCEL note: this one has the inverse)

232. EXCEL Functions Note: when need to invert the 1-tail TDIST, Use twice the area.

233. EXCEL Functions Note: when need to invert the 1-tail TDIST, Use twice the area. Area = A

234. EXCEL Functions Note: when need to invert the 1-tail TDIST, Use twice the area. Area = A Area = 2 A

235. t - Distribution HW: C23 (cont.) viii.C so that 0.05 = P{|T| > C} (3.18, 2.17, 1.98, 1.96) ix.C so that 0.99 = P{|T| < C} (5.84, 3.05, 2.61, 2.58)