ASSUMPTIONS/CONDITIONS FOR HYPOTHESIS TESTS and CONFIDENCE INTERVALS
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1 ASSUMPTIONS/CONDITIONS FOR HYPOTHESIS TESTS ad CONFIDENCE INTERVALS Oe of the importat tak whe applyig a tatitical tet (or cofidece iterval) i to check that the aumptio of the tet are ot violated. Oe-ample cofidece iterval ad z-tet o µ CONFIDENCE INTERVAL: x ± (z critical value) σ SIGNIFICANCE TEST: z = x μ 0 σ The ample mut be reaoably radom. The data mut be from a ormal ditributio or large ample (eed to check 30 ). σ mut be kow. σ The ample mut be le tha 10% of the populatio o that i valid for the tadard deviatio of the amplig ditributio of x. Oe-ample cofidece iterval ad t-tet o µ CONFIDENCE INTERVAL: x ± (t critical value) SIGNIFICANCE TEST: t = x μ 0 where degree of freedom df = - 1 I theory, the data hould be draw from a ormal ditributio or it i a large ample (eed to check that 30 ). I practice, uig the t-ditributio i ufficietly robut provided that there i little kewe ad o outlier i the data. Look at a graph of the data. The data mut be reaoably radom. The ample mut be le tha 10% of the populatio.
2 Two-ample cofidece iterval ad t-tet o µ 1 - µ 2 CONFIDENCE INTERVAL: ( x 1 x 2 ) ± t* ( ) ( ) + SIGNIFICANCE TEST: t = (x 1 x 2 ) (μ 1 μ 2 ) ( 1 ) 2 + ( 2 )2 1 2 The two ample mut be reaoably radom ad draw idepedetly or, if it i a experimet, the ubject were radomly aiged to treatmet. I theory, the data hould be draw from ormal ditributio or be large ample (check that ). I practice, uig the t-ditributio i ufficietly robut provided that there i little kewe ad o outlier i the data for each ample. Examie graph of both et of data. NOTE: There are two way to calculate the degree of freedom. Optio 1. Ue procedure baed o the t-tatitic with critical value from the t-ditributio with df equal to the maller of 1-1 ad 2-1. Thi will alway yield a coervative approximatio for df. Optio 2. The two-ample t tatitic doe ot have a t ditributio. Moreover, the exact ditributio chage a ukow populatio tadard deviatio σ 1 ad σ 2 chage. However, a excellet approximatio i available. Mot tatitical oftware ytem ad the TI-83 ue the two-ample t-tatitic where the degree of freedom are calculated i the formula below. Thi geerally i NOT a whole umber. df Optio 1 alway err o the afe ide, reportig higher p-value ad lower cofidece tha are actually true. The gap betwee what i reported ad what i true i quite mall ule the ample ize are both mall ad uequal. A the ample ize icreae, probability value baed o t with degree of freedom equal to the maller of 1-1 ad 2 1 will become more accurate.
3 Matched pair cofidece iterval ad t-tet Pairig data ofte reduce the dager of itroducig extraeou or ucotrolled factor. Pairig data ha the theoretical effect of reducig meauremet variability, which icreae the accuracy of tatitical cocluio. CONFIDENCE INTERVAL : x d ± t* d where df = -1 TEST STATISTIC : t = x d μ d d where df = -1 The ample of paired differece mut be reaoably radom. The paired differece d = x 1 - x 2 hould be approximately ormally ditributed or be a large ample (eed to check 30 ). Thi procedure i robut if there are o outlier ad little kewe i the paired differece. Examie a graph of the differece. NOTES: I matched pair where two meauremet are take o each experimetal uit, the uit erve a it ow cotrol. For matched-paired data, the tadard error of the tatitic for the matched pair tet will be maller tha the tadard error for a idepedet two-ample t-tet becaue variability withi the ample ha bee removed. It i the reao why we block i experimetal deig (radomized block)
4 Liear Regreio Tet o Slope ad Cofidece Iterval CONFIDENCE INTERVAL: b± t* SEb where df = 2 ad SE b = ( x x) 2 TEST STATISTIC: b hypotheized value t = where df = 2 ad SE b SE b = ( x x) 2 --The obervatio are idepedet. --The true relatiohip i liear. Check that the catter plot i roughly liear ad that the reidual plot ha o patter. --The tadard deviatio of the repoe y about the true lie i the ame everywhere. Look at the reidual plot ad check that the reidual have roughly the ame pread acro all the x-value. --For ay fixed value of x, the repoe y varie ormally about the true lie. Check a hitogram or templot of the reidual. NOTE: Hypothee ca pecify ay value for lope, for example, Ho : β = 1, H a : β 1 H o : β = 0 (there i o ueful liear relatiohip) H a : β 0 (there i a ueful liear relatiohip). or β > 0 or β < 0 (if eeded) Oe-ample cofidece iterval ad z-tet o p CONFIDENCE INTERVAL: ˆp ± z* p ˆ (1 p ˆ ) TEST STATISTIC: z = ˆ p p p(1 p) The ample mut be reaoably radom The ample mut be le tha 10% of the populatio The ample mut be large eough o that: p ˆ ad (1 - p ˆ ) 10 for a cofidece iterval p ad (1 - p) 10 for the igificace tet
5 Two-ample cofidece iterval ad z-tet o p 1 p 2 CONFIDENCE INTERVAL: ( p ˆ 1 p ˆ 2 ) ± z* p ˆ 1 (1 p ˆ 1 ) p + ˆ 2 (1 p ˆ 2 ) 1 2 TEST STATISTIC: z = pˆ ˆ 1 p2 x1+ x2 where pˆ c = pˆ (1 ˆ ) ˆ (1 ˆ c pc pc pc) The two ample mut be idepedetly draw ad reaoably radom or ubject were radomly aiged to two group. The ample ize mut be large eough o that: p, ˆ 1 1 ˆ 1(1 p1), p, ˆ ˆ 2(1 p2) are all five or more. (the umber of uccee ad the umber of failure mut be at leat 5) for the cofidece iterval. The ample ize mut be large eough o that: p, ˆ 1 c ˆ 1(1 p c ), p, ˆ 2 c ˆ 2(1 p c ) are all five or more. (the umber of uccee ad the umber of failure mut be at leat 5) for the igificace tet. Chi-Square Tet for Goode-of-Fit Ho : The hypotheized ditributio i correct Ha : At leat oe of the categorie/proportio i ot correct TEST STATISTIC: χ 2 = (oberved cout - expected cout)2 expected cout or χ 2 = (O - E)2 E where df = k - 1 (k i # of clae) The expected cell cout are all greater tha or equal to 5 The ample i reaoably radom Chi-Square Tet for Homogeeity of Proportio ad for Idepedece
6 H o : the true category proportio are the ame for all populatio (homogeeity). H a : the true category proportio are ot the ame for all populatio. Ho : the two variable are idepedet (alteratively, there i o aociatio) Ha : the two variable are ot idepedet (alteratively, there i a aociatio) TEST STATISTIC: χ 2 = (O - E)2 E where df = (r - 1)(c - 1) ad the expected cell cout = (row margial total)(colum margial total) gradtotal The ample mut be reaoably radom The ample ize mut be large eough o that all expected cout are at leat 1 ad o more tha 20% are le tha 5. I particular, all expected cell cout i a 2x2 table hould be 5 or more. NOTES: Fidig depedece betwee row ad colum variable doe ot imply cauatio, epecially o data from a obervatioal tudy - the typical cae whe uig a chi-quared tet for idepedece. A two-ided z-tet o p 1 p 2 will give the ame p-value a a chi-quared tet of homogeeity o a 2x2 table.
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