x z Increasing the size of the sample increases the power (reduces the probability of a Type II error) when the significance level remains fixed.

Transcription

1 ] z-tet for the mea, μ If the P-value i a mall or maller tha a pecified value, the data are tatitically igificat at igificace level. Sigificace tet for the hypothei H 0: = 0 cocerig the ukow mea of a populatio are baed o the z tatitic x z 0 coditio: Radom data come from a SRS of the populatio. Normal populatio follow a ormal ditributio or becaue of the CLT thi rule ca be broke if () if <5, ca ue z if the data are roughly ormal (ad o outlier); () if 5 < 30, ca ue z eve if the data are lightly kewed (o outlier); (3) if 30, ca ue z eve if the data are trogly kewed (o outlier). Idepedet 0% rule i atified. H 0 true H a true Reject H 0 Type I Error Accept H 0 Type II Error The power of a igificace tet meaure it ability to detect a alterative hypothei that i, reject H 0 whe H 0 i fale. The power agait a pecific alterative i the probability that the tet will reject H 0 whe the alterative i true. I a fixed level igificace tet, the igificace level i the probability of a Type I error, ad the power agait a pecific alterative i miu the probability of a Type II error for that alterative. α = probability of Type I error; β = probability of Type II error; power = β. A α decreae, β icreae, ad power decreae. Icreaig the ize of the ample icreae the power (reduce the probability of a Type II error) whe the igificace level remai fixed. ] t-tet for the mea, μ I practice we do ot kow the populatio tadard deviatio, σ. I thi cae ubtitute x (the ample tadard deviatio) for the x 0 populatio tadard deviatio. I other word, ue the tatitic t. Thi tatitic ha a approximate t-ditributio with degree of freedom. coditio: Guidelie to uig t-procedure are Radom data mut come from a SRS of the populatio Normal Populatio mut follow a ormal ditributio. Careful here () if <5, ca ue t if the data are roughly ormal (ad o outlier); () if 5 < 30, ca ue t eve if the data are lightly kewed (o outlier); (3) if 30, ca ue t eve if the data are trogly kewed (o outlier). Idepedet ample ize mut be le tha 0% of the populatio ize or the data mut be collected i uch a way o that obervatio are idepedet of oe aother. 3] z-cofidece iterval for μ A level C cofidece iterval for the mea of a ormal populatio with kow tadard deviatio, baed o a SRS of ize, i give by * x z x Here z* i choe o that the tadard ormal curve ha area C betwee -z* ad z*. Becaue of the cetral limit theorem, thi iterval i approximately correct for large ample whe the populatio i ot ormal. The umber z* i called the upper critical value of the tadard ormal ditributio. You ca fid z* for C = 90% by uig z*=ivorm(.95); if C=95% the ue z*=ivorm(.975); i geeral ue z*=ivorm(c + (-C)/)=ivorm((+C)/) Other thig beig equal, the margi of error of a cofidece iterval get maller a

2 the cofidece level C decreae, the populatio tadard deviatio decreae, ad the ample ize icreae. The ample ize required to obtai a cofidece iterval with pecified margi of error m for a ormal mea i foud by ettig * z m ad olvig for, where z* i the critical value for the deired level of cofidece. Alway roud up whe you ue thi procedure. 4] t-cofidece iterval for μ I practice, we do ot kow. Replace the tadard deviatio tatitic of x by the tadard error to get the oe-ample t x t 0 The t tatitic ha the t ditributio with - degree of freedom. coditio: Radom data mut come from a SRS of the populatio Normal Populatio mut follow a ormal ditributio. Careful here () if <5, ca ue t if the data are roughly ormal (ad o outlier); () if 5 < 30, ca ue t eve if the data are lightly kewed (o outlier); (3) if 30, ca ue t eve if the data are trogly kewed (o outlier). Idepedet ample ize mut be le tha 0% of the populatio ize or the data mut be collected i uch a way o that obervatio are idepedet of oe aother. There i a t ditributio for every poitive degree of freedom k. All are ymmetric ditributio imilar i hape to the tadard ormal ditributio, The t(k) ditributio approache the N(0, ) ditributio a k icreae. x A exact level C cofidece iterval for the mea of a ormal populatio i * x t where the area betwee t* ad t* equal C. The value t* i called a critical value, pecifically it i the upper ( - C)/ critical value of the t( ) ditributio. Comparig Two Mea The data i a two-ample problem are two idepedet SRS, each draw from a eparate ormally ditributed populatio. Tet ad cofidece iterval for the differece betwee the mea ad of the two populatio tart from the differece x x of the two ample mea. Becaue of the cetral limit theorem, the reultig procedure are approximately correct for other populatio ditributio whe the ample ize are large (at leat 30 each). Draw idepedet SRS of ize ad from two ormal populatio with parameter,, ad,. The two-ample t tatitic i x x t The tatitic t doe ot exactly have a t ditributio. For coervative iferece procedure to compare ad, ue the two ample t tatitic with the t(k) ditributio. The degree of freedom k i the maller of ad. For more accurate probability value, ue the t(k) ditributio with degree of freedom k etimated from the data. Thi i the uual procedure i tatitical oftware or the TI83. I thi cae

3 but for implicity you ca alway ue df = mi(, ) 5] The cofidece iterval for - give by df = x x ha cofidece level at leat C if t* i the upper ( - C)/ critical value for t (k) with k the maller of ad. coditio: Guidelie to uig t-procedure are a before but replace with ad. Coditio mut hold for EACH ample. t *, 6] Sigificace tet for H0: = baed o x t x 0 have a true P-value o higher tha that calculated from t(k). The guidelie for practical ue of two-ample t procedure are imilar to thoe for oe-ample t procedure. Equal ample ize are recommeded. coditio: Guidelie to uig t-procedure are a before but replace with ad. Coditio mut hold for EACH ample. 7] -proportio z-iterval for p Tet ad cofidece iterval for a populatio proportio p whe the data are a SRS of ize are baed o the ample proportio. Whe i large, ha approximately the ormal ditributio with mea p ad tadard deviatio p p. The level C cofidece iterval for p i * z where z* i the upper ( C)/ tadard ormal critical value. (The area betwee z* ad z* i C). Tet ad cofidece iterval for a populatio proportio p whe the data are a SRS of ize are baed o the ample proportio. Whe i large, ha approximately the ormal ditributio with mea p ad tadard deviatio p p. The level C cofidece iterval for p i * z where z* i the upper ( C)/ tadard ormal critical value. (The area betwee z* ad z* i C). coditio: Radom data mut come from a SRS of the populatio. Normal both p ad ( p ) mut be 0 or larger for a cofidece iterval Idepedece 0% rule mut be atified.

4 8] -proportio z-tet for H 0 : p = p 0 Tet of H 0: p = p 0 are baed o the z tatitic z p p 0 p 0 0 with P-value calculated from the tadard ormal ditributio. The coditio are idetical to the coditio eceary to carry out a proportio z-iterval. coditio: Radom data mut come from a SRS of the populatio. Normal both p 0 ad ( p 0 ) mut be 0 or larger for a igificace tet. Idepedece 0% rule mut be atified. The ample ize eeded to obtai a cofidece iterval with approximate margi of error m for a populatio proportio ivolve olvig * * p z p * m for, where p* i a gueed value for the ample proportio ad z* i the tadard ormal critical poit for the level of cofidece you wat. If you ue p* =.5 i thi formula, the margi of error of the iterval will be le tha or equal to m o matter what the value of i. 9] cofidece iterval for p p, the differece betwee two populatio proportio We wat to compare the proportio p ad p of uccee i two populatio. The compario i baed o the differece p betwee the ample proportio of uccee. Whe the ample ize ad are large eough, we ca ue z procedure ˆ becaue the amplig ditributio of i cloe to ormal. A approximate level C cofidece iterval for p p i where the tadard error of i SE = * SE z p ˆ ad z* i a tadard ormal critical value. coditio: Radom each ample i a SRS from it repective populatio Normal -- Each populatio mut be at leat te time a large a the ample take from that populatio ad the ample have to be large eough to atify p 0, ) 0, p 0, ad ) 0 for a cofidece iterval; ˆ ( ˆ Idepedet each ample atifie the 0% rule eparately. ( 0] igificace tet for H 0 : p = p Sigificace tet of H 0: p = p ue the pooled ample proportio ad the z tatitic cout of uccee i both ample combied c cout of obervatio i both ample combied z P-value come from the tadard ormal table. Coditio: Radom each ample i a SRS from it repective populatio c c c c

5 Normal -- p ˆ 0 c, ˆ ( p c ) 0, p ˆ 0 c, ad ˆ ( p c ) 0, where c i the pooled ample proportio for a igificace tet. (Accordig to our book, you ca alo ue p 0, etc. for a hypothei tet.) Idepedet each ample atifie the 0% rule eparately. ] Tet for Goode of Fit The chi-quare tet for goode of fit tet the ull hypothei that a populatio ditributio i the ame a a referece ditributio (hypotheized ditributio). The expected cout for ay variable category i obtaied by multiplyig the percet of the ditributio for each category time the ample ize. The chi-quare tatitic i X = (0 - E) /E, where the um i over variable categorie. The chi-quare tet compare the value of the tatitic X with critical value from the chi-quare ditributio with degree of freedom. A chi-quare goode of fit tet take the form: H 0: the populatio ditributio (or percet) i the ame a the hypotheized ditributio. H a: the populatio percet are differet from the hypotheized value the P-value i the area uder the chi-quare deity curve to the right of X. Large value of X are evidece agait H 0. coditio: Radom the igle ample i a SRS of the populatio Large ample all of the expected cout are 5 or larger. Idepedet the 0% rule hold FOR EACH SAMPLE SEPARATELY! ] Iferece for Two-Way Table (homogeeity & idepedece/aociatio) The chi-quare tet for homogeeity (TWO or more populatio with or more ample & ONE categorical variable. Thi i alo ued i radomized experimet with or more treatmet where the treatmet form the colum of the two-way table ad the row i the repoe (relape ye or o, heart attack ye or o, etc.) take the followig form: H 0: the ditributio (or percet) of the categorical variable for populatio i the ame a the ditributio of the categorical variable for populatio (or for experimet the ditributio of the repoe variable i the ame for all treatmet group) Ha: the ditributio (or percet) of the categorical variable for populatio i NOT the ame a the ditributio of the categorical variable for populatio (or for experimet the ditributio of the repoe variable i NOT the ame for all treatmet group The chi-quare tet for idepedece/aociatio (ONE populatio with ONE large ample ad TWO categorical variable) take the form: H0: there i o aociatio betwee the row categorical variable ad the colum categorical variable Ha: there i a aociatio betwee the row categorical variable ad the colum categorical variable OR H 0: the row categorical variable i idepedet of the colum categorical variable H a: the row categorical variable i depedet o the colum categorical variable Oe commo ue of the chi-quare tet i to compare everal populatio proportio. The ull hypothei tate that all of the populatio proportio are equal. The alterative hypothei tate that they are ot all equal but allow ay other relatiohip amog the populatio proportio. The expected cout i ay cell of a two-way table whe H 0 i true i expected cout = (row total colum total) / table total The chi-quare tatitic i X oberved cout expected cout expected cout The chi-quare tet compare the value of the tatitic X with critical value from the chi-quare ditributio with

6 df = (r - )(c - ) degree of freedom. Large value of X are evidece agait H 0 o the P-value i the area uder the chi-quare deity curve to the right of X. coditio: Radom goode of fit or aociatio -> the igle ample i a SRS of the populatio; homogeeity -> the ample are SRS from their repective populatio. Large ample all of the expected cout are 5 or larger. Idepedet the 0% rule hold FOR EACH SAMPLE SEPARATELY! If the chi-quare tet fid a tatitically igificat P-value, do a follow-up aalyi that compare the oberved cout with the expected cout ad that look for the larget compoet of chi-quare. 3] Iferece for the Slope of a Regreio Lie repreet the lope of the TRUE regreio lie that relate two quatitative variable. However, a with ay parameter, i typically ukow, therefore a ample of data poit i take to etimate it. The leat quare regreio lie i computed uig the ample data poit ad the cofidece iterval etimate for i b ± t SE b Where b i the lope of the leat quare regreio lie computed from a ample ad SE b i the tadard error for b. The formula for the tadard error, SE i give below SE t* i computed uig degree of freedom. reidual b ad x x x I tet of igificace, H 0 alway tate that there i o liear relatiohip betwee the two quatitative variable. That i, H 0: = 0. A before, the alterative hypothei H a ca be < 0, > 0, or 0. The t tatitic i computed uig the formula b 0 t = SE Whe computig the probability be ure to ue degree of freedom. coditio: ) for ay fixed x, the y value vary ormally ad are idepedet of each other. ) the mea repoe y, y, ha a liear relatiohip with x: y = + x 3) the tadard deviatio of y (call it ) i the ame for all value of x. The INFERENCE CONDITIONS ARE L.I.N.E.R L (Liear) the catterplot how a liear patter ad the reidual plot how NO curved patter I (idepedet) the obervatio are idepedet of oe aother 0% rule hold N (reidual are ormal) the ormal probability plot for the reidual how a liear patter idicatig that the reidual are approximately ormal R (radom) the data were collected radomly; SRS take b