Confidence Intervals Unknown σ

Transcription

1 Confidence Intervals Unknown σ Estimate σ Student s t-distribution Step-by-step instructions Example

2 Confidence Intervals - Known σ Standard normal distribution <latexit sha1_base64="/tkpftmrocazqq/vv7mmnjvhcuo=">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</latexit> 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sha1_base64="kifo9xyea+siw4kyh0kb9qn6xy=">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</latexit> zp and zp is p Z N0,1 p = P ( Z zp : zpp) = P ( zp Z zp ) <latexit 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sha1_base64="d1gw3hlduizg61+hpfpe/sxjeq=">aaad1hicbvjntxqxgc6sh7h+gr69tfxiqddkz5aobxiijnrqubpwihqlnc67swdtul0bsw9ga+eveo/8of4b3xnqnwfmszmo0+f5/1ojrsl63t+te03rly9dn3mrvpmrdt37s7o3ftq6phl0onaarutshkkunbzwknymrzykurytvc3q/vtzblodwwozlql1iuxebw5hd6tn5dpp54tcvex/bm0t5sq7pcqu900yhpjdbg400beuk6e3ptvmm+aga5bhkzbkbj8b1pbnocamhsuclgmb3wq67acpwqnn3ddohwogqyqkbtvgof9xwumszoiypirsn34k5yswpvlefallounedz30vlbk5ulyizojxkzctia+zzdxhh5puwqhxrcfurrvmgtqn+x4bmhclvhuhaku+dethgidyps1uxancu8icp6xddcxitx0cop/bias/hpckjhuqawp4ulutujv4whvqs61kxs9x4hygxt5ylrqjx/hcc7+mq/nqbhaaxe1h8mkoaf/wbfomjkbaypxlwchy+enkt4ixpyjzbblzejtcrrmpm1+ia0f8b7f3gqsfszewlqikrhbxhytx3vr5i5qicf//yefbjstjonittzhu1xebnlyd6x13pihetjzqko95dxkk01ooptak+hi+ew7pekf8moi1xk5q45ph5lb5o9jlltexou7i18zqcnbnygdwkiyqmt8kgeuw6pec4seqh+ul+nbybofg18eeoj11qrlpjk7j+1/uhkgf</latexit> = (zp ) ( zp ) = (zp ) <latexit 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sha1_base64="3rgaiv87txfto4m4yvvnviibcle=">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</latexit> (zp )) = (zp ) CDF of standard normal (zp ) = 1+p z0.9 = <latexit 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sha1_base64="qf8ygcitfgsfi4ejeruv5djojj8=">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</latexit> sha1_base64="hfkuubfoaa1n4y/ds3z4neolw=">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</latexit> zp = 1 1+p = ( ) <latexit sha1_base64="w9cyeb/gnggtmxyzftv5knnvgeu=">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</latexit> sha1_base64="klkbr568h8n6ocjdgvkr+e+z8q=">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</latexit> P ( Z 1.645) = 0.9 <latexit sha1_base64="he3ehp77sdsexjzfuh+0edsfjr4=">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</latexit> sha1_base64="aqjxssalsgwcs8pj7dpkpledbq=">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</latexit> <latexit sha1_base64="tyk6ifz/e9jia5dpmasdavhtg0=">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</latexit> sha1_base64="6m1rkd130ktovl8xi8m0lyq3i4=">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</latexit> 1 (0.95) = <latexit 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sha1_base64="53jlbfzvqoajrxr7jknmjm+i8=">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</latexit> <latexit sha1_base64="ny/qmtygt8wzq/qt+kuwsccswug=">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</latexit> sha1_base64="kdzjrmzekaamidtl+qencirio=">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</latexit> Sample Mean Normal X 1,X,...,X n i.i.d. known σ unknown μ X = X X n n Sample mean µ X = µ Unbiased V (X) = mean 0 n X = p n X µ X = X µ / p n N 0,1 std 1 CLT

4 <latexit sha1_base64="3jfpzdhhh0uk/1qnlx/mvrskys=">aaaecnicbvllbtqwfhu7pmrwamhjjmjaqabhsfiqkbjsvzbgwaogdq1ujwbhuzoj6tjb8dbwlj+b/clwcgrnkcf8ks67sumbawktwcn3pfssnyyoth76npxpmz587pxghevht5ytxzuwtvkzxshlpccaw3elabycv0tw4ebjuawjeiexhvv7zq+gq1zjdbnfqq9gmcwhowcgof7sgh1oxi1vspi+gq07tbg5s6s8k9hdwr3xxkrnmjsgqtiifzswldmrvhr9g7yj159thzwpsfjizo0wqsp3z93hjd1/tz0z/tfrwviwwwrqu0olk3pmm1ylgbjjsoogd9hgwyjkvkbvc/wxbpgiuaodqzk4ynnumpjesukqtovkjz+cagxuk/gqevqcbdm8gdns1lotiguaemoem6x0wcpmtyg4pda1iju1wvbzmpxwca5hbus1tawczavusozrohueumehkmlqzpnmathdt31aggoslbvamtwhgnh3kg40mzwukjeblbrsivmypl6bma8fko+/beuelhdaylkbw/zt3lp6rdnpqzubbmo/jrngdpwhx6jhzixo0rt6ajgcds9q8qoxzps+jdkqgryhzfbxzlrwu665mn7/vanjdzeipc3wrsila8mlq+9tfzgietj75umas3g81i7vxe14edmdrs00gpzhxqqru9jgjeeodxdxknh1egptovbgi+gwthfkpwko49wkji55dhyltxrdulpscv9hrdxn5odmkbvkjlkkeblpvskzsk66hjnp5af5sx41pja+nl4vh1qp6condfjxgl8/wv0j190</latexit> sha1_base64="vthvlmrzwk9cqx7mi9a+gklqtq4=">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</latexit> <latexit sha1_base64="3mwz0benshmj6xslpawvqkxujhe=">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</latexit> sha1_base64="ov9+rhnjtwfnnindswrjhd6mrro=">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</latexit> Confidence Interval p Standard normal Z N 0,1 P ( Z apple z p )=p X µ / p n N 0,1 P X µ / p apple z n p p X µ apple z p pn P p Margin of error With probability p X µ apple z p p n µ hx z p pn, X + z p pn i

5 <latexit sha1_base64="fqino7hjzzick1b3arre6lfpog=">aaaeihicbvllbtqwfhu7pmrwamhjjmjaqddrkempoitkfsdbakrbhrqphoc5e7gqmmhx6gtlh8nx8apsgefwmkalxwdn5lsztpasnx9fc65d9+4elw0vv9jbr516fkvqwvxtdv3lx1e3hpzvtsvtqbfqke0lhmshbcqt9wiyaqnla8fraxhzyr7/c+gi65krvmuibbzjljrzxhbl3dxwglggbdabhquivkb1okomhiqxuwjgxnkducsyryv7ylgdtqlc0julfw3skwwjrzm0isyysdqu7fs9vlnfece6mzry/jd1cym5waf4ztvvs5sbnilhz7gdhyqawacmtaa5nqxiklhywdpbrlcyhcmcbfjhvxunx6ouxk8ar3fpuyzly/i4j7u458yma0q9e7vtixjw3oyedcyxrwvajju05szrhdpxkjhdeg3te0lhcyqz5lm6q7tipedgaucrmbvdgjhnc/g5ogbwffmkg01cmzvkato0tjgbhqenbq/hzfzdsshfe3kzqof4e/udojoagldqipwghvuxyuo5ofgjzhxtn67nyz+kcaxtjckwojaxqkjsay+o+cioazfaclwuo54onoeimn+hjrumhmk0sibfkzk3ntgt16nor0/0/onpmujedfpqihgfteeonfeuyhfbc6+e/hutbc74apwm64seeugmyaqp7drvui3ewip8zodxdhknbn8ztcwb1nr8lhuocqxuq4idcwjjkwdmrpm/0w95mz38bdlzfkclaipfifbjkavkybjoxzif0suk+kf/kn/nt+tt6vrw+j6bzs+dco6smdx6+re+bmdl</latexit> sha1_base64="lwudtsqsoim8tqvk9w78migpegu=">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</latexit> Confidence Interval Given confidence p samples X1,, Xn Determine Critical value (z) z p = 1 1+p norm.ppf((1+p)/) Sample mean X = X X n n Margin of error z p p n h Confidence interval X σ known z p p n, X + z p p n i Problem? σ almost never known

6 Unknown σ X 1, X,..., X n <latexit sha1_base64="7tefmgvudon/stbdgziw+/enzvi=">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</latexit> sha1_base64="faoopkd4+08giw180xbveswxy4=">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</latexit> X= <latexit 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sha1_base64="fqino7hjzzick1b3arre6lfpog=">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</latexit> X Xn n X µ <latexit 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sha1_base64="yxhkmkizyku81nl4wzs8grozxs=">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</latexit> S = <latexit sha1_base64="crugjruotonjhmyey+lvo7obg8=">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</latexit> sha1_base64="ahhxsgazxsucg79kmjba9kqcbkc=">aaaewhicbvldthqxfj6fvwd9a7zui4nlgpcf7aws5ykeqaleaerlzro6tdozz3cbou3ydgts9b18fr/h+ac+gdeemuvcfprm9psc7/toe86jc8606xr+1qam67duz8zone7cvxf/wfzc4hctc5van5fcql5mnxamoguy4ddlfdas5naqh78u4wffqgkmxb45y6gf0afga5zqg65o/nurfwxtxhscu+l0x4veurrqvni4nryxli3pmbfendsmnfgi0hulzp6vw0yudsxqlm61lhfdscan4/ccsiwyi1wrbcz4sxjrkewbqfuspjlvyittuithixrfloz3qmwf90izo3mzq738o1fy1m/scptiomhek41fowchptt1qzlnbwdvjoyglytidwikaggei+rurn/japrtqfsiwfmh7lnqrymml9lsvt3dnqriwl3p3skbm+lg8gl/uwibwwijismjkakiy38zmrxvoy7ewdcdhjzjzrkzi9qkacbn1loazmsm1iprqbjsyku4rjqyevlmblfksgdzzogzdqwo4bskmnlpbdqfoeizsyaf/yr8znenlygduamvpbpdk55ijteoewaflc/oxcl/owesyzu0kjhkj6ccjih/zolselmcvf5kvqfsjol3ypiapmqkeoa3lrwxxiph9waqldxxof0t9t+96nqiibye+eujeyvvw7gp4rykxlwazz/tvniddfa4fowhw5uupugqwug/me3oojdaionxe5wf0cqdlv7km4styczybnuiaxvipznqxgnhplg6khfn7rh+tz65pq3hnnjdes98h74i17gffc/heente10u8p7xhtvbt6fsveq0+u58bq6dq55yh3qvvx/wlrqjfa==</latexit> X pµ S/ n <latexit 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sha1_base64="dwtefycm0r7lgt/pz08kjudchyg=">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</latexit> = X X pµ / n <latexit 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sha1_base64="iogetfbvm7zsjasezoepxlbvkcq=">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</latexit> 1 n 1 n P i=1 (Xi?? <latexit sha1_base64="wfx6aeslouyxvb3j+0efzw0ag=">aaaemxicbvpdttrafo7iqrj+gv5qtooyysilboseusasnieyjsagmzdlztqedhumm3u6fchkln0g73wnln0gfqhpuwvsapps9vsb7zv/9viwz6rd/lozufw9fefu5l3a/qcphzemn7ylro59ghpf0zijkczydghprurbp1uak08bvve0fvifv8hycwwffedptbnamtjmpapqqg3ddbo9jxwp+ecaueytdknrlzaflap65qjd0e4fkqcsx1nrhorihld+zjkhu985pk36xssjlzjbmyawsxykdfnztzdq1xrl+gyyjbq5fso07a/sfd5lghviyycgrcssrlsqbaduifahrselc6jnx7yz3ls9vdauj33dcizg/d3z358vfm//+pnt/wigfdzbkv1gcya8dnvvdtqwkfgamrpiou+kc0ggm0ou0g6+qy98zuagfdigk/riys3huommszaej18jnqlw/kbrpuwafy8bdq/btv8c8zrvwv6agmfvljjnyfp9inqqhwsmcjnbdelboc0htlctbqfq6rpjiiyjvpo1y7w44kjcqrzdnygggidcdcqwtrsckb1aaixexyesnefybfu1f1nigjktmq01tmfognicgssu0mui3djrss/3clnuiku+kr0k1igsselobinm0tbpowrd4ddkfazfv8qo0riv5itymhhgkdlzvrkcyxbonr/k4tj9zhccppdfiksra13ubzx5yxhcjd668a60a67hlfuc13acncri0kad/nlraru9jctced7s6ulgfk8zrat5ymrqqx38xvqxstybivvnrwyzrk33dhmxvhba0597zm1ojpwm+ul1bqca9laszatlwvp8ivllw4lretv59x16mb144a6urlqnlpjp7rzh8m3kqy=</latexit> <latexit 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sha1_base64="btttyocwkrpj0ibw++wbms3gbsw=">aaad3xicbvllbtqwfhu7pmrwakgsermkxuyvzoucrpaqigslhgu1kfi9whwndszq44dq5tzxnlcrflzrzwfa5/w01aykxbs0lujs+577suwrpe78/mbovcxuux5660r167fupm/mktd1zxhkofa6nnx5rzkejb3wkn4wnpgbwphj10b7o+3/kmxgqttt1hcyoc5uqmbgcooeh8hu85k9gbmps0qlruirxgytjf6a30mhodnejjo7pxcnnehjct4clsb5ppxhwghjfmt04kd3am+melxdatljqmr7hcthfu7ec7ma3+ydokuioi0ba4knc1mctes8kaw+ltivfgrlxlam/uq1qaafdu9gtgreqrbwoxlmzwbgrmenex8ww7rahbapgn+uiycqjw8yao0hnkoyuwxfiqm1ih+7+gkj3klddtekdxtyikn1megbqsndg6czaugn8akiomfsqdevcltjtxj8fywen0hw+suaxay0aojkqly9x8ip+kf+hcelky74k7xuzijzsf/cciazlackwtxzwghy+a1kt4ixpwdzbblzejtcqbmpkxej+0lyb7f3ayqtfszel6ikrhbxh7jx3vrlizqxbv3/xlpgkwe0mj5jusrywzqpmbkd9467klveru3nnewurlqamve0lqftebcyz0jym8thmdrfg1c8vj0sp81+snk+krvxdzzeewozhy5s+6rzrktxsdvcrbpe848eqh+ul+tt61vrs+tr4duwdnjjw3ydrpff8lisxl8a==</latexit> Nµ, Neither σ nor μ known µx = µ Sample mean <latexit sha1_base64="lraxzk0ob+n+7gpe9v557ljf5a=">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</latexit> sha1_base64="zbfyz+g3kn4p/5ug4l60tsfwlog=">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</latexit> μ=? Unbiased X = <latexit 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sha1_base64="ny/qmtygt8wzq/qt+kuwsccswug=">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</latexit> N0,1 X) Standard normal Sample variance µs = <latexit 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Almost standard S <latexit sha1_base64="qrvgkpl6iwvkqiombkf5ha8ufpw=">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</latexit> sha1_base64="ws1i8g/0vrcsbcymykubbbilqbs=">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</latexit> Student s t-distribution Bessel corrected Unbiased normal S <latexit sha1_base64="a0ugbhkbgyykvtpa3pte5ijg+m8=">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</latexit> sha1_base64="5udg6+u99wcarg9h4gkp0/b5a4=">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</latexit> r.v. n-1 degrees of freedom p n

7 <latexit sha1_base64="dp0g/izexqqfeivjdoridrpebhk=">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</latexit> sha1_base64="rqcbcr5n4/k0gbpbrqkuwoaeui=">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</latexit> <latexit sha1_base64="ag6ul3xqxkyl3cbye7vmyngox9q=">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</latexit> <latexit sha1_base64="vxsfxqznellt51s6weq/nuvtqhy=">aaaec3icbvjbtxnbfn5cvaw30ecvvcafcnyxsridem9jpigezuchinn7o5pufzpdfchkfoppvlz/cm++eczbcewmgr3znzn+87muqqpizpvbp4utuxr1/mxozcuvnbv3zufu7yilyg0q8ge3a9obizm0faxyrcfsqbjwgavohpj/xvfqwax4dvqnivoqvs87suhvqh1z3/wdrqe55ukjmoiucudexul5mkn/rlm5j9k0pzyyofsvijwzy4pwvx1ajbqz1jtis0prowpmlz/ilrg/leczpbegyctvorbxjzjtk4npbgorvow/qyq718kq005tnleaxxivg97iqlyep/nxde3vy+3u3nqveokwt4crkneso/d8vhu0lsoogzgkytniaxhe+3cajqcjzb1dvnw4nrehyo0jir9xbggpszrnsuw0crq4j1qnrmturgufyybdq96ljo55mivgoavgmq1ljc9xwwhfpbwqui4hiccc8sjyvkovi4mdhqqrqueitlud9simedffs4kwoiergletgekxycbaalgrokiesbotewnqzpepiiu8md3ql7v18qmulalo6lesoccickrso4nubznyy7cfx4do51kosburzyumlak/onjt+hlupcpdfuwsdkspqdiijucfkuxysycpjg/sjnkqu4njh+y/u/oe/uacnm0lybjliwnxzjwq8lr8bymprz1mujfx+14t/3g/7amrmkpca/iy7kturgm1ljtexrypwbttkbsbobm/czt1hab/lwb0x6go4jb7f0f6sthvzzwmp+8auu9m1wzzknn3qk7nrputjx3zrbtdslsxgm91cq9mv5tflsely8mqvolkeabm7hky/8ae56caa==</latexit> <latexit sha1_base64="vxsfxqznellt51s6weq/nuvtqhy=">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</latexit> <latexit sha1_base64="ocuwnrdwmgcgoohrdqskghsuo4o=">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</latexit> sha1_base64="m+byywzkgr0jv8imkmdrmj91lo8=">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</latexit> <latexit sha1_base64="qb5fugk8ranpj/h4u9/hrrwlbjc=">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</latexit> sha1_base64="l/1vu+djti3jv9ichsd9tia8hww=">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</latexit> Student s t-distribution T = X µ S/ p +1 Student s t-distribution,! degrees of freedom Gamma function Only dependence on t PDF T f (t) = ( +1 p ) <latexit sha1_base64="m+byywzkgr0jv8imkmdrmj91lo8=">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</latexit> ( ) 1+ t +1 Symmetric around 0 See a bit more

8 t object in scipy.stats module Degrees of freedom probability density function t.pdf(x,!) f 3 (1) from scipy.stats import t t.pdf(1,3) f1 Bell shaped Similar to Gaussian

9 <latexit sha1_base64="y8vjegojzqq1h4qm0o5ru7fsk58=">aaaev3icbvnnu9nage6xksyv0koxjkvjkyjnkfeozddodb78qkxsgbz0tumbdifnbtxsbgznt171x/bzvhnwp3j1tvqgbxemzztnn+f9tj/llnot1q/kzlxq9rs3z+5t+/cvxd/bv7bl0zmkorkllunt7ngdmbbc00h06qgcz9dnv9w1ff/d43ubmtylefpnbnacxyxekqeern/a13en6z0wuaha+kztaubp0wmyau8e9ot/6+z0isn0ng4orat77biylfijgiosesvrfbvc4ykqz5/ixkx5usfny7rkbxwys8alcbuhfz6f6bdpktdg+kxubivwnpsgkbmkxmwmiygjiexqbcl4syoxpggsxtgdszzlzbprr7atxivaknti0orn1vorrfj4vw1/bnqn/w5/e04zk5vfuaudgsyj9isknms/edvhcnvzqfhkxl8gxsgh7sgpbrfdsbrgvkcvmv7ie08ckp8ce0v8ktekotldtj+k18jlqpc0nx9apq8mzavy5edg0taa5bhav1wgioggbihuokxwscuusehiusuyygo17t+11does6ywq+jyrfq71op3gsfljbqzzcdyfbmrwngsjghmg41myakbvma6dm0jxymtaxwtwabcakq04tc1cas5ykhycekgsm+ldkvxlxbpkfrjdaeu/asug9zwau5ew8xsugnh6jxgcbs8q8uhxkiw6ilmkzlyubrxwtnvsh7hak7//h8ukok8ilmfuuicvdbs/vtxnr5zly8l7y5ogwg0gz4nmslzm/0enfya4466klvere3bnewurltflumptx15pbmpvn5ilwxx/wtjekxcxsx3l6/0vamdrkyvtd76tc3zujmoo+cx07d8z0xzqbzxtlxk5y6va+v35ufla3qsoqqkyj6kxlrhnotj3qyt8cpp4h</latexit> sha1_base64="iij/4/ozx/r9r3w/+7fytx/b+by=">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</latexit> Dependence on! As! increases f (t)! (t) standard normal distribution t-distribution!=!=5!=10!=30 Logical S constant σ Examples in notebook

10 <latexit sha1_base64="qb5fugk8ranpj/h4u9/hrrwlbjc=">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</latexit> sha1_base64="l/1vu+djti3jv9ichsd9tia8hww=">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</latexit> Analytical Argument f (t) = ( +1 ) p ( ) 1+ t +1 For small x 1+x e x As! 1+ t +1 e t +1 e t +1 e t Standard Normal Distribution Constant same ( +1 ) p ( ) 1 p

11 William Sealy Gosset Guinness Brewery World s largest Billion pints / year ~1.75 Billion bottles Quality Consistency Statisticians Trade secret Publish Pseudonym

12 <latexit sha1_base64="ghqpht+yw7sy7ms7ziem9av6yqi=">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</latexit> sha1_base64="ece553ncvdzkwibx+olun3czug=">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</latexit> <latexit sha1_base64="iee/a64ehjvkzrc3sfxrjrjts9u=">aaad0hicbvjbtxqxfc6sf1xvoi++tfyiydzkz5aodyzetfrbiyorrlohnc7zyzoo+l0bni0xleffdx/wm/xzqd/ikcd4i7qpnmzx7/vxhpowkprv7v19r069llk1dnrrwv37h56/bs3jpla4nhz7xupvtlfughyk+fvbcdmmafamerxtvebjf+gymelptsmsbgxllrgkzixcn4a7jqral9ql3dlob7nxroi8ez8ynfuhf45+e0iduemjimevaslyyqtqjk9iohdnwcam+teskssb3wa47acpwqdvwtco+wogqyqkhnrivjrp4xojyuvwhrdrfsbfcthjakoztxp3g6fdjxqzw1b8udnbongycyrhzhdumu3afmf+1wxbvmz3wag1e48cftc0c66tuypefnilvkjnsivnhcie7ijkhlhk4szgjcowjiwlooh9+6pryqavkgmsscpuzenctrueddnpkvbj6a854webuutqu/ht/jv6pd+toabqfbhs/jznglfwhx6jhziwymrh6adgca9q8ryxzrv5inlqirypvd7uzizr+769mp7+b6edgiffblouikqw1obj1ty75ikvnxj3/uuz75jkpzs8srrj6qq/ijobapwpu9jd7loxnye7/ro4uqlvtnou30whgmbdyegv0hweq9rthhi47mjfd7oj8try713cwf9ntlem+qeuu8wsuwek3xyimyqpufeke/kb/nz+ta6bh1pftmtk+dao6sidx69hfm80io</latexit> sha1_base64="lfsiy08algt5snx3svpbapcqnwm=">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</latexit> Déjà vu T Student s t-distribution,! degrees of freedom Critical value, t f (t) Probabiliity p t p, : P ( T apple t p, )=p p = P ( T apple t p, ) = P ( t p, apple T apple t p, ) =F (t p, ) 1 F (t p, )= 1+p CDF of T! t p, = F 1 1+p

13 Degrees of freedom Cumulative distribution function t.cdf(x,!) F 3 (1) from scipy.stats import t t.cdf(1,3) Inverse cdf percent point function t.ppf(x,!) F 1 3 (0.95) t.ppf(0.95,3)

14 Confidence Interval t-distribution,! degrees of freedom Probabiliity p t-statistic X µ S/ p n f n 1(t) P X µ S/ p apple t n p,n 1 = p X µ apple t p,n 1 S pn P = p margin of error With probability p X µ apple t p,n 1 S pn hx t p,n 1 S pn, X + t p,n 1 S pn i µ

15 <latexit sha1_base64="fqino7hjzzick1b3arre6lfpog=">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</latexit> sha1_base64="lwudtsqsoim8tqvk9w78migpegu=">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</latexit> Confidence Interval Given confidence p samples X1,, Xn Determine critical t t p,n 1 = F 1 n 1 1+p t.ppf((1+p)/) Sample mean X = X X n n Sample variance S = 1 n 1 P n i=1 X i X Margin of error t p,n 1 S pn σ unnecessary hx t p,n 1 S pn, X + t p,n 1 S pn i Confidence interval

16 ature African Elephant Trunk Length

17 Mature African Elephant Trunk Length 8 measurements Find 5.6, 6.07, 6.64, 5.91, 6.30, 6.55, 6.19, 5.48 feet 95% confidence interval for distribution mean Critical t tp,n 1 = 1 Fn 1 Sample mean 1+p 1 = F7 (0.975) = t.ppf(0.975,7).3646 X = Sample variance S Margin of error S tp,n 1 pn Confidence interval X S = S tp,n 1 pn, X (5.7497, ) + S tp,n 1 pn