MTH 212 Formulas page 1 out of 7. Sample variance: s = Sample standard deviation: s = s

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1 MTH Formula age out of 7 DESCRIPTIVE TOOLS Poulatio ize = N Samle ize = x x+ x x x Poulatio mea: µ = Samle mea: x = = N ( µ ) ( x x) Poulatio variace: = Samle variace: = N Poulatio tadard deviatio: = Samle tadard deviatio: = Five-umber ummary: Miimum, Q, Media, Q 3, Maximum Rage = max mi Iterquartile rage: IQR = Q 3 Q Coefficiet of Variatio: CV = 00% or CV = 00% µ P Percetile: Locatio of a Percetile: L = ( + ) where i deired ercetile, exreed i %. 00 To fid the actual value of a ercetile, iterolate if eceary. GRAPHS AND TABLES Frequecy Table LeftBoudary + RightBoudary Frequecy Cla Mar = Relative Frequecy = Total Box-lot Left Whier LW = Q. 5 IQR Right Whier: RW = Q IQR Z-core cocet To fid z-core for a ecific value: To fid value for a ecific z-core: value - mea x x t. dev x = x + z or x = µ + z x µ Emirical Rule ( Rule) Chebyhev Theorem: For ay et of obervatio, the roortio of the value that lie withi tadard deviatio of the mea i at leat, where i ay cotat greater tha.

2 MTH Formula age out of 7 PROBABILITY RULES For ay evet A: 0 P ( A) For ay amle ace: P ( S) = Coditioal Rule The robability of a evet A to occur, give that evet B ha occurred. PA ( ad B) P( A B) =, whe P ( B) 0. PB ( ) Comlemet Rule (correod to oerator NOT): For ay evet A, P( ot A) = P( A) = P( A) Geeral Multilicatio Rule (correod to oerator AND): For ay two evet A ad B, PA ( ad B) = PA ( ) PBA ( ) Secial cae for the Multilicatio Rule: If A ad B are ideedet evet, the PA ( ad B) = PA ( ) PB ( ) Geeral Additio Rule (correod to oerator OR): For ay two evet A ad B, PA ( or B) = PA ( ) + PB ( ) PA ( ad B) Secial cae for Additio Rule: If evet A ad B are mutually excluive (dijoit), the PA ( B) = PA ( ) + PB ( ). or Two evet A ad B are ideedet i equivalet to the followig: P ( A B) = P( A), P ( B A) = P( B), PA ( B) = PA ( ) PB ( ) ad That i, to chec for ideedece, you mut chec at leat oe of the roertie above. If at leat oe of thee roertie i atified, the we ca coclude that two evet are ideedet. If at leat oe of thee roertie failed, the we ca coclude that two evet are deedet. Baye Theorem Evet E, E,..., E are mutually excluive ad collectively exhautive evet. PE ( i ) are rior robabilitie. B = evet that ha occurred that might imact PE ( i ) P( E ) ( ) ( ) iadb PEi PBEi PE ( i B) = = PB ( ) PE ( ) PBE ( ) + PE ( ) PBE ( ) PE ( ) PBE ( )

3 MTH Formula age 3 out of 7 FOR ANY DISCRETE RANDOM VARIABLE If i a dicrete radom variable, the Exected (average) value of i µ= Mea( ) = E( ) = x P( x) Variace of i ( x µ ) = P( x) ad Stadard Deviatio of i Grah: value veru their robabilitie = SD( ) = Uig calculator: Create a Lit of Value ad correodig Lit of Probabilitie. You ca fid µ ad uig STAT > CALC > -Var Stat LitofValue, LitofProbabilite To aalyze the hae of the ditributio: you ca grah a ditributio (uig hitogram ymbol of grah, lot value veru their robabilitie) FOR ANY BINOMIAL RANDOM VARIABLE If i biomial (, ), the x x Probability Ditributio Fuctio i P = x) = C ( ) Mea (Exected) value of uccee i µ = ( x for x = 0,,,...,! where Cx = Cx = = x x! ( x)! Variace: = ( ) ad Stadard Deviatio: = Grah: value veru their robabilitie Uig calculator you ca fid the followig: C a uig re the go to MATH > PRB > C r P ( = a) uig DISTR > biomdf( a),, P ( a) uig DISTR > biomcdf( a),, the re a FOR ANY POISSON RANDOM VARIABLE If i Poio ( µ ), the x λ t x µ ( λ t) e µ e Probability Ditributio Fuctio i Px ( ) = P ( = x) = = for x = 0,,,... x! x! Mea (Exected) value of uccee: µ = λt Variace = ad Stadard Deviatio = λ = Exected umber of uccee i a egmet of uit ize ad t = Size of the egmet of iteret Uig calculator you ca fid the followig: P ( = a) uig DISTR > oiodf( µ, a ) P ( a) uig DISTR > oiocdf( µ, a )

4 MTH Formula age 4 out of 7 FOR ANY NORMAL VARIABLE If i Normal with the mea µ ad the tadard deviatio, the x µ Probability Deity Fuctio: f( x) = e for < x < π Uig calculator To fid robability P( LeftBoudary < < RightBoudary) we ca ue DISTR> ormalcdf ( LeftBoudary, RightBoudary, µ, ) To fid a ecific value of a variable (where area to the left of the value i ecified) To fid that value, call it a, uch that P( < a) = ecific robability We ca ue DISTR > ivnorm(ecific robability, µ, ) SAMPLING DISTRIBUTION OF THE SAMPLE MEAN, Samlig error = x µ For a radom variable with mea µ ad tadard deviatio The mea of the amlig ditributio of i Mea( ) = µ =µ The tadard deviatio (tadard error of the mea) of the amlig ditributio of i That i, SD( ) = = Shae. To determie whether or ot the hae of the amlig ditributio of i Normal, mut chec with theorem below. Theorem : If i Normal, the i Normal. Theorem : Cetral Limit Theorem For ay ditributio of, the ditributio of i aroximately ormal if amle ize i large. I fact, the larger the amle ize, the cloer the ditributio of to a Normal ditributio.. To tadardize a ecific value of x whe i give: x µ SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTION, (alo labeled a ˆ ) x Samle roortio: ˆ = =, where x = umber of uccee out of ( ) Samlig error = Mea( ) =µ = SD( ) = = Theorem: The amlig ditributio of i aroximately ormal if i large (that i, if umber of uccee ad umber of failure i 5 or more)

5 MTH Formula age 5 out of 7 Notatio: CI = Cofidece Iterval Etimate SRS = Simle Radom Samle INFERENCE ABOUT THE POPULATION MEAN, µ (baed o oe SRS) Samle tatitic i x. Oly IF i Normal (you mut how why i Normal), the Whe i ow: CI for µ i x ± zc Calculator: Z-iterval x µ 0 Tet Statitic i Calculator: Z-Tet Whe i uow: CI for µ i x ± tc with df = Calculator: T-iterval x µ 0 Tet Statitic i t = with df = Calculator: T-tet INFERENCE ABOUT THE POPULATION PROPORTION, (baed o oe SRS) x Samle tatitic i =. Note that ca alo be labeled a ˆ. Note that = x, the umber of ucce out of obervatio. Oly IF i Normal (you mut how that 5 ad ( ) 5 ), the ( ) CI for i ± zc Calculator: -ro Z iterval Oly IF i Normal (you mut how that 0 5 ad ( 0 ) 5 ), the Tet Statitic i 0 0( 0) Calculator: -ro Z-tet Determiig amle ize To fid a miimum amle ize to etimate µ with give cofidece level ad maximum allowed error: zc = Error To fid a miimum amle ize to etimate with give cofidece level ad maximum allowed error: zc = ( ) where = either reviouly ow or 0.5 if o rior ifo i give. Error

6 MTH Formula age 6 out of 7 INFERENCE ABOUT TWO POPULATION MEANS, µ µ (baed o two ideedet SRS) Samle tatitic i x x. Oly IF i Normal (you mut how why both ad are Normal), the Whe ad are both ow: ( x x) ( µ µ ) Tet Statitic i Calculator: -SamZ-Tet + Whe ad are both uow: x x µ µ Tet Statitic i t = + ( ) ( ) Calculator: -SamT-Tet (ooled = No) df = Smaller betwee ad (aroximatio by coervative method). Note: calculator doe df exactly. Whe ad are both uow, but you are alo give that = : ( x x) ( µ µ ) Tet Statitic i t = Calculator: -SamT-Tet (ooled = Ye) + ( ) + ( ) df = + ad = = ooled variace. + Note: Calculator dilay = ooled tadard deviatio. INFERENCE ABOUT TWO POPULATION MEANS, µ µ (baed o two deedet SRS, matched air). Note: µ µ =µ d Calculator directio: Put value ito Lit ad ut value ito Lit. Create oe amle of differece: D= by Lit3=Lit-Lit. Oly IF D i Normal (For mall amle, both ad had to be Normal), the d µ Tet Statitic i t = d with df = Calculator: T-Tet (Data from Lit3) d INFERENCE ABOUT TWO POPULATION PROPORTIONS, (baed o two ideedet SRS) x x Samle tatitic i =. If + =, the combied roortio i c = = + Oly IF i Normal (you mut how that c 5ad ( c) 5, c 5ad ( c) 5), the Tet Statitic: ( ) ( ) c c + ( ) Calculator: -roztet

7 MTH Formula age 7 out of 7 INFERENCE ABOUT THE POPULATION VARIANCE (baed o oe SRS) Samle tatitic = ( ) Oly IF i ormally ditributed, the follow Chi-quare ditributio with df = ( ) Tet Statitic: χ = with df = To fid the -value ue built-i chi-quare ditributio: DIST > χ cdf(lowerboud, rightboug,df) INFERENCE ABOUT TWO POPULATION VARIANCES (baed o two ideedet SRS) Oly if both variable ad are ormally ditributed, the follow F-ditributio with df umerator = ad dfdeomiator = Tet Statitic: F = with df umerator = ad dfdeomiator = Calculator: -SamFtet ANOVA: Aalyi of Variace To tet H 0: µ = µ =... = µ Oly IF All oulatio are ormally ditributed, i.e.,,..., are ormally ditributed The oulatio variace are equal, i.e. = =... = Samle are ideedet MS Betwee The Tet Statitic F = with dfumerator = ad dfdeomiator = N MS Withi where N = Calculator: Eter data ito lit, each amled data ito oe lit. The ANOVA(L, L, ) GOODNESS-OF-FIT TESTS AND CONTINGENCY ANALYSIS Tet for Ideedece RowTotal ColumTotal For each cell of a r c cotigecy table, fid ExectedCout = TableTotal ( Exected Oberved) Tet Statitic: χ = all cell Exected Calculator: Create a matrix of Oberved cout, call it A: MATRI > EDIT> elect A, tye i your ifo. (Note: you do t really eed to create a matrix of Exected cout yourelf). STAT > TESTS > Chi-Square tet. Select aroriate matrice). LINEAR REGRESSION Create a -lit ad correodig Y-lit. Cotruct catter lot (uig calculator). STAT PLOT > Chooe catter lot ymbol Fid a equatio of the leat-quare regreio lie. STAT > CALC > LiReg(ax+b) -lit, Y-lit Correlatio coefficiet ad coefficiet of determiatio will be dilayed o the cree (after erformig DiagoticO commad)