/8/009 6.3 Oe a Tw Samle Iferece fr Mea If i kw a 95% Cfiece Iterval i 96 ±.96 96.96 ± But i ever kw. If i ukw Etimate by amle taar eviati The etimate taar errr f the mea will be / Uig the etimate taar errr we have a cfiece iterval f ± ( ) The multilier ee t be bigger tha Z (e.g.,.96 fr 95% CI). The cfiece iterval ee t be wier t take it accut the ae ucertaity i uig t etimate. The crrect multilier were figure ut by a Guie Brewery wrker. What i the crrect multilier? t 00( α)% cfiece iterval whe i ukw ± t ( / ) 95% CI 00( 0.05)% cfiece iterval whe i ukw ± t ( / ) 0.975 Prertie f t itributi The value f t ee hw much ifrmati we have abut. The amut f ifrmati we have abut ee the amle ize. The ifrmati i egree f freem a fr a amle frm e rmal ulati thi will be f. t curve a z curve Quatile f t itributi t table i give i the bk Table B.4 Bth the taar rmal curve N(0,) (the z itributi), a all t(v) itributi are eity curve, ymmetric abut a mea f 0, but t itributi have mre rbability i the tail. A the amle ize icreae, thi ecreae a the t itributi mre clely arimate the z itributi. By 000 they are virtually iitiguihable frm e ather. It ee the egree f freem a well f rbability t 5 0.90.476 0 0.95.8 0 0.99.58 5 0.975.060 0.975.96
/8/009 Cfiece iterval fr the mea whe i ukw t < µ < t Eamle Nie level, 74.0 78.6 76.8 75.5 73.8 75.6 77.3 75.8 73.9 70. 8.0 73.9. Pit etimate fr the average ie level f vacuum cleaer;. 95% Cfiece iterval Sluti, 75.53. 75 Critical value (0.975 t quatile) with f t 0975.975.0 95% CI.75 75.53±.0 75.53±.75 73.78 < µ < 77.8 Eamle 8 (age 366) Failure time f 0 rig. The rmal lt lk fairly traight. (If t, try trafrmig r a ifferet itributi, e.g. Weibull) 68.3 33. 33. 0.47 0 90% CI 68.3 ±.833*0.47 68.3 ± 9. 49. t 87.5 If we were t tet H µ 50 v Ha µ 50, we wul t reject H0, ice 50 i i the cfiece iterval fr µ. T tet H µ # Aalgu t the large amle tet with z tet tatitic # z We wul have T # Determiati f reject / t reject H a well a -value are fu ue T-table with ν f We cul the tet uig 33. 0.47 0 68.3 50 8.3 t.74 0.47 0.47.383 <.74 <.833 Q(0.9) < tet tatitic < Q(.95) 0.05 < Pt ( >.74) < 0. value P( t >.74)* 0.< value < 0. But the cfiece iterval i mre ifrmative.
/8/009 O the ther ha Ather meth Rejecti Regi t 0.95,9.833, t 0.05,9.83 If we have a tet tatitic value that i either t mall (<.83) r t big (>.83), the we have trg eviece agait H 0. t.74 which i t t mall r t big (cmare t the cutff value abve/ critical value ), the we cat reject the ull. Alterative Hythee Rejecti Regi µ> µ 0 µ< µ 0 µ µ 0 z>z -α ----------------- t>t -α z<z α --------------- t<t α z>z -α/ r z<z α/ ------------------ t>t -α/ r t<t α/ Rejecti Regi meth a value meth Paire Data Fr Ha µ<µ 0, if z tet tatitic i le tha.645, the the value i le tha 0.05. Cmarig the value t 0.05 i the ame a cmarig the z value t.645. Fr t tet we ca al fi me critical value crreig t level f α that we ca cmare t ur tet tatitic. Tet tatitic i the rejecti regi i the ame a value i le tha α. 98 tuy f trace metal i Suth Iia River. 6 ram lcati 3 Tt water zic ccetrati (mg/l) Bbttm water zic (mg/l) 3 4 5 6 T 0.45 0.38 0.390 0.40 0.605 0.609 Bttm 0.430 0.66 0.567 0.53 0.707 0.76 4 5 6 6.3. Paire Mea Differece T cmare T & Bttm Water Zic frm a River Lcati Bttm T 0.430 0.45 0.66 0.38 3 0.567 0.390 4 0.53 040 0.40 5 0.707 0.605 6 0.76 0.609 That i equivalet t ak i it true that µ ifferece>0? Thi i a ecial cae f the mea f a igle clum f umber. Create a ew clum fr the ifferece betwee variable. T & Bttm Water Zic frm a River Lcati Bttm T Differece 0.430 0.45 0.05 0.66 0.38 0.08 3 0.567 0.390 0.77 4 0.53 0.40 0. 5 0.707 0.605 0.0 6 0.76 0.609 0.07 0.09 0.06
/8/009 Check rmality Orere i 0.05 0.08 0.0 0.07 0. 0.77 Z Quatile -.38-0.67-0. 0. 0.67.38 al Staar Nrma Quatile.5 0.5 - -.5 - Zic i River 0-0.5 0 0.05 0. 0.5 0. Zic Serie 6 i value f 6 5 95% Cfiece Iterval t.57 0.09 0.06 0.05 0.09 ±.57(0.05) 0.095 ± 0.064064 0.08 t 0.56 Nte Eve rmal lt frm ram rmal ata are t erfectly traight By the uual hythei tetig erective, the -value fr H µ 0 v Ha µ 0i le tha 0.05, ice µ 0 i t i a 95% cfiece iterval. Our reult wul be tatitically igificat eviece agait H µ 0. 0.09 0.06 0.05 0.09 0 t 3.68 0.05 t >.57 *0.005 < < *0.0 0.0< < 0.0 I hythei tetig te that the -value mut be le tha α t claim tatitical igificace. A tet i igificat at the α level if the H value i t i the 00(-α )% cfiece iterval. What abut α 0.0 level? -value > 0.0 D t reject H 0 0.09 ± 4.03(0.05) 0.09 ± 0.00-0.008 t 0.9 0.09 ± 0.00 Nt tatitically igificat at α 0.0level Aumti The ulati f ifferece fllw a rmal itributi. A rmal lt f ifferece,, hul be fairly traight. Nte We t ee B r T t be rmal. Rejecti regi eercie Tell whe t reject H 0 μ 0 uig a t tet. Awer wul be f the frm reject H 0 whe t <.746 r maybe reject H 0 whe t >.746 r maybe reject H 0 whe t >.746 (a) H A μ < 0, α 0.05, 0 (b) H A μ > 0, α 0.0, 8 (c) H A μ 0, α 0.0, 9
/8/009 Fi value eercie Awer wul be f the frm 0.0 < < 0.05 < 0.00 > 0.8 After fiig the -value i each cae, tell whether t reject r t reject H 0 at the α 0.0505 level. (a) H A µ > 0, 7, t -.58 6.3.3 Large Samle Cmari f Tw Mea Glue µ, Glue µ, Bth ulati rmal ieeet value fr glue Nt aire, blcke, ieeet value fr glue (b) H A µ < 0, 7, t -.58 (c) H A µ 0, 7, t -.58 i ur gue at µ µ µ? Hw much might eviate frm Var( ) Var( ) ( ) Var( ) µ Eerimet 0.. -..4 0.6 0.8 3. 0.4.8 Mea µ µ µ µ Variace A cfiece iterval fr µ µ i give by z z ± ± ( µ µ ) z A fr hythei tetig But we ever really kw. 6.3.4 Small Samle Cmari f Tw Mea Cae Bth a are etimatr f. The cfiece iterval ee t be wiee t accut fr aitial ucertaity i a a etimatr f a. Cae Aume equal variace. Cae D t aume variace are ecearily equal. But they may be. Pl a it a le, cmbie etimate f. weighte average f ( ) ( ( ) ( ) ) a, weight by f.
/8/009 ± t ± t ± t Fr the table value f t, ue f ( -) ( -) T tet H µ µ #, check if # i cfiece iterval r ue T ( ) # A cmare t T-table with f ( -) ( -). Cae D t aume Lifetime f Srig. Table 6.7 a Figure 6.5 t t ± r ± f T 4 4 ( ) ( ) ( ) # Nte With, ly f chage N rmal Quatile Srig Figure 6.5.000.500.000 0.500 0.000-0.50000 50 00 50 300 -.000 -.500 -.000 Lifetime 900 Stre 950 Stre 6 6 5 7 53 6 98 89 5 6 89 35 306 5 6 35 43 89 7 6 900 Stre 950 Stre 0 0 5.5 68.3 4.9 33. 3.57 0.47 Nte Uually lifetime are mre lgrmal tha rmal. T fllw the bk eamle, carry i time cale. Cae Aume f 9 9 8 9(4.9) 9(33.) 9 9 38.3 Var 4.9 33. 0 0 ( ) 468 7. 5.-68.3 ±.0(7.) 468 46.8 ± 36.0 0.8 t 8.8 Bae the cfiece iterval, we reject H µ µ 0 v H µ µ 0 at α 0.05 level.
/8/009 Alteratively, t tet H µ µ 0 v H µ µ 0 5. 68.6 t.7 f 9 9 8 7. 0.005005 < < 0.0 0 0.0 < < 0.0 T tet H µ µ 0 v H µ µ 0, 0.005 < < 0.0. > T tet H µ µ 0 v H µ 0 µ <, 0.99 < < 0.995. Cae Nt aumig f 6.9 7 7. Nte With, ly f chage 46.8 ±.0(7.) Wier CI